Temperature Effects on Nonlinear Vibration Behaviors of...
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Research Article Temperature Effects on Nonlinear Vibration Behaviors of Euler-Bernoulli Beams with Different Boundary Conditions Yaobing Zhao and Chaohui Huang College of Civil Engineering, Huaqiao University, Xiamen, Fujian 361021, China Correspondence should be addressed to Yaobing Zhao; [email protected] Received 14 January 2018; Revised 7 May 2018; Accepted 13 May 2018; Published 27 June 2018 Academic Editor: Pedro Galv´ ın Copyright © 2018 Yaobing Zhao and Chaohui Huang. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams with symmetric and nonsymmetric boundary conditions. It is assumed that in the considered model the temperature increases/decreases instantly, and the temperature variation is uniformly distributed along the length and the cross-section. By using the extended Hamilton’s principle, the mathematical model which takes into account thermal and mechanical loadings, represented by partial differential equations (PDEs), is established. e PDEs of the planar motion are discretized to a set of second-order ordinary differential equations by using the Galerkin method. As to three different boundary conditions, eigenvalue analyses are performed to obtain the close-form eigenvalue solutions. First four natural frequencies with thermal effects are investigated. By using the Lindstedt-Poincar´ e method and multiple scales method, the approximate solutions of the nonlinear free and forced vibrations (primary, super, and subharmonic resonances) are obtained. e influences of temperature variations on response amplitudes, the localisation of the resonance zones, and the stability of the steady-state solutions are investigated, through examining frequency response curves and excitation response curves. Numerical results show that response amplitudes, the number and the stability of nontrivial solutions, and the hardening-spring characteristics are all closely related to temperature changes. As to temperature effects on vibration behaviors of structures, different boundary conditions should be paid more attention. 1. Introduction Due to the importance in many applications in many fields, such as the industrial, civil, mechanical, automotive, aerospace, and other structural systems, a flexible beam with nonlinear characteristics has attracted more attention in the past few years. e linear and nonlinear vibration characteristics have been investigated for many years and were reviewed, e.g., by Nayfeh and Mook [1], Nayfeh and Balachandran [2], Nayfeh and Pai [3], Luongo and Zulli [4], and Lacarbonara et al. [5]. On the one hand, it is noted that beams in compression are subjected to buckle, and once the beam buckles, the beam exhibits an initially deflected static equilibrium position and looks like an arch. Due to the initial curvature, the nonlinear vibration behaviors become much more complicated [6– 8]. e nonlinear dynamics of buckled beams have been investigated by many researchers. As to these buckled and postbuckling beams, natural frequencies, mode shapes, non- linear normal modes, and nonlinear resonance responses were investigated through theoretical and experimental methods [9–16]. e geometrically nonlinear vibration of an aluminium beam hinged at both ends was investigated experimentally by Ribeiro and Carneiro [17]. e nonlinear vibration analysis of a curved beam subjected to the uniform base harmonic excitation with both quadratic and cubic nonlinearities was investigated by Huang et al. [18]. On the other hand, in many applications, these struc- tures are oſten subjected to vibration under thermal and dynamic loadings [19]. Temperature fields develop thermal stress due to the thermal expansion or contraction which influences the dynamic behavior of mechanical systems. e literature on thermal vibrations of mechanical structures is quite abundant; here, only a few papers are reviewed. e dynamic instability of a pinned beam subjected to an alternating magnetic field and thermal loads with the Hindawi Shock and Vibration Volume 2018, Article ID 9834629, 11 pages https://doi.org/10.1155/2018/9834629
Transcript of Temperature Effects on Nonlinear Vibration Behaviors of...
Research ArticleTemperature Effects on Nonlinear Vibration Behaviors ofEuler-Bernoulli Beams with Different Boundary Conditions
Yaobing Zhao and Chaohui Huang
College of Civil Engineering Huaqiao University Xiamen Fujian 361021 China
Correspondence should be addressed to Yaobing Zhao ybzhaohqueducn
Received 14 January 2018 Revised 7 May 2018 Accepted 13 May 2018 Published 27 June 2018
Academic Editor Pedro Galvın
Copyright copy 2018 Yaobing Zhao and Chaohui Huang This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
This paper is concerned with temperature effects on the modeling and vibration characteristics of Euler-Bernoulli beams withsymmetric and nonsymmetric boundary conditions It is assumed that in the consideredmodel the temperature increasesdecreasesinstantly and the temperature variation is uniformly distributed along the length and the cross-section By using the extendedHamiltonrsquos principle the mathematical model which takes into account thermal and mechanical loadings represented by partialdifferential equations (PDEs) is established The PDEs of the planar motion are discretized to a set of second-order ordinarydifferential equations by using the Galerkin method As to three different boundary conditions eigenvalue analyses are performedto obtain the close-form eigenvalue solutions First four natural frequencies with thermal effects are investigated By using theLindstedt-Poincare method and multiple scales method the approximate solutions of the nonlinear free and forced vibrations(primary super and subharmonic resonances) are obtained The influences of temperature variations on response amplitudes thelocalisation of the resonance zones and the stability of the steady-state solutions are investigated through examining frequencyresponse curves and excitation response curves Numerical results show that response amplitudes the number and the stabilityof nontrivial solutions and the hardening-spring characteristics are all closely related to temperature changes As to temperatureeffects on vibration behaviors of structures different boundary conditions should be paid more attention
1 Introduction
Due to the importance in many applications in manyfields such as the industrial civil mechanical automotiveaerospace and other structural systems a flexible beamwith nonlinear characteristics has attracted more attentionin the past few years The linear and nonlinear vibrationcharacteristics have been investigated for many years andwere reviewed eg by Nayfeh and Mook [1] Nayfeh andBalachandran [2] Nayfeh and Pai [3] Luongo and Zulli [4]and Lacarbonara et al [5]
On the one hand it is noted that beams in compressionare subjected to buckle and once the beam buckles the beamexhibits an initially deflected static equilibrium position andlooks like an arch Due to the initial curvature the nonlinearvibration behaviors become much more complicated [6ndash8] The nonlinear dynamics of buckled beams have beeninvestigated by many researchers As to these buckled and
postbuckling beams natural frequencies mode shapes non-linear normal modes and nonlinear resonance responseswere investigated through theoretical and experimentalmethods [9ndash16] The geometrically nonlinear vibration ofan aluminium beam hinged at both ends was investigatedexperimentally by Ribeiro and Carneiro [17] The nonlinearvibration analysis of a curved beam subjected to the uniformbase harmonic excitation with both quadratic and cubicnonlinearities was investigated by Huang et al [18]
On the other hand in many applications these struc-tures are often subjected to vibration under thermal anddynamic loadings [19] Temperature fields develop thermalstress due to the thermal expansion or contraction whichinfluences the dynamic behavior of mechanical systems Theliterature on thermal vibrations of mechanical structuresis quite abundant here only a few papers are reviewedThe dynamic instability of a pinned beam subjected toan alternating magnetic field and thermal loads with the
HindawiShock and VibrationVolume 2018 Article ID 9834629 11 pageshttpsdoiorg10115520189834629
2 Shock and Vibration
nonlinear strain has been studied by Wu [20 21] and effectsof frequency ratio loading factor and amplitude dampingfactor and temperature changes on the vibration behaviorshave been illustrated Manoach and Ribeiro [22 23] studiedgeometrically nonlinear vibrations of moderately thick andcurved beams under the combined action of mechanicaland thermal loads The coupled thermoelastic vibrationcharacteristics of the axially moving beam were investigatedby Guo et al [24] Treyssede [25] proposed an analyticalmodel for the vibration analysis of horizontal beams thatare self-weighted and thermally stressed Moreover due tothe relatively high possible temperature variations in beamsAvsec and Oblak [26] have developed the mathematicalmodel where fundamental thermomechanical propertiesof state are functions of temperature Thermomechanicalvibrations of a simply supported spring-mass-beam systemwere investigated analytically by Ghayesh et al [27] Three-dimensional nonlinear motion characteristics of the perfectand imperfect Timoshenko microbeams under mechanicaland thermal forces have been examined numerically byFarokhi and Ghayesh [28] Large amplitude vibrations andregular and chaotic oscillations of a Timoshenko beam underthe influence of temperature were analysed by Warminskaet al [29 30] and mechanical and thermal loadings havebeen discussedMoreover nonlinear vibration characteristicsof cross-ply composite plates in thermal environments wereinvestigated by Settimi et al [31] and Saetta et al [32]Recently based on Hamiltonrsquos principle temperature effectson the vibration behaviors of the cable-stayed-beam byintroducing two nondimensional factors were investigatedby Zhao et al [33] The nonlinear free and forced vibrationcharacteristics of the suspended cable were studied by Zhaoet al [34 35]
Furthermore the nonlinear vibration behavior of thesystem is closely related to its boundary conditions Forexample the nonlinear free and forced vibrations of abeam-mass system under five different boundary conditionswere investigated by Ozkaya et al [36] Large amplitudevibrations of rectangular plates subjected to the radicalharmonic excitation were investigated by Amabili [37] andthree boundary conditions were considered The nonlineardynamic response of an inclined Timonshenko beam withdifferent boundary conditions subjected to a varying masswith variable velocity was investigated by Mamandi et al[38]The nonlinear free vibrations of a two-layer elastic com-posite beam were studied by Lenci et al [39] and differentboundary conditions both symmetric (eg free-free fixed-fixed and hinged-hinged) and nonsymmetric (eg free-fixedand hinged-fixed) with respect to the beam midpoint wereinvestigated Furthermore by using a unified approach theplanar nonlinear vibrations of the shear indeformable beamswith eithermovable or immovable supports were investigatedby Luongo et al [40]
One of the motivations of this study is that changes in thevibration characteristics of the structure due to the damagemay be smaller than changes in ones due to variations intemperature [41]Therefore an accurate mathematical modelof the beam with thermal effects is necessary to predictits vibration characteristics To the best of our knowledge
no specific study has addressed temperature effects on thenonlinear free and forced oscillations of the beam withdifferent boundary conditions The structure of the paper isorganized as follows in Section 2 the governing equationsof an Euler-Bernoulli beam by using the extended Hamiltonrsquosprinciple are derived In addition the PDEs of the planarmotion are discretized to the second-order ordinary equa-tions via the Galerkin method Considering three differentboundary conditions eigenvalue analyses are performed toobtain close-form eigenvalue solutions Then the Lindstedt-Poincare method and multiple scales method are adopted toobtain the approximate solutions of the nonlinear free andforced resonance responses in Section 3 Some numericalresults and discussions are given to illustrate the influences oftemperature variations on response amplitudes localisationof the resonance zones and the stability of the nontrivialsolutions in Section 4 Finally at the end of the paper(Section 5) some conclusions are drawn
2 Mathematics for Nonlinear Modeling
21 Equations of Motion In this study 119906 and V denote theaxial and transverse deflections of the beam respectively 119909is the distance along the undeflected beam 119905 is the time 119871is the beam length 119898 = 120588119860 is the beamrsquos mass per unit 119864is Youngrsquos modulus 119860 is the area of cross-section 119868 is thearea moment of inertia 119875 is the applied axial compressiveload 119888119906 and 119888V are the viscous damping coefficients per unitlength Δ119879 is the temperature variation and 120572 is the thermalexpansion coefficient
The linear relation between the temperature variation andthe stress-temperature coefficient 120574 is120574 (Δ119879) = 119864120572Δ119879 (1)
and the nonlinear relation is expressed as [21]
120574 (Δ119879) = 119864120572Δ119879 + ℎ1205722Δ1198792 (2)
Here the first linear relation is adopted for the sake ofsimplicity By assuming the Lagrangian strain as the strainmeasure and introducing temperature variations the strainin the beam motion is obtained
