Vibration and buckling analysis of double-functionally graded Timoshenko beam system on Winkler-Pasternak elastic foundation Hao Deng ⇑ , KaiDong Chen, Wei Cheng, ShouGen Zhao School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China article info Article history: Received 9 August 2016 Revised 10 October 2016 Accepted 11 October 2016 Available online 19 October 2016 Keywords: Vibration and buckling Double-beam system Dynamic stiffness method Functionally graded Timoshenko beam Dynamic characteristic abstract To acquire exact solutions of double-functionally graded Timoshenko beam system on Winkler-Pasternk elastic foundation, which are benchmarks of double-beam systems in the field of engineering, motion dif- ferential equations of double-beam system are derived using Hamilton’s principle. In this paper, the exact dynamic stiffness matrix of double-functionally graded Timoshenko beam system on Winkler-Pasternak under axial loading are established and the damping of the connecting layer is also taken into consider- ation. The exact natural frequency and buckling load are obtained using Wittrick-William algorithm. To comprehensively analyze dynamic characteristics of double beam system, the effect of gradient param- eter, foundation parameters, axial loading and connecting stiffness on the frequency and buckling load is compared, and the influence of damping factor is also investigated. Finally, dynamic response of double- beam system is studied using Fourier transformation. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction Functionally graded material [1–6] is a combination of two or more than two different properties of material according to a cer- tain law. Two sides of the structure are composed of different physical properties to make the material components meet the special requirements of different conditions. The volume fraction of the component varies continuously in space. The mechanical properties of the structure can vary continuously at different posi- tion. Thus, the mutation of physical properties can be eliminated. Meanwhile, FGM can also reduce or avoid the stress concentration phenomenon in the components. Functionally graded materials have the characteristics of high strength, toughness, high temper- ature resistance and corrosion resistance, which also solves the problem that uncoordination of thermal expansion coefficient between the metal and ceramic. Functionally graded materials have presented the excellent performance in high strength, mechanical load, thermal load, or under high temperature environ- ment. Therefore, FGM is considered as the most potential compos- ite material in the fields of spacecraft, machinery industry and nuclear industry, and the research on mechanical behavior of func- tionally graded materials has become a frontier subject in modern materials science and mechanics. Free vibration of functionally graded beams was analyzed by Alshorbagy et al. [7] by using finite element method. Xiang and Yang [8] used direct analytical method to analyze the vibration of a laminated FGM Timoshenko beam. Zhu H [9] used the Fourier series-Galerkin method to investigate the functionally graded beams. Su H [10] acquired the exact solutions of the functionally graded Timoshenko beam using dynamic stiffness method. Thai HT [11] analyzed the bending and free vibration of FGB by using various higher-order shear deformation beam theories. Lai SK [12] has investigated the large amplitude vibration of FGB through accurate analytical perturbation method. Huang Y [13] used a new approach to analyze the free vibration of axially functionally graded beams with non-uniform cross section. A large amount of studies have been done about the beam resting on the Winkler- Pasternak elastic foundation. Lee [14] studied the free vibrations of non-uniform beams resting on two-parameter elastic founda- tion. Wang [15] acquired the exact solutions for Timoshenko beams on elastic foundations by using Green functions. De Rosa [16] investigated the influence of Pasternak soil on the free vibration of Euler beams. Chen W Q [17] used a mixed method to analyze bending and free vibration of beams resting on a Pasternak elastic foundation. Ying J [18] obtained the two-dimension elasticity solutions for functionally graded beams resting on elastic foundations. In recent years, some higher-order shear deformation theories and normal deformation theory were proposed. M Bourada [19] developed a simple and refined trigonometric http://dx.doi.org/10.1016/j.compstruct.2016.10.027 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: [email protected](H. Deng). Composite Structures 160 (2017) 152–168 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct
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Hao Deng ⇑, KaiDong Chen, Wei Cheng, ShouGen ZhaoSchool of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China
a r t i c l e i n f o a b s t r a c t
Article history:Received 9 August 2016Revised 10 October 2016Accepted 11 October 2016Available online 19 October 2016
Keywords:Vibration and bucklingDouble-beam systemDynamic stiffness methodFunctionally graded Timoshenko beamDynamic characteristic
To acquire exact solutions of double-functionally graded Timoshenko beam system on Winkler-Pasternkelastic foundation, which are benchmarks of double-beam systems in the field of engineering, motion dif-ferential equations of double-beam system are derived using Hamilton’s principle. In this paper, the exactdynamic stiffness matrix of double-functionally graded Timoshenko beam system on Winkler-Pasternakunder axial loading are established and the damping of the connecting layer is also taken into consider-ation. The exact natural frequency and buckling load are obtained using Wittrick-William algorithm. Tocomprehensively analyze dynamic characteristics of double beam system, the effect of gradient param-eter, foundation parameters, axial loading and connecting stiffness on the frequency and buckling load iscompared, and the influence of damping factor is also investigated. Finally, dynamic response of double-beam system is studied using Fourier transformation.
� 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Functionally graded material [1–6] is a combination of two ormore than two different properties of material according to a cer-tain law. Two sides of the structure are composed of differentphysical properties to make the material components meet thespecial requirements of different conditions. The volume fractionof the component varies continuously in space. The mechanicalproperties of the structure can vary continuously at different posi-tion. Thus, the mutation of physical properties can be eliminated.Meanwhile, FGM can also reduce or avoid the stress concentrationphenomenon in the components. Functionally graded materialshave the characteristics of high strength, toughness, high temper-ature resistance and corrosion resistance, which also solves theproblem that uncoordination of thermal expansion coefficientbetween the metal and ceramic. Functionally graded materialshave presented the excellent performance in high strength,mechanical load, thermal load, or under high temperature environ-ment. Therefore, FGM is considered as the most potential compos-ite material in the fields of spacecraft, machinery industry andnuclear industry, and the research on mechanical behavior of func-tionally graded materials has become a frontier subject in modernmaterials science and mechanics.