120576 = 120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864 (3)
The extended Hamiltonrsquos principle is given by
int11990521199051
[120575119882119890 + 120575119879 minus 120575119880] d119905 = 0 (4)
where 119879 is the kinetic energy 119880 is the elastic energy and119882119890is the work done by external loads on the system
By using Hamiltonrsquos principle one obtains [1 20 21]
12058811986012059721199061205971199052 + 119888119906 120597119906120597119905minus 119860 120597120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574 = 0
(5)
Shock and Vibration 3
1205881198601205972V1205971199052 + 119888V 120597V120597119905 + 119864119868 1205974V1205971199094minus 119860 120597120597119909 120597V120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574+ 119875 1205972V1205971199092 = 119865 (119909) cos (Ω119905)
(6)
where 119865(119909) is the spatial distribution of the harmonic load inthe V direction andΩ is the excitation frequency
Introducing the quasistatic assumptions the accelerationand velocity terms in (5) are neglected [20 21] and we obtain
119860 120597120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574 = 0 (7)
where
120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864 = constant = 120575 (8)
where 120575 is the average strain of the systems and one obtains
120575 = 1119871 int1198710 [120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864] d119909= 12119871 int1198710 ( 120597V120597119909)2 d119909 minus 120574119864
(9)
Therefore substituting (9) into (6) we obtain the follow-ing nonlinear partial differential equation of motion withoutconsidering the torsion and shear rigidities
119864119868 1205974V1205971199094 + 1205881198601205972V1205971199052 + 119888V 120597V120597119905 + 119860120574 1205972V1205971199092 + 119875 1205972V1205971199092= 1198641198602119871 1205972V1205971199092 int1198710 ( 120597V120597119909)2 d119909 + 119865 (119909) cos (Ω119905)
(10)
For the sake of simplicity the following nondimensionalquantities are introduced
Vlowast = V119871 119909lowast = 119909119871 119905lowast = 119905radic 1198751205881198601198712 V119891 = radic 1198641198681198751198712 120574lowast = 119860120574119875 V1 = radic119864119860119875
119865lowast = 119865119871119875 Ωlowast = Ωradic1205881198601198712119875 119888lowastV = 119888Vradic 1198712120588119860119875
(11)
Substituting (11) into (10) and neglecting the asterisks oneobtains
V21198911205974V1205971199094 + 1205972V1205971199052 + 119888V 120597V120597119905 + 1205972V1205971199092 + 120574 1205972V1205971199092= 12V21 1205972V1205971199092 int10 ( 120597V120597119909)2 d119909 + 119865 (119909) cos (Ω119905)
(12)
22 Linear Analysis Dropping the damping excitation andnonlinear terms in (12) the following linear equation isobtained
V21198911205974V1205971199094 + 1205972V1205971199052 + (120574 + 1) 1205972V1205971199092 = 0 (13)
and one lets
V (119909 119905) = 120601 (119909) 119890119894120596119905 (14)
where 119894 denotes the imaginary unit 120601(119909) is the normalizedundamped mode shape and 120596 is the natural frequency
Substituting (14) into (13) one obtains
V21198911205974V1205971199094 + (120574 + 1) 1205972V1205971199092 minus 1205962120601 = 0 (15)
The corresponding eigenvalue equation is expressed asfollows
V21198911198634 + (120574 + 1)1198632 minus 1205962 = 0 (16)
Accordingly one lets
120579 = radicradic(1 + 120574)2 + 4V21198911205962 + (1 + 120574)2V2119891
120573 = radicradic(1 + 120574)2 + 4V21198911205962 minus (1 + 120574)2V2
119891
(17)
The homogeneous solution of (16) is given by
120601 (119909) = 1198881 cos (120579119909) + 1198882 sin (120579119909) + 1198883 cosh (120573119909)+ 1198884 sinh (120573119909) (18)
In the following sections three different symmetric andnonsymmetric boundary conditions are chosen and investi-gated hinged-hinged hinged-fixed and fixed-fixed
4 Shock and Vibration
Case A (hinged-hinged) As to the hinged-hinged boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 12059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=1 = 0
(19)
Substituting (19) into (18) we obtain the following coeffi-cient matrix
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus1205792 cos (120579) minus1205792 sin (120579) 1205733 cosh (120573) 1205733 sinh (120573)
]]]]]](20)
where the determinant of the coefficientmatrix in (20) equalszero we have
(1205792 + 1205733) sin (120579) sinh (120573) = 0 (21)
Hence the natural frequency of the beam with hinged-hinged boundary conditions could be obtained by solving thefollowing equation
sin (120579) = 0 (22)
and the corresponding mode shapes are
120601119894 (119909) = 119862119894 sin (120579119894) (23)
where 119862119894 is determined by normalization conditions
Case B (hinged-fixed) As to the hinged-fixed boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (24)
Substituting (24) into (18) the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](25)
Similarly the following equation is obtained
minus (1205792 + 1205732) [minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573)]= 0 (26)
Hence the natural frequency could be obtained by solv-ing the following transcendental equation
minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573) = 0 (27)
with the following mode shapes120601119894 (119909)= 119862119894 [ sinh (120573119894) sin (120579119894119909) minus sin (120579119894) sinh (120573119894119909)
sin (120579119894) ] (28)
where 119862119894 is also determined by normalization conditions
Case C (fixed-fixed) As to the fixed-fixed boundary condi-tions we have 120601 (0) = 120601 (1) = 0120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (29)
Similarly the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)0 120579 0 120573minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](30)
and we obtain the transcendental equation asminus 2120579120573 + 2120579120573 cos (120579) cosh (120573)+ (1205792 minus 1205732) sin (120579) sinh (120573) = 0 (31)
where the natural frequency could be obtained and themodeshapes are
120601119894 (119909) = 119862119894 [(cosh (120573119894119909) minus cos (120579119894119909))sdot 120573119894 sin (120579119894) minus 120579119894 sinh (120573119894)120579119894 cosh (120573119894) minus 120579119894 cos (120579119894) + sinh (120573119894119909) minus 120573119894120579119894sdot sin (120579119894)]
(32)
where 119862119894 is also determined by normalization conditions
23 Discrete Model The Galerkin procedure is used todiscretize (12) into a set of second-order ordinary differentialequations
V (119909 119905) = infinsum119899=1
120601119899 (119909) 119902119899 (119905) (33)
where 119902119899(119905) is the generalized coordinate and 120601119899(119909) is thelinear vibration mode shape
Substituting (33) into (12) multiplying the result by120601119899(119909)integrating the outcome over the domain and finally usingthe orthonormality condition yield119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905)
+ infinsum119894=1
infinsum119895=1
infinsum119896=1
Γ119899119894119895119896119902119894 (119905) 119902119895 (119905) 119902119896 (119905) = 119891119899 cos (Ω119905) (34)
Shock and Vibration 5
where1205962119899= (120574 + 1) int1
012060110158401015840119899 (119909) 120601119899 (119909) d119909
+ V2119891 int101206011015840101584010158401015840119899 (119909) 120601119899 (119909) d119909
(35)
Γ119899119894119895119896= minus12V2119891 int10 12060110158401015840119894 (119909) 120601119899 (119909) (int10 120601119895 (119909) 120601119896 (119909) d119909) d119909 (36)
119891119899 = int10119865 (119909) 120601119899 (119909) d119909 (37)
120583119899 = 12 int10 119888V120601119899 (119909) d119909 (38)
3 Perturbation Analysis
Perturbation methods are used to investigate the nonlinearfree primary super and subharmonic resonance responsesof the beam with thermal effects subjected to the planarexcitation and no internal resonance is considered in thisstudy For the sake of simplicity only the single-modediscretization equation is considered119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905) + Γ1198991198991198991198991199023119899 (119905)= 119891119899 cos (Ω119905) (39)
31 Nonlinear Free Vibration This section begins with theapproximate series solutions for the nonlinear free vibrationsobtained with the Lindstedt-Poincare method [42]
Firstly a new independent variable is introduced whichis = 120596119899119905 (40)
where 120596119899 is the natural frequencyTherefore by neglecting the damping and forcing terms
(39) is transformed into
119902119899 + 119902119899 + Γ1198991198991198991198991205962119899 1199023119899 = 0 (41)
where the coefficient of the linear term is equal to unity sincethe time is nondimensionalized with respect to the linearvibration frequency
By assuming an expansion for 119902119899 = 120576119902119899 (where 120576 is a smallfinite nondimensional parameter) and omitting the tilde weobtain 119902119899 + 119902119899 + 1205762 Γ1198991198991198991198991205962119899 1199023119899 = 0 (42)
Following the method of Lindstedt-Poincare we seek thefourth-order approximate solution to (42) by letting
119902119899 ( 120576) = 4sum119898=0
120576119898119902119899119898 (119905) (43)
and a strained time coordinate 120591 is introduced120591 = (1 + 4sum
119898=1
120576119898120572119898) (44)
Then we could obtain the relation between the nonlinearfrequencyΩ119899 and the linear one 120596119899 as followsΩ119899 minus 120596119899120596119899 = 4sum
119898=1
120576119898120572119898 (45)
Substituting (43)-(44) into (42) and equating the coeffi-cients 120576119898 on both sides the nonlinear ordinary equations arereduced to a set of linearized equationsThen the polar formis introduced and the secular terms are set to zero Finally theseries solutions of 119902119899119898 and 120572119898 are obtained based on whichthe fourth-order series solutions of the frequency amplituderelationship and displacement are as follows
Ω119899 = 120596119899 [1 + 38 Γ1198991198991198991198991205962119899 1198862 minus 15256 Γ21198991198991198991198991205964119899 1198864] (46)
where 119886 is the actual nondimensional response amplitude
32 Primary Resonances First of all we order the dampingexcitation and nonlinearity term as follows [43]2120583119899 119902119899 (119905) = 2120576120583119899 119902119899 (119905) 119891119899 cos (Ω119899119905) = 120576119891119899 cos (Ω119899119905) Γ119899119899119899119899 1199023119899 (119905) = 120576Γ119899119899119899119899 1199023119899 (119905)
(47)
Substituting (47) into (39) the governing equationbecomes 119902119899 (119905) + 1205962119899119902119899 (119905) + 2120576120583119899 119902119899 (119905) + 120576Γ1198991198991198991198991199023119899 (119905)= 120576119891119899 cos (Ω119905) (48)
In the case of the primary resonance we introduce adetuning parameter 120590119899 which quantitatively describes thenearness of Ω119899 to 120596119899 Ω119899 = 120596119899 + 120576120590119899 (49)
where 120576 is a small finite parameterWe express the solution in terms of different time scales
as 119902119899 (119905 120576) = 1199061198990 (1198790 1198791) + 1205761199061198991 (1198790 1198791) + sdot sdot sdot (50)
where 1198790 = 119905 and 1198791 = 120576119905Substituting (49)-(50) into (48) equating the coefficients
of 1205760 and 1205761 on both sides following the method of multiplescales finally the frequency response equation is obtained
120590119899 = 38 Γ119899119899119899119899120596119899 1198862119899 plusmn radic 1198912119899412059621198991198862119899 minus 1205832119899 (51)
where 119886119899 is the actual nondimensional response amplitudeTherefore the first approximation to the steady-state
solution is obtained119902119899 (119905) = 119886119899 cos (Ω119899119905 minus 120574119899) + 119874 (120576) (52)
where 119905 is the actual time scale and 120574119899 is the phase
6 Shock and Vibration
Table 1 Physical parameters and material properties
Parameter Physical meaning Value Unit120588 density 7781 kgm3119871 length 2032 m119864 Youngrsquos modulus 207 GPa119860 area of cross-section 2581 times 10minus3 m2120572 thermal expansion coefficient 10 times 10minus5 ∘C119875 axial load 11281 kN119868 area moment of inertia 5550 times 10minus7 m4
33 Subharmonic Resonances In order to investigate the sub-harmonic resonance a detuning parameter 120590119899 is introducedΩ119899 = 3120596119899 + 120576120590119899 (53)
where 120576 is a small but finite parameterSimilarly the frequency response equation can be
obtained by using the multiple scales method
91205832119899 + (120590119899 minus 9Γ119899119899119899119899Λ2119899120596119899 minus 9Γ1198991198991198991198998120596119899 1198862119899)2
= 81Γ2119899119899119899119899Λ2119899161205962119899 1198862119899 (54)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (55)
The first approximation to the steady-state solution isgiven by
119902119899 (119905) = 119886119899 cos [13 (Ω119899119905 minus 120574119899)] + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (56)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase34 Super Harmonic Resonances In this case to express thenearness of Ω119899 to (13)120596119899 we also introduce the detuningparameter 120590119899 defined by
3Ω119899 = 120596119899 + 120576120590119899 (57)
where 120576 is a small but finite parameterFollowing the step of the multiple scales method a
frequency response equation can be obtained
120590119899 = 3Γ119899119899119899119899Λ2119899120596119899 + 3Γ1198991198991198991198998120596119899 1198862119899 plusmn radicΓ2119899119899119899119899Λ611989912059621198991198862119899 minus 1205832119899 (58)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (59)
Therefore the first approximation in this case is obtained
119902119899 (119905) = 119886119899 cos (3Ω119899119905 minus 120574119899) + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (60)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase4 Numerical Examples and Discussions
In the following numerical analysis material properties andphysical parameters of the beam are chosen in Table 1 Thenondimensional temperature variation factor 120574 is chosen as[minus10 +10] and the corresponding dimensional temperaturevariation Δ119879 is [minus2083∘C +2083∘C] Moreover the nondi-mensional factor V119891 in (11) is 05 in this study
41 Natural Frequencies By using the numerical methodthe transcendental equations (see (22) (27) and (31)) canbe solved and the corresponding natural frequencies areobtained In order to investigate temperature effects onnatural frequencies with different boundary conditions afrequency variation factor 119877120596 is defined
119877120596 = 120596Δ119879 minus 12059601205960 (61)
where 120596Δ119879 and 1205960 are the natural frequencies with andwithout thermal effects respectively
Figure 1 shows effects of temperature variations on thefirst four mode frequencies in the case of three differentboundary conditions As shown in these figures with theincrease (decrease) in the temperature the mode frequenciesall decrease (increase) In previous studies it is shown thatan increase in temperature leads to a decrease in naturalfrequencies [41] and the conclusion in this study is in goodagreement with some of the previous findings
Furthermore the changing of natural frequencies withtemperature was found to be dependent on the order of themode and boundary conditions With the increase of theorder the natural frequency becomes less sensitive to thetemperature variation To be more specific in the case of120574 = +10 as to three different boundary conditions thevariation factors 119877120596 of these first order mode frequenciesare 436 (hinged-hinged) 130 (hinged-fixed) and 57
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
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2 Shock and Vibration
nonlinear strain has been studied by Wu [20 21] and effectsof frequency ratio loading factor and amplitude dampingfactor and temperature changes on the vibration behaviorshave been illustrated Manoach and Ribeiro [22 23] studiedgeometrically nonlinear vibrations of moderately thick andcurved beams under the combined action of mechanicaland thermal loads The coupled thermoelastic vibrationcharacteristics of the axially moving beam were investigatedby Guo et al [24] Treyssede [25] proposed an analyticalmodel for the vibration analysis of horizontal beams thatare self-weighted and thermally stressed Moreover due tothe relatively high possible temperature variations in beamsAvsec and Oblak [26] have developed the mathematicalmodel where fundamental thermomechanical propertiesof state are functions of temperature Thermomechanicalvibrations of a simply supported spring-mass-beam systemwere investigated analytically by Ghayesh et al [27] Three-dimensional nonlinear motion characteristics of the perfectand imperfect Timoshenko microbeams under mechanicaland thermal forces have been examined numerically byFarokhi and Ghayesh [28] Large amplitude vibrations andregular and chaotic oscillations of a Timoshenko beam underthe influence of temperature were analysed by Warminskaet al [29 30] and mechanical and thermal loadings havebeen discussedMoreover nonlinear vibration characteristicsof cross-ply composite plates in thermal environments wereinvestigated by Settimi et al [31] and Saetta et al [32]Recently based on Hamiltonrsquos principle temperature effectson the vibration behaviors of the cable-stayed-beam byintroducing two nondimensional factors were investigatedby Zhao et al [33] The nonlinear free and forced vibrationcharacteristics of the suspended cable were studied by Zhaoet al [34 35]
Furthermore the nonlinear vibration behavior of thesystem is closely related to its boundary conditions Forexample the nonlinear free and forced vibrations of abeam-mass system under five different boundary conditionswere investigated by Ozkaya et al [36] Large amplitudevibrations of rectangular plates subjected to the radicalharmonic excitation were investigated by Amabili [37] andthree boundary conditions were considered The nonlineardynamic response of an inclined Timonshenko beam withdifferent boundary conditions subjected to a varying masswith variable velocity was investigated by Mamandi et al[38]The nonlinear free vibrations of a two-layer elastic com-posite beam were studied by Lenci et al [39] and differentboundary conditions both symmetric (eg free-free fixed-fixed and hinged-hinged) and nonsymmetric (eg free-fixedand hinged-fixed) with respect to the beam midpoint wereinvestigated Furthermore by using a unified approach theplanar nonlinear vibrations of the shear indeformable beamswith eithermovable or immovable supports were investigatedby Luongo et al [40]
One of the motivations of this study is that changes in thevibration characteristics of the structure due to the damagemay be smaller than changes in ones due to variations intemperature [41]Therefore an accurate mathematical modelof the beam with thermal effects is necessary to predictits vibration characteristics To the best of our knowledge
no specific study has addressed temperature effects on thenonlinear free and forced oscillations of the beam withdifferent boundary conditions The structure of the paper isorganized as follows in Section 2 the governing equationsof an Euler-Bernoulli beam by using the extended Hamiltonrsquosprinciple are derived In addition the PDEs of the planarmotion are discretized to the second-order ordinary equa-tions via the Galerkin method Considering three differentboundary conditions eigenvalue analyses are performed toobtain close-form eigenvalue solutions Then the Lindstedt-Poincare method and multiple scales method are adopted toobtain the approximate solutions of the nonlinear free andforced resonance responses in Section 3 Some numericalresults and discussions are given to illustrate the influences oftemperature variations on response amplitudes localisationof the resonance zones and the stability of the nontrivialsolutions in Section 4 Finally at the end of the paper(Section 5) some conclusions are drawn
2 Mathematics for Nonlinear Modeling
21 Equations of Motion In this study 119906 and V denote theaxial and transverse deflections of the beam respectively 119909is the distance along the undeflected beam 119905 is the time 119871is the beam length 119898 = 120588119860 is the beamrsquos mass per unit 119864is Youngrsquos modulus 119860 is the area of cross-section 119868 is thearea moment of inertia 119875 is the applied axial compressiveload 119888119906 and 119888V are the viscous damping coefficients per unitlength Δ119879 is the temperature variation and 120572 is the thermalexpansion coefficient
The linear relation between the temperature variation andthe stress-temperature coefficient 120574 is120574 (Δ119879) = 119864120572Δ119879 (1)
and the nonlinear relation is expressed as [21]
120574 (Δ119879) = 119864120572Δ119879 + ℎ1205722Δ1198792 (2)
Here the first linear relation is adopted for the sake ofsimplicity By assuming the Lagrangian strain as the strainmeasure and introducing temperature variations the strainin the beam motion is obtained
120576 = 120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864 (3)
The extended Hamiltonrsquos principle is given by
int11990521199051
[120575119882119890 + 120575119879 minus 120575119880] d119905 = 0 (4)
where 119879 is the kinetic energy 119880 is the elastic energy and119882119890is the work done by external loads on the system
By using Hamiltonrsquos principle one obtains [1 20 21]
12058811986012059721199061205971199052 + 119888119906 120597119906120597119905minus 119860 120597120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574 = 0
(5)
Shock and Vibration 3
1205881198601205972V1205971199052 + 119888V 120597V120597119905 + 119864119868 1205974V1205971199094minus 119860 120597120597119909 120597V120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574+ 119875 1205972V1205971199092 = 119865 (119909) cos (Ω119905)
(6)
where 119865(119909) is the spatial distribution of the harmonic load inthe V direction andΩ is the excitation frequency
Introducing the quasistatic assumptions the accelerationand velocity terms in (5) are neglected [20 21] and we obtain
119860 120597120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574 = 0 (7)
where
120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864 = constant = 120575 (8)
where 120575 is the average strain of the systems and one obtains
120575 = 1119871 int1198710 [120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864] d119909= 12119871 int1198710 ( 120597V120597119909)2 d119909 minus 120574119864
(9)
Therefore substituting (9) into (6) we obtain the follow-ing nonlinear partial differential equation of motion withoutconsidering the torsion and shear rigidities
119864119868 1205974V1205971199094 + 1205881198601205972V1205971199052 + 119888V 120597V120597119905 + 119860120574 1205972V1205971199092 + 119875 1205972V1205971199092= 1198641198602119871 1205972V1205971199092 int1198710 ( 120597V120597119909)2 d119909 + 119865 (119909) cos (Ω119905)
(10)
For the sake of simplicity the following nondimensionalquantities are introduced
Vlowast = V119871 119909lowast = 119909119871 119905lowast = 119905radic 1198751205881198601198712 V119891 = radic 1198641198681198751198712 120574lowast = 119860120574119875 V1 = radic119864119860119875
119865lowast = 119865119871119875 Ωlowast = Ωradic1205881198601198712119875 119888lowastV = 119888Vradic 1198712120588119860119875
(11)
Substituting (11) into (10) and neglecting the asterisks oneobtains
V21198911205974V1205971199094 + 1205972V1205971199052 + 119888V 120597V120597119905 + 1205972V1205971199092 + 120574 1205972V1205971199092= 12V21 1205972V1205971199092 int10 ( 120597V120597119909)2 d119909 + 119865 (119909) cos (Ω119905)
(12)
22 Linear Analysis Dropping the damping excitation andnonlinear terms in (12) the following linear equation isobtained
V21198911205974V1205971199094 + 1205972V1205971199052 + (120574 + 1) 1205972V1205971199092 = 0 (13)
and one lets
V (119909 119905) = 120601 (119909) 119890119894120596119905 (14)
where 119894 denotes the imaginary unit 120601(119909) is the normalizedundamped mode shape and 120596 is the natural frequency
Substituting (14) into (13) one obtains
V21198911205974V1205971199094 + (120574 + 1) 1205972V1205971199092 minus 1205962120601 = 0 (15)
The corresponding eigenvalue equation is expressed asfollows
V21198911198634 + (120574 + 1)1198632 minus 1205962 = 0 (16)
Accordingly one lets
120579 = radicradic(1 + 120574)2 + 4V21198911205962 + (1 + 120574)2V2119891
120573 = radicradic(1 + 120574)2 + 4V21198911205962 minus (1 + 120574)2V2
119891
(17)
The homogeneous solution of (16) is given by
120601 (119909) = 1198881 cos (120579119909) + 1198882 sin (120579119909) + 1198883 cosh (120573119909)+ 1198884 sinh (120573119909) (18)
In the following sections three different symmetric andnonsymmetric boundary conditions are chosen and investi-gated hinged-hinged hinged-fixed and fixed-fixed
4 Shock and Vibration
Case A (hinged-hinged) As to the hinged-hinged boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 12059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=1 = 0
(19)
Substituting (19) into (18) we obtain the following coeffi-cient matrix
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus1205792 cos (120579) minus1205792 sin (120579) 1205733 cosh (120573) 1205733 sinh (120573)
]]]]]](20)
where the determinant of the coefficientmatrix in (20) equalszero we have
(1205792 + 1205733) sin (120579) sinh (120573) = 0 (21)
Hence the natural frequency of the beam with hinged-hinged boundary conditions could be obtained by solving thefollowing equation
sin (120579) = 0 (22)
and the corresponding mode shapes are
120601119894 (119909) = 119862119894 sin (120579119894) (23)
where 119862119894 is determined by normalization conditions
Case B (hinged-fixed) As to the hinged-fixed boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (24)
Substituting (24) into (18) the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](25)
Similarly the following equation is obtained
minus (1205792 + 1205732) [minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573)]= 0 (26)
Hence the natural frequency could be obtained by solv-ing the following transcendental equation
minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573) = 0 (27)
with the following mode shapes120601119894 (119909)= 119862119894 [ sinh (120573119894) sin (120579119894119909) minus sin (120579119894) sinh (120573119894119909)
sin (120579119894) ] (28)
where 119862119894 is also determined by normalization conditions
Case C (fixed-fixed) As to the fixed-fixed boundary condi-tions we have 120601 (0) = 120601 (1) = 0120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (29)
Similarly the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)0 120579 0 120573minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](30)
and we obtain the transcendental equation asminus 2120579120573 + 2120579120573 cos (120579) cosh (120573)+ (1205792 minus 1205732) sin (120579) sinh (120573) = 0 (31)
where the natural frequency could be obtained and themodeshapes are