Free vibration of functionally graded beams was analyzed byAlshorbagy et al. [7] by using finite element method. Xiang andYang [8] used direct analytical method to analyze the vibrationof a laminated FGM Timoshenko beam. Zhu H [9] used the Fourierseries-Galerkin method to investigate the functionally gradedbeams. Su H [10] acquired the exact solutions of the functionallygraded Timoshenko beam using dynamic stiffness method. ThaiHT [11] analyzed the bending and free vibration of FGB by usingvarious higher-order shear deformation beam theories. Lai SK[12] has investigated the large amplitude vibration of FGB throughaccurate analytical perturbation method. Huang Y [13] used a newapproach to analyze the free vibration of axially functionallygraded beams with non-uniform cross section. A large amount ofstudies have been done about the beam resting on the Winkler-Pasternak elastic foundation. Lee [14] studied the free vibrationsof non-uniform beams resting on two-parameter elastic founda-tion. Wang [15] acquired the exact solutions for Timoshenkobeams on elastic foundations by using Green functions. De Rosa[16] investigated the influence of Pasternak soil on the freevibration of Euler beams. Chen W Q [17] used a mixed method toanalyze bending and free vibration of beams resting on a Pasternakelastic foundation. Ying J [18] obtained the two-dimensionelasticity solutions for functionally graded beams resting on elasticfoundations. In recent years, some higher-order shear deformationtheories and normal deformation theory were proposed.M Bourada [19] developed a simple and refined trigonometric
H. Deng et al. / Composite Structures 160 (2017) 152–168 153
higher-order beam theory to analyze bending and vibration offunctionally graded beams. H Hebali [20] developed a new quasi-three-dimensional hyperbolic shear deformation theory for thebending and free vibration analysis of functionally graded plates.M Bennoun [21] applied a new five-variable refined plate theoryfor the free vibration analysis of functionally graded sandwichplates. SA Yahia [22] developed various higher-order shear defor-mation plate theories for wave propagation in functionally gradedplates. Z Belabed [23] proposed an efficient and simple higherorder shear and normal deformation theory for functionally gradedmaterial plates. A Mahi [24] developed a new hyperbolic sheardeformation theory for bending and free vibration analysis of iso-tropic, functionally graded, sandwich and laminated compositeplates. A Hamidi [25] presented a simple but accurate sinusoidalplate theory for thermomechanical bending analysis of function-ally graded sandwich plates. High-order beam theories are ableto represent the section warping in the deformed configurationand have higher accuracy for high order modes, while acquiringthe exact solutions is complex and difficult. For simplicity,Timoshenko beam theory is considered in this paper. Timoshenkobeam theory requires shear correction factors and is suitable forcomposite beams for which the shear correction factors can bedetermined.
Double-beam system which consists of two parallel beams con-nected by elastic layer is an important technological extension inindustrial field such as double-beam cranes, double-beam spec-trometers, double-beam interferometers, etc. Double-beam sys-tems are elements of various devices as well as of mechanical,civil, and aircraft structures. Examples also include aircraft wingspars, double-beam cranes, railway tracks resting on a foundation,bridge spans, pipelines, and trusses. Twin beams can also beencountered in mechanical systems on a smaller scale, for examplelinear guideways used in plotters or during such technological pro-cesses as cutting. Meanwhile, two-beam system such as floating-slab tracks is widely used to control the vibration from under-ground trains. Functionally graded material can resist high temper-ature and reduce stress concentration phenomenon, which has awide range of applications in the high-temperature environment.Therefore, functionally graded double beam system has a broadprospect of application.
S. Kukla [26] studied the free vibration of the two-beam systemconnected by many translational springs. Z. Oniszczuk [27] studiedthe free vibration of elastically connected simply supported double-beam complex system. Recently the double-beam system has beenused as a new type of vibration absorber to control the vibration of abeam-type structure. Aida [28] et al. analyzed the vibration controlof beams by using two-beam system. Hussein [29] modeledfloating-slab tracks by using two-beam system and the layer wassimulated by the springs and dampers. Shamalta M [30] analyzedthe dynamic response of an embedded railway track by analyticalmethod. Zhang et al. [31] investigated the vibration and bucklingof a double-beam system under compressive axial loading. Jun Li[32] established an exact dynamic stiffness matrix to compute thenatural frequency for an elastically connected three-beam system,which is composed of three parallel beams of uniform propertieswith uniformly distributed-connecting springs among them. ArieiA [33] studied the transverse vibration of a multiple-Timoshenkobeam system with intermediate elastic connections. Simsek M[34] et al. studied nonlocal effects in the forced vibration of an elas-tically connected double-carbon nanotube system. Palmeri [35]used a novel state-space form to study transverse vibrations ofdouble-beam systems made of two outer elastic beams continu-ously joined by an inner viscoelastic layer. M. Abu [36] studiedthe dynamic response of a double Euler-Bernoulli beam under amoving constant load.W.-R [37] has investigated bending vibrationof axially loaded Timoshenko beams with locally distributed
Kelvin-Voigt damping. Recently, Simsek [38] studied the dynamicsof elastically connected double-functionally graded beam systemswith different boundary conditions under action of a moving loadbased on the Euler beam theory.
As mentioned earlier, these studies primarily focused on beamsystem made of homogeneous material. Inhomogeneous material,such as functionally graded material, is rarely involved. At thesame time, main solution method these studies based on dependson numerical solution, such as finite element method, but the pre-cision of numerical method cannot be gauranteed. Thus, it is essen-tial to acquire exact solutions which can be benchmarks of double-beam systems made of functionally graded materials. Elastic foun-dation model can be used to simulate the interactions betweenbeam system and elastic medium, B Bouderba [39] delt with thethermomechanical bending response of functionally graded platesresting on Winkler-Pasternak elastic foundations. Meziane [40]presented an efficient and simple refined shear deformation theoryfor the vibration and buckling of exponentially graded materialsandwich plate resting on elastic foundations under variousboundary conditions. M Zidi [41] studied the bending response offunctionally graded material plate resting on elastic foundation.ND Duc [42] investigated the nonlinear dynamic response ofeccentrically stiffened functionally graded double curved shallowshells resting on elastic foundations. ND Duc [43] studied the non-linear response of thick functionally graded double-curved shallowpanels resting on elastic foundations and subjected to thermal andthermomechanical loads. ND Duc [44] studied the nonlinearresponse of panels resting on elastic foundations, which accountfor higher order transverse shear deformation and panel-foundation interaction. ND Duc [45] studied nonlinear responseof imperfect eccentrically stiffened FGM cylindrical panels on elas-tic foundation subjected to mechanical loads. ND Duc [46] devel-oped an analytical approach to investigate the nonlinear staticbuckling and post-buckling for imperfect eccentrically stiffenedfunctionally graded thin circular cylindrical shells surrounded onelastic foundation with ceramic–metal–ceramic layers. ND Duc[47] analyzed nonlinear stability of the imperfect FGM cylindricalpanel reinforced by eccentrically stiffeners on elastic foundations.ND Duc [48] investigated the linear stability analysis of eccentri-cally stiffened FGM conical shell panels reinforced by mechanicaland thermal loads on elastic foundation. ND Duc [49] used thirdshear deformation shell theory to investigate nonlinear thermaldynamic behavior of imperfect functionally graded circular cylin-drical shells eccentrically reinforced by outside stiffeners and sur-rounded on elastic foundations. Recently, Duc ND [50] studiednonlinear thermo-electro-mechanical dynamic response of sheardeformable piezoelectric Sigmoid functionally graded sandwichcircular cylindrical shells on elastic foundations. However, thestudies about the double-functionally graded beams system rest-ing on Winkler-Pasternak elastic foundations are very rare.Because of the widely application of functionally gradedTimoshenko beam in engineering field, the author formulates theexact dynamic stiffness matrix of the two-beam system restingon Winkler-Pasternak elastic foundation under the axial loads,and the damping of connection layer is also taken into considera-tion when analyzing the dynamic response of the structure. Finally,buckling of double-functionally graded Timoshenko beam systemis also analyzed. The concept of the dynamic stiffness matrixmethod was first proposed by Kolousek [51] in the early 1940 s.This method is a powerful tool to solve the problem of structuralvibration in engineering, especially in the need to obtain higherorder natural frequency and higher accuracy. The dynamic stiffnessmatrix method [52–58] is also commonly referred to as an exactmethod. Thus, the dynamic stiffness matrix of the two-beam sys-tem is established to analyze the effect of parameters on dynamiccharacteristic and buckling of the system.
154 H. Deng et al. / Composite Structures 160 (2017) 152–168
2. Theory and formulation
2.1. 1motion differential equations
Fig. 1 shows that the two functionally graded Timoshenkobeams are connected by the elastic connection. This two-beam sys-tem rests on the elastic foundation under axial forces N. The Win-kler elastic layer between the two beams can be simulated by thetension spring, and the spring stiffness of per unit length is K. Theelastic foundation is made of two layers which are Winkler layerand shearing layer. The two functionally graded Timoshenkobeams have the same length L, width b and thickness h. Assumingthat the material properties of the two beams are the same, and thematerial properties are elastic modulus EðzÞ, shear modulus GðzÞ,poisson’s ratio m and mass density qðzÞ. For the common FGMbeam, the bottom surface is made of metal materials and the topsurface is made of ceramic materials. The middle part of beam isthe mixture of two materials. Because the values of possion’s ratiom of metal materials and ceramic materials are close, the Poisson’sratio v of the beam is regarded as a constant value. And the mate-rial properties varying along the thickness direction satisfy power-law [1] form except for poisson’s ratio m. Therefore, the propertiesof the functionally graded Timoshenko beam need to satisfy:
PðzÞ ¼ ðPt � PbÞVt þ Pb ð1Þwhere PðzÞ is the material property including elastic modulus, shearmodulus and mass density. Pt is the property at the top surface, and
zy
NN
N N
Winkler layer Shear layer
Winkler layer
1 midpoint
2 midpoint
Fig. 1. An elastically connected double-functionally graded Timoshenko beamsystem on the elastic foundation under the axial forces.
-0.5 -0.4 -0.3 -0.2 -0.1 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Thickness co
Vol
ume
frac
tion
func
tion
Fig. 2. The effect of the gradient index
Pb is the properties at the bottom surface. where Vt is the volumefraction of material component. Assuming the origin of coordinatesis established on the central axis of the functionally gradedTimoshenko beam, volume fraction Vt which is the power functionalong the thickness direction can be given by power-law [1]:
Vt ¼ zhþ 12
� �k
ðk P 0Þ ð2Þ
where k is the gradient index. Fig. 2 shows the influence of gradientindex k on the volume fraction Vt .
According to the theory of functionally graded Timoshenko, vand w are the displacements of the point on the neutral axis alongthe y and z directions. uy anduz are the displacements of the pointon the cross section perpendicular to the neutral axis along they and z directions. Thus,
uyðy; z; tÞ ¼ vðy; tÞ � z/ðy; tÞ ð3Þ
uzðy; z; tÞ ¼ wðy; tÞ ð4Þwhere / is the bending rotation of the cross-sections. According tothe theory of elasticity, the relationship between strain and dis-placement is given as:
eyycyz
( )¼
@uy@y
@uy@z þ @uz
@y
8<:
9=; ¼
@v@y � z � @/
@y
@w@y � /
( )ð5Þ
where eyy is the normal strains and cyz is shear strain. According toHooke’s law:
ryy
syz
� �¼ EðzÞ 0
0 GðzÞ
� � eyycyz
" #ð6Þ
where ryy; syz are the normal and shear stresses of the functionallygraded Timoshenko beam. Using the above constitutive relation-ships, the strain energy Ps and kinetic energy T of double-functionally graded Timoshenko beam systems can be expressed as:
Ps ¼ 12
X2i¼1
Z L
0A0
@v i
@y
� �2
� 2A1@v i
@y@/i
@yþ A2
@/i
@y
� �2"
þ A3@wi
@y
� �2
� 2@wi
@y/i þ /2
i
#dy ð7Þ
0.1 0.2 0.3 0.4 0.5ordinate z/h
k=1k=5k=10k=0.1k=0.2k=0.5k=2
k on the volume fraction function.
H. Deng et al. / Composite Structures 160 (2017) 152–168 155
T ¼12
X2i¼1
Z L
0I0 ð@wi
@tÞ2
þð@v i
@tÞ2
!�2I1
@v i
@t
� �� ð@/i
@tÞþ I2 � ð@/i
@tÞ2
" #dy
ð8Þwhere subscripts 1 and 2 denote the upper beam and lower beam.And
Ii ¼RziqðzÞdA; Ai ¼
RziEðzÞdAði ¼ 0;1;2Þ
A3 ¼ R jGðzÞdA ð9Þ
where j is the shear coefficient andj ¼ 5=6. The work done by axialforceN can be expressed as:
Pp ¼ 12
X2i¼1
Z L
0
�Nð@wi
@yÞ2�dy ð10Þ
The elastic potential energy of elastic foundation is donoted by
Pf ¼ b2
Z L
0
�Kww2
2 þ Ksð@w2
@xÞ2�dx ð11Þ
where Kw;Ks are the parameters of the elastic foundation. Potentialenergy induced by the elastic layer between the beams can beexpressed as
Where t1 and t2 are the time interval, and d is the usual variationaloperator. The governing differential equations of motion can beobtained by Eq. (13) through the application of symboliccomputation:
�I0€v1 þ A0v 001 þ I1€/1 � A1/
001 ¼ 0 ð14Þ
�I0 €w1 þ A3w001 � A3/
01 � Kðw1 �w2Þ � Nw00
1 ¼ 0 ð15Þ
I1€v1 � A1v 001 þ A3w0
1 � I2€/1 þ A2/001 � A3/1 ¼ 0 ð16Þ
�I0€v2 þ A0v 002 þ I1€/2 � A1/
002 ¼ 0 ð17Þ
�I0 €w2 þ A3w002 � A3/
02 þ Kðw1 �w2Þ �Nw00
2 þ ðKsw002 � Kww2Þ � b ¼ 0
ð18Þ
I1€v2 � A1v 002 þ A3w0
2 � I2€/2 þ A2/002 � A3/2 ¼ 0 ð19Þ
The axial force F1; F2, shear force S1; S2 and bending momentM1;M2 are obtained as:
where Vi;Wi and Ui are amplitudes of v i;wi;/i, andx is the circularfrequency. Differential equations of state space can be obtained bysubstituting Eq. (21) into Eqs. (14)–(19):
@v@y
¼ A � v ð22Þ
where v is state space vector, and v ¼ ½W1;U1;V1; S1;M1; F1;
W2;U2;V2; S2;M2; F2�T . T denotes transpose and Si;Mi; Fiði ¼ 1;2Þare the amplitudes of shear force, bending moment and axialforce.where
where
a1 ¼ �A�13 ; a2 ¼ A0
A21 � A0A2
; a3 ¼ A1
A21 � A0A2
; a4 ¼ A2
A21 � A0A2
a5 ¼ x2I0; a6 ¼ x2I2; a7 ¼ �x2I1ð23Þ
kiði ¼ 1 . . .12Þ are the twelve eigenvalues of matrix A. Thus, Eq. (22)can be solved by using differential equation theory:
ViðyÞ ¼X12j¼1
Pijekiy; WiðyÞ ¼X12j¼1
Qijekiy; UiðyÞ ¼
X12j¼1
Rijekiyði ¼ 1;2Þ
ð24Þwhere Pij;Qij;Rij are the constants, and the relationship between theconstants can be obtained as:
ary conditions for forces and displacements at the end of the beamare applied. According to Fig. 3, the boundary conditions for forcesand displacements are:
ð45ÞAccording to Eq. (44) and Eq. (38), the relationship between forcevector F and displacement vector d can be derived as:
F ¼ K � d ð46Þwhere K is dynamic stiffness matrix of two-beam system and can beexpressed as:
K ¼ A � B�1 ð47ÞIt is clear that the dynamic stiffness matrix K is frequency
dependent. The formation method of the global dynamic stiffnessmatrix which can be obtained through assembling elementdynamic stiffness matrix is similar to the finite element method.To compute the frequencies and mode shapes of the two-beamsystem, the well-established algorithm of Wittrick-Williams needto be applied.