120601119894 (119909) = 119862119894 [(cosh (120573119894119909) minus cos (120579119894119909))sdot 120573119894 sin (120579119894) minus 120579119894 sinh (120573119894)120579119894 cosh (120573119894) minus 120579119894 cos (120579119894) + sinh (120573119894119909) minus 120573119894120579119894sdot sin (120579119894)]
(32)
where 119862119894 is also determined by normalization conditions
23 Discrete Model The Galerkin procedure is used todiscretize (12) into a set of second-order ordinary differentialequations
V (119909 119905) = infinsum119899=1
120601119899 (119909) 119902119899 (119905) (33)
where 119902119899(119905) is the generalized coordinate and 120601119899(119909) is thelinear vibration mode shape
Substituting (33) into (12) multiplying the result by120601119899(119909)integrating the outcome over the domain and finally usingthe orthonormality condition yield119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905)
+ infinsum119894=1
infinsum119895=1
infinsum119896=1
Γ119899119894119895119896119902119894 (119905) 119902119895 (119905) 119902119896 (119905) = 119891119899 cos (Ω119905) (34)
Shock and Vibration 5
where1205962119899= (120574 + 1) int1
012060110158401015840119899 (119909) 120601119899 (119909) d119909
+ V2119891 int101206011015840101584010158401015840119899 (119909) 120601119899 (119909) d119909
(35)
Γ119899119894119895119896= minus12V2119891 int10 12060110158401015840119894 (119909) 120601119899 (119909) (int10 120601119895 (119909) 120601119896 (119909) d119909) d119909 (36)
119891119899 = int10119865 (119909) 120601119899 (119909) d119909 (37)
120583119899 = 12 int10 119888V120601119899 (119909) d119909 (38)
3 Perturbation Analysis
Perturbation methods are used to investigate the nonlinearfree primary super and subharmonic resonance responsesof the beam with thermal effects subjected to the planarexcitation and no internal resonance is considered in thisstudy For the sake of simplicity only the single-modediscretization equation is considered119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905) + Γ1198991198991198991198991199023119899 (119905)= 119891119899 cos (Ω119905) (39)
31 Nonlinear Free Vibration This section begins with theapproximate series solutions for the nonlinear free vibrationsobtained with the Lindstedt-Poincare method [42]
Firstly a new independent variable is introduced whichis = 120596119899119905 (40)
where 120596119899 is the natural frequencyTherefore by neglecting the damping and forcing terms
(39) is transformed into
119902119899 + 119902119899 + Γ1198991198991198991198991205962119899 1199023119899 = 0 (41)
where the coefficient of the linear term is equal to unity sincethe time is nondimensionalized with respect to the linearvibration frequency
By assuming an expansion for 119902119899 = 120576119902119899 (where 120576 is a smallfinite nondimensional parameter) and omitting the tilde weobtain 119902119899 + 119902119899 + 1205762 Γ1198991198991198991198991205962119899 1199023119899 = 0 (42)
Following the method of Lindstedt-Poincare we seek thefourth-order approximate solution to (42) by letting
119902119899 ( 120576) = 4sum119898=0
120576119898119902119899119898 (119905) (43)
and a strained time coordinate 120591 is introduced120591 = (1 + 4sum
119898=1
120576119898120572119898) (44)
Then we could obtain the relation between the nonlinearfrequencyΩ119899 and the linear one 120596119899 as followsΩ119899 minus 120596119899120596119899 = 4sum
119898=1
120576119898120572119898 (45)
Substituting (43)-(44) into (42) and equating the coeffi-cients 120576119898 on both sides the nonlinear ordinary equations arereduced to a set of linearized equationsThen the polar formis introduced and the secular terms are set to zero Finally theseries solutions of 119902119899119898 and 120572119898 are obtained based on whichthe fourth-order series solutions of the frequency amplituderelationship and displacement are as follows
Ω119899 = 120596119899 [1 + 38 Γ1198991198991198991198991205962119899 1198862 minus 15256 Γ21198991198991198991198991205964119899 1198864] (46)
where 119886 is the actual nondimensional response amplitude
32 Primary Resonances First of all we order the dampingexcitation and nonlinearity term as follows [43]2120583119899 119902119899 (119905) = 2120576120583119899 119902119899 (119905) 119891119899 cos (Ω119899119905) = 120576119891119899 cos (Ω119899119905) Γ119899119899119899119899 1199023119899 (119905) = 120576Γ119899119899119899119899 1199023119899 (119905)
(47)
Substituting (47) into (39) the governing equationbecomes 119902119899 (119905) + 1205962119899119902119899 (119905) + 2120576120583119899 119902119899 (119905) + 120576Γ1198991198991198991198991199023119899 (119905)= 120576119891119899 cos (Ω119905) (48)
In the case of the primary resonance we introduce adetuning parameter 120590119899 which quantitatively describes thenearness of Ω119899 to 120596119899 Ω119899 = 120596119899 + 120576120590119899 (49)
where 120576 is a small finite parameterWe express the solution in terms of different time scales
as 119902119899 (119905 120576) = 1199061198990 (1198790 1198791) + 1205761199061198991 (1198790 1198791) + sdot sdot sdot (50)
where 1198790 = 119905 and 1198791 = 120576119905Substituting (49)-(50) into (48) equating the coefficients
of 1205760 and 1205761 on both sides following the method of multiplescales finally the frequency response equation is obtained
120590119899 = 38 Γ119899119899119899119899120596119899 1198862119899 plusmn radic 1198912119899412059621198991198862119899 minus 1205832119899 (51)
where 119886119899 is the actual nondimensional response amplitudeTherefore the first approximation to the steady-state
solution is obtained119902119899 (119905) = 119886119899 cos (Ω119899119905 minus 120574119899) + 119874 (120576) (52)
where 119905 is the actual time scale and 120574119899 is the phase
6 Shock and Vibration
Table 1 Physical parameters and material properties
Parameter Physical meaning Value Unit120588 density 7781 kgm3119871 length 2032 m119864 Youngrsquos modulus 207 GPa119860 area of cross-section 2581 times 10minus3 m2120572 thermal expansion coefficient 10 times 10minus5 ∘C119875 axial load 11281 kN119868 area moment of inertia 5550 times 10minus7 m4
33 Subharmonic Resonances In order to investigate the sub-harmonic resonance a detuning parameter 120590119899 is introducedΩ119899 = 3120596119899 + 120576120590119899 (53)
where 120576 is a small but finite parameterSimilarly the frequency response equation can be
obtained by using the multiple scales method
91205832119899 + (120590119899 minus 9Γ119899119899119899119899Λ2119899120596119899 minus 9Γ1198991198991198991198998120596119899 1198862119899)2
= 81Γ2119899119899119899119899Λ2119899161205962119899 1198862119899 (54)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (55)
The first approximation to the steady-state solution isgiven by
119902119899 (119905) = 119886119899 cos [13 (Ω119899119905 minus 120574119899)] + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (56)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase34 Super Harmonic Resonances In this case to express thenearness of Ω119899 to (13)120596119899 we also introduce the detuningparameter 120590119899 defined by
3Ω119899 = 120596119899 + 120576120590119899 (57)
where 120576 is a small but finite parameterFollowing the step of the multiple scales method a
frequency response equation can be obtained
120590119899 = 3Γ119899119899119899119899Λ2119899120596119899 + 3Γ1198991198991198991198998120596119899 1198862119899 plusmn radicΓ2119899119899119899119899Λ611989912059621198991198862119899 minus 1205832119899 (58)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (59)
Therefore the first approximation in this case is obtained
119902119899 (119905) = 119886119899 cos (3Ω119899119905 minus 120574119899) + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (60)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase4 Numerical Examples and Discussions
In the following numerical analysis material properties andphysical parameters of the beam are chosen in Table 1 Thenondimensional temperature variation factor 120574 is chosen as[minus10 +10] and the corresponding dimensional temperaturevariation Δ119879 is [minus2083∘C +2083∘C] Moreover the nondi-mensional factor V119891 in (11) is 05 in this study
41 Natural Frequencies By using the numerical methodthe transcendental equations (see (22) (27) and (31)) canbe solved and the corresponding natural frequencies areobtained In order to investigate temperature effects onnatural frequencies with different boundary conditions afrequency variation factor 119877120596 is defined
119877120596 = 120596Δ119879 minus 12059601205960 (61)
where 120596Δ119879 and 1205960 are the natural frequencies with andwithout thermal effects respectively
Figure 1 shows effects of temperature variations on thefirst four mode frequencies in the case of three differentboundary conditions As shown in these figures with theincrease (decrease) in the temperature the mode frequenciesall decrease (increase) In previous studies it is shown thatan increase in temperature leads to a decrease in naturalfrequencies [41] and the conclusion in this study is in goodagreement with some of the previous findings
Furthermore the changing of natural frequencies withtemperature was found to be dependent on the order of themode and boundary conditions With the increase of theorder the natural frequency becomes less sensitive to thetemperature variation To be more specific in the case of120574 = +10 as to three different boundary conditions thevariation factors 119877120596 of these first order mode frequenciesare 436 (hinged-hinged) 130 (hinged-fixed) and 57
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
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Shock and Vibration 3
1205881198601205972V1205971199052 + 119888V 120597V120597119905 + 119864119868 1205974V1205971199094minus 119860 120597120597119909 120597V120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574+ 119875 1205972V1205971199092 = 119865 (119909) cos (Ω119905)
(6)
where 119865(119909) is the spatial distribution of the harmonic load inthe V direction andΩ is the excitation frequency
Introducing the quasistatic assumptions the accelerationand velocity terms in (5) are neglected [20 21] and we obtain
119860 120597120597119909 119864[120597119906120597119909 + 12 ( 120597V120597119909)2] minus 120574 = 0 (7)
where
120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864 = constant = 120575 (8)
where 120575 is the average strain of the systems and one obtains
120575 = 1119871 int1198710 [120597119906120597119909 + 12 ( 120597V120597119909)2 minus 120574119864] d119909= 12119871 int1198710 ( 120597V120597119909)2 d119909 minus 120574119864
(9)
Therefore substituting (9) into (6) we obtain the follow-ing nonlinear partial differential equation of motion withoutconsidering the torsion and shear rigidities
119864119868 1205974V1205971199094 + 1205881198601205972V1205971199052 + 119888V 120597V120597119905 + 119860120574 1205972V1205971199092 + 119875 1205972V1205971199092= 1198641198602119871 1205972V1205971199092 int1198710 ( 120597V120597119909)2 d119909 + 119865 (119909) cos (Ω119905)
(10)
For the sake of simplicity the following nondimensionalquantities are introduced
Vlowast = V119871 119909lowast = 119909119871 119905lowast = 119905radic 1198751205881198601198712 V119891 = radic 1198641198681198751198712 120574lowast = 119860120574119875 V1 = radic119864119860119875
119865lowast = 119865119871119875 Ωlowast = Ωradic1205881198601198712119875 119888lowastV = 119888Vradic 1198712120588119860119875
(11)
Substituting (11) into (10) and neglecting the asterisks oneobtains
V21198911205974V1205971199094 + 1205972V1205971199052 + 119888V 120597V120597119905 + 1205972V1205971199092 + 120574 1205972V1205971199092= 12V21 1205972V1205971199092 int10 ( 120597V120597119909)2 d119909 + 119865 (119909) cos (Ω119905)
(12)
22 Linear Analysis Dropping the damping excitation andnonlinear terms in (12) the following linear equation isobtained
V21198911205974V1205971199094 + 1205972V1205971199052 + (120574 + 1) 1205972V1205971199092 = 0 (13)
and one lets
V (119909 119905) = 120601 (119909) 119890119894120596119905 (14)
where 119894 denotes the imaginary unit 120601(119909) is the normalizedundamped mode shape and 120596 is the natural frequency
Substituting (14) into (13) one obtains
V21198911205974V1205971199094 + (120574 + 1) 1205972V1205971199092 minus 1205962120601 = 0 (15)
The corresponding eigenvalue equation is expressed asfollows
V21198911198634 + (120574 + 1)1198632 minus 1205962 = 0 (16)
Accordingly one lets
120579 = radicradic(1 + 120574)2 + 4V21198911205962 + (1 + 120574)2V2119891
120573 = radicradic(1 + 120574)2 + 4V21198911205962 minus (1 + 120574)2V2
119891
(17)
The homogeneous solution of (16) is given by
120601 (119909) = 1198881 cos (120579119909) + 1198882 sin (120579119909) + 1198883 cosh (120573119909)+ 1198884 sinh (120573119909) (18)
In the following sections three different symmetric andnonsymmetric boundary conditions are chosen and investi-gated hinged-hinged hinged-fixed and fixed-fixed
4 Shock and Vibration
Case A (hinged-hinged) As to the hinged-hinged boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 12059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=1 = 0
(19)
Substituting (19) into (18) we obtain the following coeffi-cient matrix
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus1205792 cos (120579) minus1205792 sin (120579) 1205733 cosh (120573) 1205733 sinh (120573)
]]]]]](20)
where the determinant of the coefficientmatrix in (20) equalszero we have
(1205792 + 1205733) sin (120579) sinh (120573) = 0 (21)