2.3. Dynamic stiffness formulation using precise integration method
Formation of dynamic stiffness matrix in 2.2 section is an exactmethod. However, stiffness matrix can also be obtained in a moreconvenient method using precise integration method. According toEq. (22), v can be computed as following:
vðyÞ ¼ eAyvð0Þ ð48ÞAlthough above form is concise, precise and efficient computa-
tion of matrix exponent is difficult. Matrix exponent can be com-puted by precise integration method, while computationalstability cannot be guaranteed, especially for special kind of matri-ces. This method may lead to inaccurate results or error results.Recently, Tan and Wu [62] proposed an efficient method to com-pute matrix exponent, which combines PIM and Pade approxima-tion. The procedure of computing the exponential matrix can begiven as follows.
T ¼ Iþ TaðNþ1Þ ð54ÞThe computational accuracy of PIM can be determined by
parameters p, q and N. These parameters can be determined byadaptive algorithms of selecting parameters [54]. Eq. (48) can alsobe expressed as:
X�FR
� �¼ T11 T12
T21 T22
� �XL
FL
� �ð55Þ
where XR; FR are the state vectors of the displacement and force atthe right end of the element and XL; FL are the vectors at the leftend of the element. After transformation, Eq. (55) can take the fol-lowing form,
FL
FR
� �¼ K11 K12
K21 K22
� �XL
XR
� �ð56Þ
where
K11 ¼ �T�112 � T11;K12 ¼ T�1
12 ;K21 ¼ T21 þ T22T�112 T11
K22 ¼ �T22T�122
ð57Þ
Therefore, dynamic stiffness matrix can be obtained by Eq. (57).Although this is a numerical method, results are close to exactsolutions if appropriate parameters are selected.
2.4. Winkler layer between two beams with viscous damping
When the viscous damping of the Winkler layer between twobeams is taken into consideration, the motion differentialequations can also be obtained:
�I0€v1 þ A0v 001 þ I1€/1 � A1/
001 ¼ 0 ð58Þ
�I0 €w1 þ A3w001 � A3/
01 � Kðw1 �w2Þ � Nw00
1 � cð _w1 � _w2Þ ¼ 0
ð59Þ
I1€v1 � A1v 001 þ A3w0
1 � I2€/1 þ A2/001 � A3/1 ¼ 0 ð60Þ
�I0€v2 þ A0v 002 þ I1€/2 � A1/
002 ¼ 0 ð61Þ
� I0 €w2 þ A3w002 � A3/
02 þ Kðw1 �w2Þ � Nw00
2 þ ðKsw002 � Kww2Þ
� bþ cð _w1 � _w2Þ ¼ 0 ð62Þ
I1€v1 � A1v 001 þ A3w0
1 � I2€/1 þ A2/001 � A3/1 ¼ 0 ð63Þ
where c is a viscous damping factor per unit length. The dynamicstiffness of the two-beam system can be obtained through replacingthe stiffness K with the complex stiffness K þ icx.
2.5. Frequency response function of two-beam system
The frequency response function matrix H(x) of two-beam sys-tem can be obtained by inverse of global dynamic stiffness matrixD(x):
½HðxÞ� ¼ ½DðxÞ��1 ð64Þ
2.6. Boundary conditions of two-beam system
Boundary conditions of two-beam can be derived by Eq. (13). Inthis paper, the boundary conditions of hinged-hinged, clamped-hinged, clamped-clamped, clamped-free are considered. For differ-ent conditions, the force and displacement of two-beam systemshould satisfy:
Clamped : V1 ¼ W1 ¼ U1 ¼ V2 ¼ W2 ¼ U2 ¼ 0
158 H. Deng et al. / Composite Structures 160 (2017) 152–168
Hinged : V1 ¼ W1 ¼ M1 ¼ V2 ¼ W2 ¼ M2 ¼ 0 ð65Þ
Free : F1 ¼ S1 þ N@W1
@y¼ M1 ¼ 0; F2 ¼ S2 þ N
@W2
@y¼ M2 ¼ 0
Fig. 4. The double beam system under moving load.
2.7. Application of the Wittrick-William algorithm [59]
For free vibration, after applying boundary conditions, the glo-bal dynamic stiffness matrix K(x) needs to satisfy [59]:
KðxÞd ¼ 0 ð66Þwhere d is modal shape vector. In order to obtain the natural fre-quency of the structure, the dynamic stiffness matrix K(x) shouldsatisfy:
jKðxÞj ¼ 0 ð67ÞWittrick-William algorithm (W-W) is an efficient algorithm for
solving Eq. (54). This algorithm does not directly compute the nat-ural frequency of the structure. In fact, this method is a countingalgorithm. According to Wittrick-William algorithm, the numberof frequencies which are lower than the given value x⁄ can begiven by
Jðx�Þ ¼ J0ðx�Þ þ s Kðx�Þf g ð68Þwhere J0ðx�Þ is the total number of frequencies of individual ele-mentwith fixed boundary conditionswhich are lower than the givenvaluex⁄. s{K(x⁄)} is the number of negative elements on the leadingdiagonal of KD; KD is the upper triangular matrix which can beobtained through the Gauss elimination method, and J; J0; s Kðx�Þf gare integers. The number of natural frequencies in any frequencyrange can be obtained by Wittrick-William algorithm. The dichot-omy is the simplest way to solve the structural frequency. Howeverthe computational efficiency of the dichotomy is low. Recently, YuanSi [60] have proposed a new method named Second order mode-finding method to compute the frequencies and mode shapes effi-ciently. Due to the limited space of the article, the detailed compu-tational procedure can be found in the literature [60].
2.8. Buckling analysis of two-beam system
For the buckling analysis of two-beam system, the governingequations of two-beam system can be obtained through settingthe inertia term to zero in Eqs. (14)–(19). Thus,
A0v 001 � A1/
001 ¼ 0 ð69Þ
A3w001 � A3/
01 � Kðw1 �w2Þ � Nw00
1 ¼ 0 ð70Þ
�A1v 001 þ A3w0
1 þ A2/001 � A3/1 ¼ 0 ð71Þ
A0v 002 � A1/
002 ¼ 0 ð72Þ
A3w002 � A3/
02 þ Kðw1 �w2Þ � Nw00
2 þ ðKsw002 � Kww2Þ � b ¼ 0 ð73Þ
�A1v 001 þ A3w0
1 þ A2/001 � A3/1 ¼ 0 ð74Þ
The static stiffness matrix KðNÞ of the two-beam system underthe axial forces is obtained by setting the frequency term x2 tozero in dynamic stiffness matrix. Similarly, the global stiffnessmatrix can be assembled by element stiffness matrix. By applyingboundary conditions, which is similar to the finite elementmethod, global stiffness matrix KðNÞ should satisfy [59]:
KðNÞd ¼ 0 ð75Þwhere d is the displacement vector. To obtain the buckling load N ofthe double-beam system, the stiffness matrix KðNÞ should satisfy:
jKðNÞj ¼ 0 ð76ÞIt is clear that the lowest positive solution of Eq. (76) is the crit-
ical buckling load of the two-beam system, and Eq. (76) can also besolved by the Wittrick-William algorithm mentioned in 2.7.
2.9. Dynamic characteristics of double beam system under movingload
In this section, we analyze the dynamic characteristics of dou-ble functionally graded beam system under moving load. The stan-dard linear solid model [35] is applied to simulate the dynamicbehavior of the viscoelastic inner layer, which is made of Maxwell’selement and elastic spring as shown in Fig. 4. According to litera-ture [35], the complex-valued stiffness between two layers canbe expressed as:
kinnðxÞ ¼ K0 þ K1isx
1þ isxð77Þ
where i ¼ffiffiffiffiffiffiffi�1
pis the imaginary unit and s is the relaxation time of
the viscoelastic material. The relationship between force and dis-placement can be obtained as [35]:
Finn ¼ Kinn � _drðtÞ ¼Z t
�1Kinnðt � sÞ _drðsÞds ð78Þ
where ⁄ denotes the convolution operator, the over-dot representsderivative with respect to time t. drðtÞ is the pertinent displacement.
KinnðtÞ is the relaxation function and can be derived as,
KinnðtÞ ¼ F�1 1ix
kinnðxÞ� �
ð79Þ
where F�1 represents inverse Fourier transform. The moving load onthe upper beam can be expressed as follows
Fðx; tÞ ¼ f � eiXtdðx� vtÞ ð80Þwhere f is moving force amplitude and d represents dirac function.X and v are harmonic excitation frequency and moving speed.External force potential energy can be expressed as:
Uext ¼ �Z L
0Fðx; tÞ �w1ðx; tÞdx ð81Þ
Using Hamilton’s principle, we can obtain
dZ t2
t1
ðT � U � UextÞdt ¼ 0 ð82Þ
Through the symbolic operation, Eq. (82) can be written asfollowing:
�I0€v1 þ A0v 001 þ I1€/1 � A1/
001 ¼ 0 ð83Þ
�I0 €w1 þ A3w001 � A3/
01 � KinnðtÞ � ð _w1 � _w2Þ � Nw00
1 þ Fðx; tÞ ¼ 0
ð84Þ
H. Deng et al. / Composite Structures 160 (2017) 152–168 159
I1€v1 � A1v 001 þ A3w0
1 � I2/001 þ A2/
001 � A3/1 ¼ 0 ð85Þ
�I0€v2 þ A0v 002 þ I1€/2 � A1/
002 ¼ 0 ð86Þ
�I0 €w2þA3w002�A3/
02þ KinnðtÞ � ð _w1� _w2Þ�Nw00
2þðKsw002�Kww2Þ �b¼0
ð87Þ
I1€v2 � A1v 002 þ A3w0
2 � I2/002 þ A2/
002 � A3/2 ¼ 0 ð88Þ
where * denotes the convolution operator. Transforming aboveequations from the time domain t to the frequencyx, we can obtain
@v@x
¼ �A � v þ F ð89Þ
where F ¼ ½0;0; 0; Fðx;xÞ;0; 0;0;0; 0;0;0; 0�T and v ¼ ½W1;U1;V1; S1;
M1; F1;W2;U2;V2; S2;M2; F2�T . For example, W1ðx;xÞ can be trans-formed from w1ðx; tÞ using Fourier transform. �A can be obtainedfrom Eq. (22) by replacing variable K with kinnðxÞ. After transforma-tion from the space-frequency domain ðx;xÞ to the wavenumber-frequency ðn;xÞ, Eq. (75) can be transformed into such form
in � E � �v ¼ �Aðn;xÞ � �v þ �Fðn;xÞ ð90Þ
where �Fðn;xÞ ¼ ½0;0;0;4p2dðxþ nv �XÞ; 0;0;0; 0;0;0; 0;0�T . E is12 � 12 unit matrix, Eq. (90) can also be written as
ðin � E� �Aðn;xÞÞ � �v ¼ �Fðn;xÞ ð91Þwhere i is
ffiffiffiffiffiffiffi�1
p. Solving for v from above equation
�v ¼ ðin � E� �Aðn;xÞÞ�1�Fðn;xÞ ð92ÞTherefore, ~v can be computed through transforming above
equation from wavenumber-frequency to wavenumber-time,
~v ¼ ðin � E� �Aðn;x ¼ X� nvÞÞ�1~F ð93Þ
where ~F ¼ ð0;0; 0;2p � eiðX�nvÞt;0;0;0;0;0; 0;0;0ÞT . Thus, v can bederived using Fourier transformation from wavenumber-timedomain to space-time domain
v ¼ 12p
Z þ1
�1~v � einxdn ð94Þ
Equation above can be solved using fast discrete Fourier inversetransform. For example,
vðx; tÞ ¼ 12p
Z þ1
�1Vðn; tÞdn � Dn
2pXN
j¼�Nþ1
Wðnj; tÞeinjx ð95Þ
where
nk ¼ ðk� NÞDn� Dn2
; k ¼ 1;2; . . . ;2N ð96Þ
Substituting Eq. (96) into Eq. (95),
vðxm; tÞ � Dn2pX2Nk¼1
Vðnk; tÞeinkxm ð97Þ
where
xm ¼ ðm� NÞDx;m ¼ 1;2; � � � ;2N ð98Þand
Dx � Dn ¼ pN
ð99Þ
Thus,
Dx ¼ pNDn
ð100Þ
nkxm ¼ � N � 12
� �ðm� NÞp
Nþ ðk� 1Þðm� NÞp
Nð101Þ
Using Eq. (101) and Eq. (97), vðxm; tÞ can expressed as following:
vðxm; tÞ � Db2p
e�iðN�12Þðm�NÞp=NX2N
k¼1
Vðnk; tÞeiðk�1Þðm�NÞp=N ð102Þ
Eq. (94) can be computed following above procedure. We needto mention that the number of sampling points should be largeenough to obtain accurate results. To aviod aliasing, sampling fre-quency 2p=Dn should satisfy Nyquist criterion [53]. However,when moving speed increases, sampling frequency must increaseto aviod aliasing. Therefore, convergence of results should betested.