Hence the natural frequency of the beam with hinged-hinged boundary conditions could be obtained by solving thefollowing equation
sin (120579) = 0 (22)
and the corresponding mode shapes are
120601119894 (119909) = 119862119894 sin (120579119894) (23)
where 119862119894 is determined by normalization conditions
Case B (hinged-fixed) As to the hinged-fixed boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (24)
Substituting (24) into (18) the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](25)
Similarly the following equation is obtained
minus (1205792 + 1205732) [minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573)]= 0 (26)
Hence the natural frequency could be obtained by solv-ing the following transcendental equation
minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573) = 0 (27)
with the following mode shapes120601119894 (119909)= 119862119894 [ sinh (120573119894) sin (120579119894119909) minus sin (120579119894) sinh (120573119894119909)
sin (120579119894) ] (28)
where 119862119894 is also determined by normalization conditions
Case C (fixed-fixed) As to the fixed-fixed boundary condi-tions we have 120601 (0) = 120601 (1) = 0120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (29)
Similarly the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)0 120579 0 120573minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](30)
and we obtain the transcendental equation asminus 2120579120573 + 2120579120573 cos (120579) cosh (120573)+ (1205792 minus 1205732) sin (120579) sinh (120573) = 0 (31)
where the natural frequency could be obtained and themodeshapes are
120601119894 (119909) = 119862119894 [(cosh (120573119894119909) minus cos (120579119894119909))sdot 120573119894 sin (120579119894) minus 120579119894 sinh (120573119894)120579119894 cosh (120573119894) minus 120579119894 cos (120579119894) + sinh (120573119894119909) minus 120573119894120579119894sdot sin (120579119894)]
(32)
where 119862119894 is also determined by normalization conditions
23 Discrete Model The Galerkin procedure is used todiscretize (12) into a set of second-order ordinary differentialequations
V (119909 119905) = infinsum119899=1
120601119899 (119909) 119902119899 (119905) (33)
where 119902119899(119905) is the generalized coordinate and 120601119899(119909) is thelinear vibration mode shape
Substituting (33) into (12) multiplying the result by120601119899(119909)integrating the outcome over the domain and finally usingthe orthonormality condition yield119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905)
+ infinsum119894=1
infinsum119895=1
infinsum119896=1
Γ119899119894119895119896119902119894 (119905) 119902119895 (119905) 119902119896 (119905) = 119891119899 cos (Ω119905) (34)
Shock and Vibration 5
where1205962119899= (120574 + 1) int1
012060110158401015840119899 (119909) 120601119899 (119909) d119909
+ V2119891 int101206011015840101584010158401015840119899 (119909) 120601119899 (119909) d119909
(35)
Γ119899119894119895119896= minus12V2119891 int10 12060110158401015840119894 (119909) 120601119899 (119909) (int10 120601119895 (119909) 120601119896 (119909) d119909) d119909 (36)
119891119899 = int10119865 (119909) 120601119899 (119909) d119909 (37)
120583119899 = 12 int10 119888V120601119899 (119909) d119909 (38)
3 Perturbation Analysis
Perturbation methods are used to investigate the nonlinearfree primary super and subharmonic resonance responsesof the beam with thermal effects subjected to the planarexcitation and no internal resonance is considered in thisstudy For the sake of simplicity only the single-modediscretization equation is considered119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905) + Γ1198991198991198991198991199023119899 (119905)= 119891119899 cos (Ω119905) (39)
31 Nonlinear Free Vibration This section begins with theapproximate series solutions for the nonlinear free vibrationsobtained with the Lindstedt-Poincare method [42]
Firstly a new independent variable is introduced whichis = 120596119899119905 (40)
where 120596119899 is the natural frequencyTherefore by neglecting the damping and forcing terms
(39) is transformed into
119902119899 + 119902119899 + Γ1198991198991198991198991205962119899 1199023119899 = 0 (41)
where the coefficient of the linear term is equal to unity sincethe time is nondimensionalized with respect to the linearvibration frequency
By assuming an expansion for 119902119899 = 120576119902119899 (where 120576 is a smallfinite nondimensional parameter) and omitting the tilde weobtain 119902119899 + 119902119899 + 1205762 Γ1198991198991198991198991205962119899 1199023119899 = 0 (42)
Following the method of Lindstedt-Poincare we seek thefourth-order approximate solution to (42) by letting
119902119899 ( 120576) = 4sum119898=0
120576119898119902119899119898 (119905) (43)
and a strained time coordinate 120591 is introduced120591 = (1 + 4sum
119898=1
120576119898120572119898) (44)
Then we could obtain the relation between the nonlinearfrequencyΩ119899 and the linear one 120596119899 as followsΩ119899 minus 120596119899120596119899 = 4sum
119898=1
120576119898120572119898 (45)
Substituting (43)-(44) into (42) and equating the coeffi-cients 120576119898 on both sides the nonlinear ordinary equations arereduced to a set of linearized equationsThen the polar formis introduced and the secular terms are set to zero Finally theseries solutions of 119902119899119898 and 120572119898 are obtained based on whichthe fourth-order series solutions of the frequency amplituderelationship and displacement are as follows
Ω119899 = 120596119899 [1 + 38 Γ1198991198991198991198991205962119899 1198862 minus 15256 Γ21198991198991198991198991205964119899 1198864] (46)
where 119886 is the actual nondimensional response amplitude
32 Primary Resonances First of all we order the dampingexcitation and nonlinearity term as follows [43]2120583119899 119902119899 (119905) = 2120576120583119899 119902119899 (119905) 119891119899 cos (Ω119899119905) = 120576119891119899 cos (Ω119899119905) Γ119899119899119899119899 1199023119899 (119905) = 120576Γ119899119899119899119899 1199023119899 (119905)
(47)
Substituting (47) into (39) the governing equationbecomes 119902119899 (119905) + 1205962119899119902119899 (119905) + 2120576120583119899 119902119899 (119905) + 120576Γ1198991198991198991198991199023119899 (119905)= 120576119891119899 cos (Ω119905) (48)
In the case of the primary resonance we introduce adetuning parameter 120590119899 which quantitatively describes thenearness of Ω119899 to 120596119899 Ω119899 = 120596119899 + 120576120590119899 (49)
where 120576 is a small finite parameterWe express the solution in terms of different time scales
as 119902119899 (119905 120576) = 1199061198990 (1198790 1198791) + 1205761199061198991 (1198790 1198791) + sdot sdot sdot (50)
where 1198790 = 119905 and 1198791 = 120576119905Substituting (49)-(50) into (48) equating the coefficients
of 1205760 and 1205761 on both sides following the method of multiplescales finally the frequency response equation is obtained
120590119899 = 38 Γ119899119899119899119899120596119899 1198862119899 plusmn radic 1198912119899412059621198991198862119899 minus 1205832119899 (51)
where 119886119899 is the actual nondimensional response amplitudeTherefore the first approximation to the steady-state
solution is obtained119902119899 (119905) = 119886119899 cos (Ω119899119905 minus 120574119899) + 119874 (120576) (52)
where 119905 is the actual time scale and 120574119899 is the phase
6 Shock and Vibration
Table 1 Physical parameters and material properties
Parameter Physical meaning Value Unit120588 density 7781 kgm3119871 length 2032 m119864 Youngrsquos modulus 207 GPa119860 area of cross-section 2581 times 10minus3 m2120572 thermal expansion coefficient 10 times 10minus5 ∘C119875 axial load 11281 kN119868 area moment of inertia 5550 times 10minus7 m4
33 Subharmonic Resonances In order to investigate the sub-harmonic resonance a detuning parameter 120590119899 is introducedΩ119899 = 3120596119899 + 120576120590119899 (53)
where 120576 is a small but finite parameterSimilarly the frequency response equation can be
obtained by using the multiple scales method
91205832119899 + (120590119899 minus 9Γ119899119899119899119899Λ2119899120596119899 minus 9Γ1198991198991198991198998120596119899 1198862119899)2
= 81Γ2119899119899119899119899Λ2119899161205962119899 1198862119899 (54)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (55)
The first approximation to the steady-state solution isgiven by
119902119899 (119905) = 119886119899 cos [13 (Ω119899119905 minus 120574119899)] + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (56)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase34 Super Harmonic Resonances In this case to express thenearness of Ω119899 to (13)120596119899 we also introduce the detuningparameter 120590119899 defined by
3Ω119899 = 120596119899 + 120576120590119899 (57)
where 120576 is a small but finite parameterFollowing the step of the multiple scales method a
frequency response equation can be obtained
120590119899 = 3Γ119899119899119899119899Λ2119899120596119899 + 3Γ1198991198991198991198998120596119899 1198862119899 plusmn radicΓ2119899119899119899119899Λ611989912059621198991198862119899 minus 1205832119899 (58)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (59)
Therefore the first approximation in this case is obtained
119902119899 (119905) = 119886119899 cos (3Ω119899119905 minus 120574119899) + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (60)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase4 Numerical Examples and Discussions
In the following numerical analysis material properties andphysical parameters of the beam are chosen in Table 1 Thenondimensional temperature variation factor 120574 is chosen as[minus10 +10] and the corresponding dimensional temperaturevariation Δ119879 is [minus2083∘C +2083∘C] Moreover the nondi-mensional factor V119891 in (11) is 05 in this study
41 Natural Frequencies By using the numerical methodthe transcendental equations (see (22) (27) and (31)) canbe solved and the corresponding natural frequencies areobtained In order to investigate temperature effects onnatural frequencies with different boundary conditions afrequency variation factor 119877120596 is defined
119877120596 = 120596Δ119879 minus 12059601205960 (61)
where 120596Δ119879 and 1205960 are the natural frequencies with andwithout thermal effects respectively
Figure 1 shows effects of temperature variations on thefirst four mode frequencies in the case of three differentboundary conditions As shown in these figures with theincrease (decrease) in the temperature the mode frequenciesall decrease (increase) In previous studies it is shown thatan increase in temperature leads to a decrease in naturalfrequencies [41] and the conclusion in this study is in goodagreement with some of the previous findings
Furthermore the changing of natural frequencies withtemperature was found to be dependent on the order of themode and boundary conditions With the increase of theorder the natural frequency becomes less sensitive to thetemperature variation To be more specific in the case of120574 = +10 as to three different boundary conditions thevariation factors 119877120596 of these first order mode frequenciesare 436 (hinged-hinged) 130 (hinged-fixed) and 57
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
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4 Shock and Vibration
Case A (hinged-hinged) As to the hinged-hinged boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 12059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=1 = 0
(19)
Substituting (19) into (18) we obtain the following coeffi-cient matrix
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus1205792 cos (120579) minus1205792 sin (120579) 1205733 cosh (120573) 1205733 sinh (120573)
]]]]]](20)
where the determinant of the coefficientmatrix in (20) equalszero we have
(1205792 + 1205733) sin (120579) sinh (120573) = 0 (21)
Hence the natural frequency of the beam with hinged-hinged boundary conditions could be obtained by solving thefollowing equation
sin (120579) = 0 (22)
and the corresponding mode shapes are
120601119894 (119909) = 119862119894 sin (120579119894) (23)
where 119862119894 is determined by normalization conditions
Case B (hinged-fixed) As to the hinged-fixed boundaryconditions we have
120601 (0) = 120601 (1) = 012059721206011205971199092 100381610038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (24)
Substituting (24) into (18) the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)minus1205792 0 1205732 0minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](25)
Similarly the following equation is obtained
minus (1205792 + 1205732) [minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573)]= 0 (26)
Hence the natural frequency could be obtained by solv-ing the following transcendental equation
minus120573 cosh (120573) sin (120579) + 120579 cos (120579) sinh (120573) = 0 (27)
with the following mode shapes120601119894 (119909)= 119862119894 [ sinh (120573119894) sin (120579119894119909) minus sin (120579119894) sinh (120573119894119909)
sin (120579119894) ] (28)
where 119862119894 is also determined by normalization conditions
Case C (fixed-fixed) As to the fixed-fixed boundary condi-tions we have 120601 (0) = 120601 (1) = 0120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=0 = 120597120601120597119909 10038161003816100381610038161003816100381610038161003816119909=1 = 0 (29)
Similarly the coefficient matrix is
[[[[[[
1 0 1 0cos (120579) sin (120579) cosh (120573) sinh (120573)0 120579 0 120573minus120579 sin (120579) 120579 cos (120579) 120573 sinh (120573) 120573 cosh (120573)
]]]]]](30)