2.10. Interval analysis of double beam system under moving load withuncertain-but-bounded parameters
To analyze dynamic response of double beam system undermoving load with uncertain parameters, Eq. (93) can be rewrittenas:
Z � v ¼ F ð103Þwhere Z can be expressed as
Z ¼ in � E� �Aðn;x ¼ X� nvÞ ð104ÞAssume that parameters of beam system are uncertain-but-
bounded vector a, which can be expressed as:
aI ¼ ½alower ;aupper� ð105Þwhere alower and aupper represent the lower and upper bounds of theuncertain parameter vector a. Z and v are complex matrix and com-plex vector respectively. Thus, Eq. (103) can be rewritten as:
ðZreal þ i � ZimagÞ � ðvreal þ i � vimagÞ ¼ Freal þ i � Fimag ð106ÞThe above equation can be expressed as:
Zreal �Zimag
�Zimag Zreal
� � vreal
vimag
� �¼ Freal
Fimag
� �ð107Þ
With the interval extension, Eq. (107) can be rewritten as:
ZðaIÞvI ¼ F ð108Þwhere vI is the theoretical solution set, which can be expressed as:
vI ¼ ½vl; vu� ð109ÞTo solve Eq. (108), interval matrix ZðaIÞ can be approximated by
using Taylor series expansion:
ZðaIÞ ¼ ZðamÞ þXki¼1
DaIi@ZðamÞ@ai
¼ Zm þ DZI ð110Þ
where am is the interval mean value of the interval vector aI. aIi rep-resents interval parameter aiði ¼ 1 � � � kÞ. DaIi can be expressed as:
DaIi ¼ ½�Dai;Dai� ð111ÞTherefore, we can obtain:
am ¼ ðalower þ aupperÞ=2Da ¼ ðaupper � alowerÞ=2 ð112ÞInserting Eq. (110) into Eq. (108),
ðZm þ DZIÞ � ðvm þ DvIÞ ¼ F ð113ÞThus,
vm þ DvI ¼ ðZm þ DZIÞ�1F ð114Þ
Table 1Comparison of frequencies ðrad=sÞ under the S-S boundary condition.
160 H. Deng et al. / Composite Structures 160 (2017) 152–168
If the spectral radius of matrix ðZmÞ�1DZ is less than 1, equationabove can be rewritten as follows using Neumann series:
vm þ DvI ¼ ðZmÞ�1Fþ
X1c¼1
ðZmÞ�1ð�DZIðZmÞ�1ÞcF ð115Þ
Ignoring the higher order terms, the equation above can be sim-plified as follows:
vm ¼ ðZmÞ�1F ð116Þ
DvI ¼ �ðZmÞ�1DZIvm ð117Þ
Substituting Eq. (110) into Eq. (117), we can obtain:
DvI ¼ �ðZmÞ�1 Xk
i¼1
DaIi@ZðamÞ@ai
!vm ð118Þ
It is clear that DvI is a monotonic function with respect to DaIi.Therefore, the lower and upper bounds can be expressed as:
vlower ¼ vm � ðZmÞ�1 Xk
i¼1
DaIi@ZðamÞ@ai
vm
��������
!���������� ð119Þ
vupper ¼ vm þ ðZmÞ�1 Xk
i¼1
DaIi@ZðamÞ@ai
vm
��������
!���������� ð120Þ
where j � j represents the absolute value. The interval of vðx; tÞ canbe obtained through Eq. (102).The derivative of matrix Z withrespect to ai can be expressed as:
@ZðamÞ@ai
¼ � @�AðamÞ@ai
ð121Þ
3. Numerical results and discussion
To verify the correctness of the results in this paper, numericalresults of free vibration of simply supported beams arecompared with literature [27]. In literature [27], the material oftwo-beam system is homogeneous. The values of parametersare: E ¼ 1� 1010 N=m2;A ¼ 5� 10�2, I ¼ 4� 10�4, L ¼ 10 m,q ¼ 2� 103 kg=m3, Ks ¼ Kw ¼ 0, where A and I are area andmoment of inertia. The results of first three natural frequenciesare compared in table 1. From table 1, it is observed that the pre-sent numerical results are in good agreement with literature [27].
To analyze the buckling and dynamic characteristics oftwo-functionally graded Timoshenko beam system, thegeometrical properties of two-beam system are as follows:b ¼ 0:1 m;h ¼ 0:1 m. The material properties of metal are:
Table 2The fundamental non-dimensional natural frequency k of two-beam system under CC bou
The effects of the slenderness ratio L=h and gradient parameterk on the fundamental non-dimensional natural frequency of two-beam system are showed in Tables 2–5. The stiffness of Winklerelastic layer between two beams is K ¼ 105 N=m and the axialforce N ¼ 0. The parameters of the foundation are Ks ¼ Kw ¼ 0.According to Tables 2–5, for different slenderness ratioL=hðL ¼ 1 mÞ, the fundamental frequency decreases with theincrease of gradient parameter k under different boundary condi-tions. It is noticeable that increase of gradient parameter k can bothincrease the bending stiffness and the mass of the two-beam sys-tem, while the effect of the mass increase is more significant com-pared to bending stiffness, and the boundary condition has greatinfluence on the fundamental frequency. The slenderness ratioobviously affects non-dimensional natural frequency as shown inTables 2–5. In general, with the increase of slenderness ratio,non-dimensional natural frequency decreases rapidly.
It is clear that the Winkler layer between the two beams greatlyaffects the frequencies corresponding to reverse modes of doublebeam, but have little effect on other forms of vibration modes.Fig. 5 shows the influence of the stiffness K of connecting layeron the fundamental frequency of two-beam system under differentboundary conditions. The length of two-beam system is L ¼ 5 m.Parameters of the foundation are Ks ¼ Kw ¼ 0:01 Gpa and the axialforce N ¼ 0. The other parameters are the same as before. It is seenthat the increase of the stiffness of the connecting layer canincrease the fundamental frequency under different boundary con-ditions. C; F;H denote clamped, free and hinged.
Fig. 4 plots the effect of elastic foundation parameters Kw;Ks onthe non-dimensionless fundamental frequencykof the two-beamsystem under the boundary condition CAC and HAH. The lengthof two-beam system is L ¼ 5 m and gradient parameter k ¼ 1.The stiffness of elastic layer is K ¼ 105 N=mand the axial force isN ¼ 0. The range of Kw;Ks is 0.01–0.1 Gpa. It is clear that with
stiffness of Winkler elastic layer between two beams(N/m)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
k=0.2k=0.5k=1k=2k=5
1 2 3 4 5 6 7 8 9 10
x 105
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
stiffness of Winkler elastic layer between two beams(N/m)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
k=0.2k=0.5k=1k=2k=5
(c) H-H (d) C-H
1 2 3 4 5 6 7 8 9 10
x 105
4.5
5
5.5
6
6.5
7
7.5
8
stiffness of Winkler elastic layer between two beams(N/m)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
k=0.2k=0.5k=1k=2k=5
1 2 3 4 5 6 7 8 9 10
x 105
6
6.5
7
7.5
8
8.5
9
9.5
10
stiffness of Winkler elastic layer between two beams(N/m)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
k=0.2k=0.5k=1k=2k=5
(b) C-C
Fig. 5. The influence of the stiffness K on fundamental frequency of two-beam system: (a) clamped-free (b) clamped-clamped(c) hinged-hinged (d) clamped-hinged.
H. Deng et al. / Composite Structures 160 (2017) 152–168 161
(a) C-C (b) H-H
00.02
0.040.06
0.080.1
00.02
0.040.06
0.080.1
10.01
10.015
10.02
10.025
10.03
10.035
10.04
Kw(Gpa)Ks(Gpa)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
00.02
0.040.06
0.080.1
00.02
0.040.06
0.080.1
5.27
5.28
5.29
5.3
5.31
5.32
Kw(Gpa)Ks(Gpa)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
Fig. 6. The influence of the foundation parameters Kw;Ks on fundamental frequency of two-beam system: (a) clamped-clamped (b) simple-simple.
(a) C-C
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 105
7
8
9
10
11
12
13
compressive axial force(N)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
k=0.2k=0.5k=1k=2k=5
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 105
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
compressive axial force(N)
non-
dim
ensi
onal
fun
dam
enta
l fre
quen
cy
k=0.2k=0.5k=1k=2k=5
(c) H-H
Fig. 7. The influence of the compressive axial force on fundamental frequency of two-beam system: (a) clamped-clamped (b) hinged-hinged.
162 H. Deng et al. / Composite Structures 160 (2017) 152–168
the increase of the parameter Kw;Ks, the fundamental frequency ofthe two-beam system will increase, because the increase of thestiffness of the elastic foundation will increase the stiffness ofthe whole system. However, fundamental frequency keeps almostconstant when the values of the Kw;Ks approximate 0:1 Gpa. Fig. 6shows that the effects of Winkler layer and shear layer on funda-mental frequency are close. Fig. 7 plots the effect of compressiveaxial force on the fundamental frequency of the two-beam systemunder CAC and HAH boundary conditions. Fig. 7 shows thatincreasing the compressive axial force lead to the decrease of sys-tem stiffness.
Figs. 8 and 9 shows that the effect of viscous damping factor con the frequency response function of the two-beam system underCAC and HAH boundary conditions, where foundation parametersare Ks ¼ 0:01 Gpa;Kw ¼ 0, gradient parameter is k ¼ 1, and theaxial force is N ¼ 0. Figs. 8 and 9 show the origin FRF of transversedisplacement at position 1 and the cross-point FRF at position 2under the excitation at position 1 along the z direction. Position1 and position 2 denote the midpoints of two beams as shown inFig. 1. It is observed that the damping factor has a significant effecton the second-order frequency, because second-order mode is areverse mode of two beams and the value of the frequency increasewith the increase of the damping factor c. However, for fundamen-
tal frequency, the damping factor only affects the peak value of fre-quency response function. Because the mode shape of fundamentalfrequency is a similar mode of two beams along the same direction,the damping factor has a little influence on the value of fundamen-tal frequency. It is obvious that the amplitude of the cross-pointFRF decreases compared to the origin FRF due to the damping layerbetween the two beams. Similar mode denotes deflections of twobeams towards the same direction, while reverse mode denotesdeflections of two beams towards opposite direction as shownbelow:
upperbeam
lowerbeam
upperbeam
lowerbeam
reversesimilar
In order to investigate the effect of gradient parameter k ondynamic characteristic of two-beam system with damping, the lossfactor g of the system can be defined as [61]
(a) origin FRF
0 50 100 150 200 250 300-140
-130
-120
-110
-100
-90
-80
-70
-60
Circular frequency(rad/s)
disp
lace
men
t F
RF
(db)
180 200 220-120
-115
-110
-105
-100
c=0c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/m
0 50 100 150 200 250 300-100
-80
-60
-40
-20
0
20
40
60
80
100
Circular frequency(rad/s)
phas
e(de
g)
c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/mc=0
(b) cross-point FRF
0 50 100 150 200 250 300-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
Circular frequency(rad/s)
disp
lace
men
t F
RF
(db)
c=0c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/m
0 50 100 150 200 250 300-100
-80
-60
-40
-20
0
20
40
60
80
100
Circular frequency(rad/s)
phas
e(de
g)
c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/mc=0
Fig. 8. The influence of the damping factor c on the amplitude and phase of FRF under CAC boundary condition: (a) origin FRF (b) cross-point FRF.
H. Deng et al. / Composite Structures 160 (2017) 152–168 163
g ¼ Imðx2nÞ
Reðx2nÞ
ð123Þ
where x2n is the nth complex natural frequency. Fig. 10 shows the
influence of gradient parameter k on the loss factor g of the systemunder CAC and H-H boundary conditions. Damping factor c is103Ns=m. The stiffness of elastic layer is K ¼ 105 N=m and founda-tion parameters are Ks ¼ 0:01 Gpa and Kw ¼ 0. It can be observedthat the increase of gradient parameter k can increase the loss factorof the two-beam system, and the parameters of foundation havegreat effect on the loss factor.
3.2. Buckling analysis of two-beam system
In this section, numerical results of buckling analysis of two-beam system are analyzed. The length of the beam is L ¼ 5 m ,the geometrical properties of two-beam system are as follows:b ¼ 0:1 m;h ¼ 0:1 m. The material properties of metal are:Em ¼ 70 Gpa, mm ¼ 0:23, qm ¼ 2700 kg=m3. The material propertiesof ceramic are: Ec ¼ 380 Gpa, mc ¼ 0:23 Gpa, qc ¼ 3800 kg=m3 andshear coefficient j ¼ 5=6. The non-dimensional buckling load canbe defined as:
p� ¼ pL2
p2EmIð124Þ
where I is cross sectional moment of inertia.
Tables 6–8 show that the buckling load decreases with theincrease of the gradient parameter k under different boundary con-ditions. It can also be observed that the increase of the connectingstiffness K will increase the buckling load. For different foundationparameters, the effect of stiffness Ks of shear layer on buckling loadis similar to the effect of stiffness Kw of Winkler layer according toTables 6–8.