and we obtain the transcendental equation asminus 2120579120573 + 2120579120573 cos (120579) cosh (120573)+ (1205792 minus 1205732) sin (120579) sinh (120573) = 0 (31)
where the natural frequency could be obtained and themodeshapes are
120601119894 (119909) = 119862119894 [(cosh (120573119894119909) minus cos (120579119894119909))sdot 120573119894 sin (120579119894) minus 120579119894 sinh (120573119894)120579119894 cosh (120573119894) minus 120579119894 cos (120579119894) + sinh (120573119894119909) minus 120573119894120579119894sdot sin (120579119894)]
(32)
where 119862119894 is also determined by normalization conditions
23 Discrete Model The Galerkin procedure is used todiscretize (12) into a set of second-order ordinary differentialequations
V (119909 119905) = infinsum119899=1
120601119899 (119909) 119902119899 (119905) (33)
where 119902119899(119905) is the generalized coordinate and 120601119899(119909) is thelinear vibration mode shape
Substituting (33) into (12) multiplying the result by120601119899(119909)integrating the outcome over the domain and finally usingthe orthonormality condition yield119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905)
+ infinsum119894=1
infinsum119895=1
infinsum119896=1
Γ119899119894119895119896119902119894 (119905) 119902119895 (119905) 119902119896 (119905) = 119891119899 cos (Ω119905) (34)
Shock and Vibration 5
where1205962119899= (120574 + 1) int1
012060110158401015840119899 (119909) 120601119899 (119909) d119909
+ V2119891 int101206011015840101584010158401015840119899 (119909) 120601119899 (119909) d119909
(35)
Γ119899119894119895119896= minus12V2119891 int10 12060110158401015840119894 (119909) 120601119899 (119909) (int10 120601119895 (119909) 120601119896 (119909) d119909) d119909 (36)
119891119899 = int10119865 (119909) 120601119899 (119909) d119909 (37)
120583119899 = 12 int10 119888V120601119899 (119909) d119909 (38)
3 Perturbation Analysis
Perturbation methods are used to investigate the nonlinearfree primary super and subharmonic resonance responsesof the beam with thermal effects subjected to the planarexcitation and no internal resonance is considered in thisstudy For the sake of simplicity only the single-modediscretization equation is considered119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905) + Γ1198991198991198991198991199023119899 (119905)= 119891119899 cos (Ω119905) (39)
31 Nonlinear Free Vibration This section begins with theapproximate series solutions for the nonlinear free vibrationsobtained with the Lindstedt-Poincare method [42]
Firstly a new independent variable is introduced whichis = 120596119899119905 (40)
where 120596119899 is the natural frequencyTherefore by neglecting the damping and forcing terms
(39) is transformed into
119902119899 + 119902119899 + Γ1198991198991198991198991205962119899 1199023119899 = 0 (41)
where the coefficient of the linear term is equal to unity sincethe time is nondimensionalized with respect to the linearvibration frequency
By assuming an expansion for 119902119899 = 120576119902119899 (where 120576 is a smallfinite nondimensional parameter) and omitting the tilde weobtain 119902119899 + 119902119899 + 1205762 Γ1198991198991198991198991205962119899 1199023119899 = 0 (42)
Following the method of Lindstedt-Poincare we seek thefourth-order approximate solution to (42) by letting
119902119899 ( 120576) = 4sum119898=0
120576119898119902119899119898 (119905) (43)
and a strained time coordinate 120591 is introduced120591 = (1 + 4sum
119898=1
120576119898120572119898) (44)
Then we could obtain the relation between the nonlinearfrequencyΩ119899 and the linear one 120596119899 as followsΩ119899 minus 120596119899120596119899 = 4sum
119898=1
120576119898120572119898 (45)
Substituting (43)-(44) into (42) and equating the coeffi-cients 120576119898 on both sides the nonlinear ordinary equations arereduced to a set of linearized equationsThen the polar formis introduced and the secular terms are set to zero Finally theseries solutions of 119902119899119898 and 120572119898 are obtained based on whichthe fourth-order series solutions of the frequency amplituderelationship and displacement are as follows
Ω119899 = 120596119899 [1 + 38 Γ1198991198991198991198991205962119899 1198862 minus 15256 Γ21198991198991198991198991205964119899 1198864] (46)
where 119886 is the actual nondimensional response amplitude
32 Primary Resonances First of all we order the dampingexcitation and nonlinearity term as follows [43]2120583119899 119902119899 (119905) = 2120576120583119899 119902119899 (119905) 119891119899 cos (Ω119899119905) = 120576119891119899 cos (Ω119899119905) Γ119899119899119899119899 1199023119899 (119905) = 120576Γ119899119899119899119899 1199023119899 (119905)
(47)
Substituting (47) into (39) the governing equationbecomes 119902119899 (119905) + 1205962119899119902119899 (119905) + 2120576120583119899 119902119899 (119905) + 120576Γ1198991198991198991198991199023119899 (119905)= 120576119891119899 cos (Ω119905) (48)
In the case of the primary resonance we introduce adetuning parameter 120590119899 which quantitatively describes thenearness of Ω119899 to 120596119899 Ω119899 = 120596119899 + 120576120590119899 (49)
where 120576 is a small finite parameterWe express the solution in terms of different time scales
as 119902119899 (119905 120576) = 1199061198990 (1198790 1198791) + 1205761199061198991 (1198790 1198791) + sdot sdot sdot (50)
where 1198790 = 119905 and 1198791 = 120576119905Substituting (49)-(50) into (48) equating the coefficients
of 1205760 and 1205761 on both sides following the method of multiplescales finally the frequency response equation is obtained
120590119899 = 38 Γ119899119899119899119899120596119899 1198862119899 plusmn radic 1198912119899412059621198991198862119899 minus 1205832119899 (51)
where 119886119899 is the actual nondimensional response amplitudeTherefore the first approximation to the steady-state
solution is obtained119902119899 (119905) = 119886119899 cos (Ω119899119905 minus 120574119899) + 119874 (120576) (52)
where 119905 is the actual time scale and 120574119899 is the phase
6 Shock and Vibration
Table 1 Physical parameters and material properties
Parameter Physical meaning Value Unit120588 density 7781 kgm3119871 length 2032 m119864 Youngrsquos modulus 207 GPa119860 area of cross-section 2581 times 10minus3 m2120572 thermal expansion coefficient 10 times 10minus5 ∘C119875 axial load 11281 kN119868 area moment of inertia 5550 times 10minus7 m4
33 Subharmonic Resonances In order to investigate the sub-harmonic resonance a detuning parameter 120590119899 is introducedΩ119899 = 3120596119899 + 120576120590119899 (53)
where 120576 is a small but finite parameterSimilarly the frequency response equation can be
obtained by using the multiple scales method
91205832119899 + (120590119899 minus 9Γ119899119899119899119899Λ2119899120596119899 minus 9Γ1198991198991198991198998120596119899 1198862119899)2
= 81Γ2119899119899119899119899Λ2119899161205962119899 1198862119899 (54)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (55)
The first approximation to the steady-state solution isgiven by
119902119899 (119905) = 119886119899 cos [13 (Ω119899119905 minus 120574119899)] + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (56)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase34 Super Harmonic Resonances In this case to express thenearness of Ω119899 to (13)120596119899 we also introduce the detuningparameter 120590119899 defined by
3Ω119899 = 120596119899 + 120576120590119899 (57)
where 120576 is a small but finite parameterFollowing the step of the multiple scales method a
frequency response equation can be obtained
120590119899 = 3Γ119899119899119899119899Λ2119899120596119899 + 3Γ1198991198991198991198998120596119899 1198862119899 plusmn radicΓ2119899119899119899119899Λ611989912059621198991198862119899 minus 1205832119899 (58)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (59)
Therefore the first approximation in this case is obtained
119902119899 (119905) = 119886119899 cos (3Ω119899119905 minus 120574119899) + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (60)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase4 Numerical Examples and Discussions
In the following numerical analysis material properties andphysical parameters of the beam are chosen in Table 1 Thenondimensional temperature variation factor 120574 is chosen as[minus10 +10] and the corresponding dimensional temperaturevariation Δ119879 is [minus2083∘C +2083∘C] Moreover the nondi-mensional factor V119891 in (11) is 05 in this study
41 Natural Frequencies By using the numerical methodthe transcendental equations (see (22) (27) and (31)) canbe solved and the corresponding natural frequencies areobtained In order to investigate temperature effects onnatural frequencies with different boundary conditions afrequency variation factor 119877120596 is defined
119877120596 = 120596Δ119879 minus 12059601205960 (61)
where 120596Δ119879 and 1205960 are the natural frequencies with andwithout thermal effects respectively
Figure 1 shows effects of temperature variations on thefirst four mode frequencies in the case of three differentboundary conditions As shown in these figures with theincrease (decrease) in the temperature the mode frequenciesall decrease (increase) In previous studies it is shown thatan increase in temperature leads to a decrease in naturalfrequencies [41] and the conclusion in this study is in goodagreement with some of the previous findings
Furthermore the changing of natural frequencies withtemperature was found to be dependent on the order of themode and boundary conditions With the increase of theorder the natural frequency becomes less sensitive to thetemperature variation To be more specific in the case of120574 = +10 as to three different boundary conditions thevariation factors 119877120596 of these first order mode frequenciesare 436 (hinged-hinged) 130 (hinged-fixed) and 57
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
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Shock and Vibration 5
where1205962119899= (120574 + 1) int1
012060110158401015840119899 (119909) 120601119899 (119909) d119909
+ V2119891 int101206011015840101584010158401015840119899 (119909) 120601119899 (119909) d119909
(35)
Γ119899119894119895119896= minus12V2119891 int10 12060110158401015840119894 (119909) 120601119899 (119909) (int10 120601119895 (119909) 120601119896 (119909) d119909) d119909 (36)
119891119899 = int10119865 (119909) 120601119899 (119909) d119909 (37)
120583119899 = 12 int10 119888V120601119899 (119909) d119909 (38)
3 Perturbation Analysis
Perturbation methods are used to investigate the nonlinearfree primary super and subharmonic resonance responsesof the beam with thermal effects subjected to the planarexcitation and no internal resonance is considered in thisstudy For the sake of simplicity only the single-modediscretization equation is considered119902119899 (119905) + 1205962119899119902119899 (119905) + 2120583119899 119902119899 (119905) + Γ1198991198991198991198991199023119899 (119905)= 119891119899 cos (Ω119905) (39)
31 Nonlinear Free Vibration This section begins with theapproximate series solutions for the nonlinear free vibrationsobtained with the Lindstedt-Poincare method [42]
Firstly a new independent variable is introduced whichis = 120596119899119905 (40)
where 120596119899 is the natural frequencyTherefore by neglecting the damping and forcing terms
(39) is transformed into
119902119899 + 119902119899 + Γ1198991198991198991198991205962119899 1199023119899 = 0 (41)
where the coefficient of the linear term is equal to unity sincethe time is nondimensionalized with respect to the linearvibration frequency
By assuming an expansion for 119902119899 = 120576119902119899 (where 120576 is a smallfinite nondimensional parameter) and omitting the tilde weobtain 119902119899 + 119902119899 + 1205762 Γ1198991198991198991198991205962119899 1199023119899 = 0 (42)
Following the method of Lindstedt-Poincare we seek thefourth-order approximate solution to (42) by letting
119902119899 ( 120576) = 4sum119898=0
120576119898119902119899119898 (119905) (43)
and a strained time coordinate 120591 is introduced120591 = (1 + 4sum
119898=1
120576119898120572119898) (44)
Then we could obtain the relation between the nonlinearfrequencyΩ119899 and the linear one 120596119899 as followsΩ119899 minus 120596119899120596119899 = 4sum
119898=1
120576119898120572119898 (45)
Substituting (43)-(44) into (42) and equating the coeffi-cients 120576119898 on both sides the nonlinear ordinary equations arereduced to a set of linearized equationsThen the polar formis introduced and the secular terms are set to zero Finally theseries solutions of 119902119899119898 and 120572119898 are obtained based on whichthe fourth-order series solutions of the frequency amplituderelationship and displacement are as follows
Ω119899 = 120596119899 [1 + 38 Γ1198991198991198991198991205962119899 1198862 minus 15256 Γ21198991198991198991198991205964119899 1198864] (46)
where 119886 is the actual nondimensional response amplitude
32 Primary Resonances First of all we order the dampingexcitation and nonlinearity term as follows [43]2120583119899 119902119899 (119905) = 2120576120583119899 119902119899 (119905) 119891119899 cos (Ω119899119905) = 120576119891119899 cos (Ω119899119905) Γ119899119899119899119899 1199023119899 (119905) = 120576Γ119899119899119899119899 1199023119899 (119905)
(47)
Substituting (47) into (39) the governing equationbecomes 119902119899 (119905) + 1205962119899119902119899 (119905) + 2120576120583119899 119902119899 (119905) + 120576Γ1198991198991198991198991199023119899 (119905)= 120576119891119899 cos (Ω119905) (48)