3.3. Dynamic characteristic analysis of double-beam system undermoving load
In this section, the geometry dimensions of double beam systemare b ¼ 0:1 m;h ¼ 0:1 m for both beams. The material properties ofmetal are Em ¼ 70 Gpa, mm ¼ 0:23 and qm ¼ 2700 kg=m3. Similarly,the material properties of ceramic are Ec ¼ 380 Gpa, mc ¼ 0:23qc ¼ 3800 kg=m3 and shear coefficient j ¼ 5=6. Winkler-type equi-librium modulus is K0 ¼ 30 kN=m2, and Maxwell’s parameter isK1 ¼ 5K0. Relaxation time is s ¼ 0:2 s for inner layer. The elasticfoundation parameters are Ks ¼ Kw ¼ 0:01 Gpa. Axial load is N ¼ 0.Characteristic parameters of moving load are f ¼ 1000 N andX ¼ 0. (a) v ¼ 50 m=s;t ¼ 0:1 s (b) v ¼ 200 m=s;t ¼ 0:1 s
According to Fig. 11, displacement responses corresponding todifferent speeds are shown. It is noticeable that the track deflectionis close to deformation of the structure under static load, when thespeed of moving load is far lower than critical speed. However,
(a) origin FRF
0 50 100 150 200 250 300-140
-130
-120
-110
-100
-90
-80
-70
-60
Circular frequency(rad/s)
disp
lace
men
t F
RF
(db)
c=0c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/m
0 50 100 150 200 250 300-100
-80
-60
-40
-20
0
20
40
60
80
100
Circular frequency(rad/s)
phas
e(de
g)
c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/mc=0
(b) cross-point FRF
0 50 100 150 200 250 300-170
-160
-150
-140
-130
-120
-110
-100
-90
-80
-70
Circular frequency(rad/s)
disp
lace
men
t F
RF
(db)
c=0c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/m
0 50 100 150 200 250 300-100
-80
-60
-40
-20
0
20
40
60
80
100
Circular frequency(rad/s)
phas
e(de
g)c=500Ns/mc=1000Ns/mc=1500Ns/mc=2000Ns/mc=0
Fig. 9. The influence of the damping factor c on the amplitude and phase of FRF under H-H boundary condition: (a) origin FRF (b) cross-point FRF.
Fig. 10. The influence of the gradient k on the loss factor: (a) clamped-clamped(b) hinged-hinged.
164 H. Deng et al. / Composite Structures 160 (2017) 152–168
when the speed of load exceeds critical speed, structure begins tovibrate in wide area as shown in Fig. 11(b). In Fig. 11, the dottedline represents lower beam and the solid line denotes upper beam.
It is clear that different values of gradient parameter have a signif-icant effect on the deflection of double-beam system. Displace-ment of upper beam in time domain can be seen in Fig. 12.
Table 6The non-dimensional buckling load P⁄ of two-beam system under CC boundary conditions.
KðN=mÞ k ¼ 0:1 k ¼ 0:2 k ¼ 0:5 k ¼ 1 k ¼ 2 k ¼ 5 k ¼ 10
Fig. 12. Displacement response of upper beam in time domain: (a) v ¼ 50 m=s;x ¼ 20 m (b)v ¼ 200 m=s;x ¼ 20 m.
(a) different parameter k (b) different relaxation time ( 1)k
100 150 200 250 300 350 4001
2
3
4
5
6
7
8
9
10
11x 10
-3
velocity(m/s)
max
imum
dis
plac
emen
t(m
)
k=0.1k=1k=10
100 150 200 250 300 350 4000
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
velocity(m/s)
max
miu
m d
ispl
acem
ent(
m)
160 180 2000.008
0.01
0.012
0.014
0.016
relaxation time 0.5srelaxation time 0.2srelaxation time 1s
Fig. 13. Maximum track vibration displacement corresponding to different moving speeds.
(a) wavenumber domain (b) space domain
-3 -2 -1 0 1 2 3-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
wavenumber (rad/m)
disp
lace
men
t (m
)
-4000 -3000 -2000 -1000 0 1000 2000 3000
-1.5
-1
-0.5
0
0.5
1
1.5
x 10-3
x(m)
disp
lace
men
t(m
)
1000 1500 2000 2500-5
0
5x 10
-4
Fig. 14. Transverse displacement response of upper beam ðv ¼ 200 m=s;k ¼ 0:1; t ¼ 10 sÞ: (a) wavenumber domain (b) space domain.
166 H. Deng et al. / Composite Structures 160 (2017) 152–168
(a) wavenumber domain (b) space domain
-3 -2 -1 0 1 2 3-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
wavenumber (rad/m)
disp
lace
men
t(m
)
-500 0 500 1000 1500 2000 2500 3000-8
-6
-4
-2
0
2
4
6
x 10-4
x(m)
disp
lace
men
t(m
)
14001600180020002200
-1
0
1
2
x 10-4
Fig. 15. Transverse displacement response of upper beam ðv ¼ 180 m=s;k ¼ 0:1; t ¼ 10 sÞ: (a) wavenumber domain (b) space domain.
H. Deng et al. / Composite Structures 160 (2017) 152–168 167
Fig. 13 shows maximum transverse displacement of upperbeam corresponding to different speeds, where peaks of curvesrepresent critical speed. As we can see from the Fig. 13(a), whenthe moving speed approximates critical speed of the track, ampli-tude of structure will increase sharply. Meanwhile, with theincrease of the gradient k, critical speed will increase significantly,and the vibration amplitude will decrease. In Fig. 13(b), it is foundthat relaxation time s only affects the value of peak, while it has noeffect on the critical speed. Figs. 14 and 15 present transverse dis-placement response of upper beam, where Fig. 15a and b representresponse in wavenumber domain and space domain respectively.According to Fig. 15, the critical speed of track is 189 m/s whenk ¼ 0:1. Therefore, the range of response is relatively narrow whenmoving speed less than critical speed. In comparison, track willvibrate in large spatial range as show in Fig. 14b when movingspeed (200 m/s) exceeds critical speed.
4. Conclusion
In this paper, To acquire exact solutions of double-functionallygraded Timoshenko beam system on Winkler-Pasternk elasticfoundation, which are benchmarks in the field of engineering, theexact dynamic stiffness matrix of the double-functionally gradedTimoshenko beam system on Winkler-Pasternak foundation underaxial loading is established. The results obtained by dynamicstiffness method are in great agreement with previous studies.Thus, the correctness and effectiveness of the method was demon-strated. After comprehensive study of this beam system, we canconclude that:
1. With the increase of gradient parameter k, the fundamental fre-quency of the two-beam system decreases under differentboundary conditions.
2. The stiffness of connecting layer has great influence on thereverse modes of two beams, and the increase of the stiffnesscan significantly increase the frequencies of the two-beamsystem.
3. the increase of the parameters of elastic foundation willincrease the frequencies of two beam system and the effectsof Winkler layer and shear layer on fundamental frequencyare close.
4. The damping factor of connecting layer has great influence onthe FRF, with the increase of the damping factor, the peaks ofthe FRF will decrease. And damping factor greatly affects thefrequencies of reverse modes.
5. The increase of gradient parameter k can increase the loss factorof the two-beam system.
6. The buckling load decreases with the increase of the gradientparameter k under different boundary conditions.
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