In the case of the primary resonance we introduce adetuning parameter 120590119899 which quantitatively describes thenearness of Ω119899 to 120596119899 Ω119899 = 120596119899 + 120576120590119899 (49)
where 120576 is a small finite parameterWe express the solution in terms of different time scales
as 119902119899 (119905 120576) = 1199061198990 (1198790 1198791) + 1205761199061198991 (1198790 1198791) + sdot sdot sdot (50)
where 1198790 = 119905 and 1198791 = 120576119905Substituting (49)-(50) into (48) equating the coefficients
of 1205760 and 1205761 on both sides following the method of multiplescales finally the frequency response equation is obtained
120590119899 = 38 Γ119899119899119899119899120596119899 1198862119899 plusmn radic 1198912119899412059621198991198862119899 minus 1205832119899 (51)
where 119886119899 is the actual nondimensional response amplitudeTherefore the first approximation to the steady-state
solution is obtained119902119899 (119905) = 119886119899 cos (Ω119899119905 minus 120574119899) + 119874 (120576) (52)
where 119905 is the actual time scale and 120574119899 is the phase
6 Shock and Vibration
Table 1 Physical parameters and material properties
Parameter Physical meaning Value Unit120588 density 7781 kgm3119871 length 2032 m119864 Youngrsquos modulus 207 GPa119860 area of cross-section 2581 times 10minus3 m2120572 thermal expansion coefficient 10 times 10minus5 ∘C119875 axial load 11281 kN119868 area moment of inertia 5550 times 10minus7 m4
33 Subharmonic Resonances In order to investigate the sub-harmonic resonance a detuning parameter 120590119899 is introducedΩ119899 = 3120596119899 + 120576120590119899 (53)
where 120576 is a small but finite parameterSimilarly the frequency response equation can be
obtained by using the multiple scales method
91205832119899 + (120590119899 minus 9Γ119899119899119899119899Λ2119899120596119899 minus 9Γ1198991198991198991198998120596119899 1198862119899)2
= 81Γ2119899119899119899119899Λ2119899161205962119899 1198862119899 (54)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (55)
The first approximation to the steady-state solution isgiven by
119902119899 (119905) = 119886119899 cos [13 (Ω119899119905 minus 120574119899)] + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (56)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase34 Super Harmonic Resonances In this case to express thenearness of Ω119899 to (13)120596119899 we also introduce the detuningparameter 120590119899 defined by
3Ω119899 = 120596119899 + 120576120590119899 (57)
where 120576 is a small but finite parameterFollowing the step of the multiple scales method a
frequency response equation can be obtained
120590119899 = 3Γ119899119899119899119899Λ2119899120596119899 + 3Γ1198991198991198991198998120596119899 1198862119899 plusmn radicΓ2119899119899119899119899Λ611989912059621198991198862119899 minus 1205832119899 (58)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (59)
Therefore the first approximation in this case is obtained
119902119899 (119905) = 119886119899 cos (3Ω119899119905 minus 120574119899) + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (60)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase4 Numerical Examples and Discussions
In the following numerical analysis material properties andphysical parameters of the beam are chosen in Table 1 Thenondimensional temperature variation factor 120574 is chosen as[minus10 +10] and the corresponding dimensional temperaturevariation Δ119879 is [minus2083∘C +2083∘C] Moreover the nondi-mensional factor V119891 in (11) is 05 in this study
41 Natural Frequencies By using the numerical methodthe transcendental equations (see (22) (27) and (31)) canbe solved and the corresponding natural frequencies areobtained In order to investigate temperature effects onnatural frequencies with different boundary conditions afrequency variation factor 119877120596 is defined
119877120596 = 120596Δ119879 minus 12059601205960 (61)
where 120596Δ119879 and 1205960 are the natural frequencies with andwithout thermal effects respectively
Figure 1 shows effects of temperature variations on thefirst four mode frequencies in the case of three differentboundary conditions As shown in these figures with theincrease (decrease) in the temperature the mode frequenciesall decrease (increase) In previous studies it is shown thatan increase in temperature leads to a decrease in naturalfrequencies [41] and the conclusion in this study is in goodagreement with some of the previous findings
Furthermore the changing of natural frequencies withtemperature was found to be dependent on the order of themode and boundary conditions With the increase of theorder the natural frequency becomes less sensitive to thetemperature variation To be more specific in the case of120574 = +10 as to three different boundary conditions thevariation factors 119877120596 of these first order mode frequenciesare 436 (hinged-hinged) 130 (hinged-fixed) and 57
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
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6 Shock and Vibration
Table 1 Physical parameters and material properties
Parameter Physical meaning Value Unit120588 density 7781 kgm3119871 length 2032 m119864 Youngrsquos modulus 207 GPa119860 area of cross-section 2581 times 10minus3 m2120572 thermal expansion coefficient 10 times 10minus5 ∘C119875 axial load 11281 kN119868 area moment of inertia 5550 times 10minus7 m4
33 Subharmonic Resonances In order to investigate the sub-harmonic resonance a detuning parameter 120590119899 is introducedΩ119899 = 3120596119899 + 120576120590119899 (53)
where 120576 is a small but finite parameterSimilarly the frequency response equation can be
obtained by using the multiple scales method
91205832119899 + (120590119899 minus 9Γ119899119899119899119899Λ2119899120596119899 minus 9Γ1198991198991198991198998120596119899 1198862119899)2
= 81Γ2119899119899119899119899Λ2119899161205962119899 1198862119899 (54)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (55)
The first approximation to the steady-state solution isgiven by
119902119899 (119905) = 119886119899 cos [13 (Ω119899119905 minus 120574119899)] + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (56)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase34 Super Harmonic Resonances In this case to express thenearness of Ω119899 to (13)120596119899 we also introduce the detuningparameter 120590119899 defined by
3Ω119899 = 120596119899 + 120576120590119899 (57)
where 120576 is a small but finite parameterFollowing the step of the multiple scales method a
frequency response equation can be obtained
120590119899 = 3Γ119899119899119899119899Λ2119899120596119899 + 3Γ1198991198991198991198998120596119899 1198862119899 plusmn radicΓ2119899119899119899119899Λ611989912059621198991198862119899 minus 1205832119899 (58)
where
Λ 119899 = 12 1198911198991205962119899 minus Ω2119899 (59)
Therefore the first approximation in this case is obtained
119902119899 (119905) = 119886119899 cos (3Ω119899119905 minus 120574119899) + 1198911198991205962119899 minus Ω2119899 cos (Ω119899119905)+ 119874 (120576) (60)
where 119886119899 is the actual nondimensional response amplitude 119905is the actual time scale and 120574119899 is the phase4 Numerical Examples and Discussions
In the following numerical analysis material properties andphysical parameters of the beam are chosen in Table 1 Thenondimensional temperature variation factor 120574 is chosen as[minus10 +10] and the corresponding dimensional temperaturevariation Δ119879 is [minus2083∘C +2083∘C] Moreover the nondi-mensional factor V119891 in (11) is 05 in this study
41 Natural Frequencies By using the numerical methodthe transcendental equations (see (22) (27) and (31)) canbe solved and the corresponding natural frequencies areobtained In order to investigate temperature effects onnatural frequencies with different boundary conditions afrequency variation factor 119877120596 is defined
119877120596 = 120596Δ119879 minus 12059601205960 (61)
where 120596Δ119879 and 1205960 are the natural frequencies with andwithout thermal effects respectively
Figure 1 shows effects of temperature variations on thefirst four mode frequencies in the case of three differentboundary conditions As shown in these figures with theincrease (decrease) in the temperature the mode frequenciesall decrease (increase) In previous studies it is shown thatan increase in temperature leads to a decrease in naturalfrequencies [41] and the conclusion in this study is in goodagreement with some of the previous findings
Furthermore the changing of natural frequencies withtemperature was found to be dependent on the order of themode and boundary conditions With the increase of theorder the natural frequency becomes less sensitive to thetemperature variation To be more specific in the case of120574 = +10 as to three different boundary conditions thevariation factors 119877120596 of these first order mode frequenciesare 436 (hinged-hinged) 130 (hinged-fixed) and 57
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 7R
04
02
00
minus02
minus04
minus10 minus05 00 05 10
1
2
3 4
(a)R
02
01
00
minus02
minus01
minus10 minus05 00 05 10
1
2
3 4
(b)
R
008
004
000
minus008
minus004
minus10 minus05 00 05 10
1
2
3 4
(c)
Figure 1 Temperature effects on the first four natural frequencies (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(Ω1minus1)1
20
15
10
05
00
0000 0005 0010 0015
=minus10= 0
=+10
(a)
(Ω1minus1)1
06
04
02
00
0000 0005 0010 0015
=minus10= 0
=+10
(b)
(Ω1minus1)1
03
02
01
00
0000 0005 0010 0015
=minus10= 0
=+10
(c)
Figure 2 Frequency response curves of nonlinear free vibrations (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
(fixed-fixed) respectively However as to the fourth-ordermode frequencies the corresponding variation ratios 119877120596are 13 11 and 09 respectively Moreover we canobserve a slightly higher sensitivity to warming than coolingin the case of the hinged-hinged boundary condition asshown in Figure 1(a) However in the case of the fixed-fixed boundary condition (Figure 1(c)) the asymmetryphenomenon between the cooling and warming conditionsalmost disappears It is concluded that temperature effectsare very closely related to the boundary conditions and thenatural frequency decreases with an increase in temperatureIn addition it should be pointed out that as to these threecases temperature effects on mode shapes can be negligible
42 Nonlinear Free and Forced Vibrations Neglecting thedamping and excitation terms the nonlinear free vibrationsof the beam with thermal effects are investigated Figure 2illustrates temperature effects on the nonlinear free vibrationbehaviors in the case of three different boundary conditionsFrequency response curves all show the hardening-springcharacteristics due to the cubic nonlinearity term The hard-ening behavior tends to increase for a positive temperature
change and to decrease for a negative one Moreover withthe enhancement of the constraint conditions (from hinged-hinged hinged-fixed to fixed-fixed) temperature effects onthe vibration behaviors are reduced
Temperature effects on the nonlinear responses to theprimary super and subharmonic excitations are studied inthe following in terms of frequency response curves andexcitation response curves respectively Moreover by usingthe multiple scales method the stability of these solutionsis determined by examining the eigenvalues of the Jacobianmatrix evaluated at the equilibrium point
In the nonlinear resonant cases the nondimensionalexcitation amplitude is chosen as 005 and the nondimen-sional damping coefficient is assumed to be independentof the temperature variation (120583119899 = 001) Figure 3 showstemperature effects on frequency response curves of theprimary resonance and three different boundary conditionsare considered There are almost three steady-state solutionsone unstable plotted in dashed lines and two stable of lowandhigh amplitudes plotted in continuous lines respectivelyThe stable solution is physically determined by the initialcondition of the system finally In the primary resonance
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
8 Shock and Vibration
0020
0015
0010
0005
0000
minus2 0 2 4
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
minus2 minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 3 Frequency response curves of primary resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
0015
0010
0005
0000
f
0000 0004 0008 0012
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(a)
0015
0010
0005
0000
f
000 001 002 003
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0015
0010
0005
0000
f
000 002 004 006
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 4 Excitation response curves of primary resonances (120590 = 20) (a) hinged-hinged (b) hinged-fixed (c) fixed-fixedthe system exhibits the hardening-spring behavior due to thecubic nonlinearity term and all these curves bend to the rightside As to the hinged-hinged boundary condition as shownin Figure 3(a) temperature effects are obvious Specificallyin the range of the large response amplitude the responseamplitude decreases with the increase in the temperature Onthe contrary as to the small response amplitude the responseamplitude increases with the increase in the temperatureAs to the hinged-fixed and fixed-fixed boundary conditionsthe temperature effect on the frequency response curvesis negligible in the range of small response amplitudesHowever as to the large response amplitudes the curvebends to the right more in the warming condition As to theboundary conditions from hinged-hinged to fixed-fixed it is
noted that the temperature effect on the resonance charactersis reduced
Figure 4 shows the variation of the response amplitudewith the excitation amplitude in the case of the primaryresonance under three boundary conditions Actually it isknown that depending on the value of detuning parameter120590 some curves are multivalued while others are single-value In this study only the case of 120590 = 20 is inves-tigated As shown in these figures the multivaluedness offrequency response curves due to the nonlinearity leads tothe jump phenomenon in the excitation response curvesThere are significant quantitative changes of these curves dueto temperature variations Moreover it is noted that in therange of large and small response amplitudes temperature
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
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wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 9
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
0008
0006
0004
0002
0000
00 05 10 15 20
(a)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(b)
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
00 05 10 15 20
(c)
Figure 5 Frequency response curves of subharmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
effects are different Specifically as to the small responseamplitude with the increase in the temperature the responseamplitude increases slightly On the contrary as to the largeone the response amplitude decreases significantly in thesame condition
Temperature effects on the subharmonic resonance areshown in Figure 5 In this case the stable trivial solution(119886119899 equiv 0) is not shown and only two nontrivial solutions arestudied in which one is stable and another is unstable It isnoted that there is no jump phenomenon in the case of thesubharmonic resonance As shown in Figure 5(a) as to thehinged-hinged boundary condition the temperature effectis the most visible and the threshold response amplitudedecreases with an increase temperature and increases with adecrease one The same conclusion could be observed fromthe Figure 5(b) in the case of the hinged-fixed boundarycondition In the last case (Figure 5(c)) it is shown that thetemperature effect on the frequency response curve is notobvious
Moreover excitation response curves of the subhar-monic resonance in the case of the hinged-hinged boundarycondition when 120590 = 20 are illustrated in Figure 6 Itis noted that there are significant quantitative changes ofthese curves due to temperature variations Considering thetemperature variations the same response amplitude 119886 isinduced by very different excitation amplitude 119891 Moreoverunder the excitationwith the same amplitude119891 very differentresponse amplitudes are obtained due to the temperaturevariations
Figure 7 illustrates temperature effects on frequencyresponse curves in the case of the super harmonic resonanceand three boundary conditions are also included Due tothe effect of the cubic nonlinearity the system exhibitsthe hardening behavior Generally as same as the previousresonant cases the hardening behavior also increases with
f
00 05 10 15
0008
0006
0004
0002
0000
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
Figure 6 Excitation response curves of subharmonic resonances inthe case of hinged-hinged boundary condition (120590 = 20)an increase in temperature and decreases with a decreasein temperature However it should be pointed out that inthe case of the super harmonic resonance (Figure 7(a)) nounstable solution is obtained and no jump phenomenon isobserved in the warming condition (120574 = +10) As shown inFigure 7(b) with the increase in the temperature the curvebends to the right more and the system exhibits strongerhardening-spring behavior It should be pointed out thatin the case of the cooling condition (120574 = minus10) the peakvalue of the frequency response curve decreases significantlyIn the last case as shown in Figure 7(c) as to the systemwith fixed-fixed boundary condition temperature effects onthe super harmonic resonance are much more obvious thanthe ones on the primary resonance and frequency responsecurves are shifted to the right more with the increase intemperature
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
10 Shock and Vibration
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
(a)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(b)
0012
0008
0004
0000
minus1 0 1 2
= minus10 (stable)= minus10 (unstable)= 0 (ＭＮ＜Ｆ)
= 0 (ＯＨＭＮ＜Ｆ)
= +10 (ＭＮ＜Ｆ)
= +10 (ＯＨＭＮ＜Ｆ)
(c)
Figure 7 Frequency response curves of super harmonic resonances (a) hinged-hinged (b) hinged-fixed (c) fixed-fixed
5 Summary and Conclusions
Apositive (resp negative) temperature change does decrease(resp increase) natural frequencies and temperature effectsonmode shapes can be negligible Temperature effects on thevibration behaviors are closely related to the mode order andboundary conditions and one can observe a slightly highersensitivity to warming than cooling As to the nonlinearvibration characteristics with thermal effects no matter theprimary super or subharmonic resonant cases the systemexhibits the hardening-spring behavior due to the cubicnonlinearity term The hardening behavior increases with anincrease in temperature and it decreases with a decrease intemperature Only some quantitative changes are observeddue to the temperature changes by examining the frequencyresponse curves and excitation response curvesThe responseamplitude the stability and the number of the nontrivialsolutions and the range of the resonance response are allclosely related to temperature variations Moreover as totemperature effects on vibration characteristics of the beamdifferent symmetric and nonsymmetric boundary conditionsshould be paid more attention
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research described in this paper was financially sup-ported by the National Natural Science Foundation of China
(11602089) and the Promotion Program for Young andMiddle-Aged Teacher in Science and Technology Research ofHuaqiao University (ZQN-YX505)
References
[1] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons New York NY USA 1979
[2] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995
[3] A H Nayfeh and P F Pai Linear and Nonlinear StructuralMechanics Wiley-Interscience New York NY USA 2004
[4] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley amp Sons Inc Hoboken USA 2013
[5] W Lacarbonara Nonlinear structural mechanics SpringerBerlin 2013
[6] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 1 Formulationrdquo International Journal ofNon-Linear Mechanics vol 44 no 6 pp 623ndash629 2009
[7] D Zulli R Alaggio and F Benedettini ldquoNon-linear dynamicsof curved beams Part 2 numerical analysis and experimentsrdquoInternational Journal of Non-LinearMechanics vol 44 no 6 pp630ndash643 2009
[8] F Benedettini R Alaggio and D Zulli ldquoNonlinear couplingand instability in the forced dynamics of a non-shallow archTheory and experimentsrdquo Nonlinear Dynamics vol 68 no 4pp 505ndash517 2012
[9] A H Nayfeh W Kreider and T J Anderson ldquoInvestigation ofnatural frequencies and mode shapes of buckled beamsrdquo AIAAJournal vol 33 no 6 pp 1121ndash1126 1995
[10] W Kreider and A Nayfeh ldquoExperimental investigation ofsingle-mode responses in a fixed-fixed buckled beamrdquo Nonlin-ear Dynamics vol 15 pp 155ndash177 1998
[11] A H Nayfeh W Lacarbonara and C-M Chin ldquoNonlinearnormal modes of buckled beams three-to-one and one-to-one
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Shock and Vibration 11
internal resonancesrdquoNonlinearDynamics vol 18 no 3 pp 253ndash273 1999
[12] W Lestari and S Hanagud ldquoNonlinear vibration of buckledbeams Some exact solutionsrdquo International Journal of Solids andStructures vol 38 no 26-27 pp 4741ndash4757 2001
[13] S A Emam and A H Nayfeh ldquoNonlinear Responses of Buck-led Beams to Subharmonic-Resonance Excitationsrdquo NonlinearDynamics vol 35 no 2 pp 105ndash122 2004
[14] A H Nayfeh and S A Emam ldquoExact solution and stabilityof postbuckling configurations of beamsrdquo Nonlinear Dynamicsvol 54 no 4 pp 395ndash408 2008
[15] S A Emam and A H Nayfeh ldquoNon-linear response of buckledbeams to 11 and 31 internal resonancesrdquo International Journalof Non-Linear Mechanics vol 52 pp 12ndash25 2013
[16] S A Emam and M M Abdalla ldquoSubharmonic parametricresonance of simply supported buckled beamsrdquo NonlinearDynamics vol 79 no 2 pp 1443ndash1456 2014
[17] P Ribeiro and R Carneiro ldquoExperimental detection of modalinteraction in the non-linear vibration of a hinged-hingedbeamrdquo Journal of Sound andVibration vol 277 no 4-5 pp 943ndash954 2004
[18] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011
[19] S Maleki and A Maghsoudi-Barmi ldquoEffects of Concur-rent Earthquake and Temperature Loadings on Cable-StayedBridgesrdquo International Journal of Structural Stability andDynamics vol 16 no 6 2016
[20] G Y Wu ldquoThe analysis of dynamic instability and vibrationmotions of a pinned beam with transverse magnetic fields andthermal loadsrdquo Journal of Sound and Vibration vol 284 no 1-2pp 343ndash360 2005
[21] G-Y Wu ldquoThe analysis of dynamic instability on the largeamplitude vibrations of a beam with transverse magnetic fieldsand thermal loadsrdquo Journal of Sound and Vibration vol 302 no1-2 pp 167ndash177 2007
[22] E Manoach and P Ribeiro ldquoCoupled thermoelastic largeamplitude vibrations of Timoshenko beamsrdquo International Jour-nal of Mechanical Sciences vol 46 no 11 pp 1589ndash1606 2004
[23] P Ribeiro and E Manoach ldquoThe effect of temperature on thelarge amplitude vibrations of curved beamsrdquo Journal of Soundand Vibration vol 285 no 4-5 pp 1093ndash1107 2005
[24] X-X Guo Z-M Wang Y Wang and Y-F Zhou ldquoAnalysis ofthe coupled thermoelastic vibration for axially moving beamrdquoJournal of Sound and Vibration vol 325 no 3 pp 597ndash6082009
[25] F Treyssede ldquoVibration analysis of horizontal self-weightedbeams and cables with bending stiffness subjected to thermalloadsrdquo Journal of Sound and Vibration vol 329 no 9 pp 1536ndash1552 2010
[26] J Avsec andM Oblak ldquoThermal vibrational analysis for simplysupported beam and clamped beamrdquo Journal of Sound andVibration vol 308 no 3ndash5 pp 514ndash525 2007
[27] M H Ghayesh S Kazemirad M A Darabi and P WooldquoThermo-mechanical nonlinear vibration analysis of a spring-mass-beam systemrdquoArchive of AppliedMechanics vol 82 no 3pp 317ndash331 2012
[28] H Farokhi andMH Ghayesh ldquoThermo-mechanical dynamicsof perfect and imperfect Timoshenko microbeamsrdquo Interna-tional Journal of Engineering Science vol 91 pp 12ndash33 2015
[29] A Warminska E Manoach and J Warminski ldquoNonlineardynamics of a reduced multimodal Timoshenko beam sub-jected to thermal and mechanical loadingsrdquoMeccanica vol 49no 8 pp 1775ndash1793 2014
[30] A Warminska E Manoach J Warminski and S SamborskildquoRegular and chaotic oscillations of a Timoshenko beamsubjected to mechanical and thermal loadingsrdquo ContinuumMechanics and Thermodynamics vol 27 no 4-5 pp 719ndash7372015
[31] V Settimi E Saetta and G Rega ldquoLocal and global nonlineardynamics of thermomechanically coupled composite plates inpassive thermal regimerdquo Nonlinear Dynamics vol 93 no 1 pp167ndash187 2018
[32] E Saetta V Settimi and G Rega ldquoNonlinear vibrations ofsymmetric cross-ply laminates via thermomechanically cou-pled reduced order modelsrdquo Procedia Engineering vol 199 pp802ndash807 2017
[33] Y Zhao Z Wang X Zhang and L Chen ldquoEffects of temper-ature variation on vibration of a cable-stayed beamrdquo Interna-tional Journal of Structural Stability and Dynamics vol 17 no10 1750123 18 pages 2017
[34] Y Zhao J Peng Y Zhao and L Chen ldquoEffects of temperaturevariations on nonlinear planar free and forced oscillations atprimary resonances of suspended cablesrdquo Nonlinear Dynamicsvol 89 no 4 pp 2815ndash2827 2017
[35] Y Zhao C Huang L Chen and J Peng ldquoNonlinear vibrationbehaviors of suspended cables under two-frequency excitationwith temperature effectsrdquo Journal of Sound and Vibration vol416 pp 279ndash294 2018
[36] E Ozkaya M Pakdemirli and H R Oz ldquoNon-liner vibrationsof a beam-mass system under different boundary conditionsrdquoJournal of Sound andVibration vol 199 no 4 pp 679ndash696 1997
[37] M Amabili ldquoNonlinear vibrations of rectangular plates withdifferent boundary conditions theory and experimentsrdquo Com-puters amp Structures vol 82 no 31-32 pp 2587ndash2605 2004
[38] AMamandiM H Kargarnovin and S Farsi ldquoAn investigationon effects of traveling mass with variable velocity on nonlineardynamic response of an inclined Timoshenko beamwith differ-ent boundary conditionsrdquo International Journal of MechanicalSciences vol 52 no 12 pp 1694ndash1708 2010
[39] S Lenci F Clementi and JWarminski ldquoNonlinear free dynam-ics of a two-layer composite beam with different boundaryconditionsrdquoMeccanica vol 50 no 3 pp 675ndash688 2015
[40] A Luongo G Rega and F Vestroni ldquoOn nonlinear dynamics ofplanar shear indeformable beamsrdquo Journal of Applied Mechan-ics vol 53 no 3 pp 619ndash624 1986
[41] Y Xia B Chen S Weng Y-Q Ni and Y-L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012
[42] Y Zhao C Sun Z Wang and L Wang ldquoApproximate seriessolutions for nonlinear free vibration of suspended cablesrdquoShock andVibration vol 2014 Article ID 795708 12 pages 2014
[43] Y Zhao C Sun Z Wang and L Wang ldquoAnalytical solutionsfor resonant response of suspended cables subjected to externalexcitationrdquo Nonlinear Dynamics vol 78 no 2 pp 1017ndash10322014
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
