Sung-Ryol So ,,Hoyong Yun ,Yongho Ri ,Ryongsik O ,Yong-Il Yun

a Institute of Advanced Science,Kim Il Sung University,Pyongyang,Democratic People’s Republic of Korea

b Institute of Science,Kimchaek University of Technology,Pyongyang,Democratic People’s Republic of Korea

c Department of Engineering Machine,Pyongyang University of Mechanical Engineering,Pyongyang,Democratic People’s Republic of Korea

d Department of Transport Mechanical Engineering,Pyongyang University of Mechanical Engineering,Pyongyang,Democratic People’s Republic of Korea

Abstract Investigation on vibration of laminated composite beam (LCB) is an important issue owing to its wide use as fundamental component.In the present work,we study the free vibration of arbitrarily LCB with generalized elastic boundary condition (BC) by using Haar wavelet discretization method (HWDM).Timoshenko beam theory is utilized to model the free vibration of LCB.The LCB is first split into several segments,and then the displacement for each segment is obtained from the Haar wavelet series and their integral.Hamilton’s principle is applied to construct governing equations and the artificial spring boundary technique is adopted to obtain the general elastic boundary and continuity conditions at two ends of LCB.The vibration characteristics of beam with different fiber orientations and lamina numbers is investigated and its displacements are compared with those in previous works.Numerical results are shown graphically and demonstrate the validation of our method.

Keywords:Laminated beam;Haar wavelet discretization method;Elastic boundary condition;Free vibration;Artificial spring boundary technique.

1.Introduction

Composite laminated structures for their high strength are being widely used in the fields such as aerospace,high-speed turbine machinery,shipbuilding,etc.As a fundamental component of structure,beams are contained in nearly every engineering structure.In particular,LCBs are often utilized in complex working environments,and the material fatigue owing to violent vibrations leads to the failure and collapse of structure.Therefore,based on the deep understanding of the structural vibration,the LCB should be designed to improve its performance.

Up to now,many researchers have proposed different methods for the vibration analysis of LCB.The LCB are made of several laminas tightly bonded in geometry,so the linear elastic theory can be applied to the study of three-dimensional structure of LCB [1].Some scholars investigated the vibration features of LCB by using first-order shear deformation beam theory (FSDBT),classical beam theory (CBT) and higherorder shear deformation beam theory (HSDBT) [2–8].Since the CBT usually neglects the transverse shear deformation effect,it is capable of underestimating the deflection and overestimating the natural frequency for moderately thick beams[9,10].Later,the CBT was developed into the FSDBT which considers the transverse shear deformation in beam bending.Because of violations of stress-free BCs on top and bottom surfaces in FSDBT,the shear factor should be corrected to weaken the discrepancy between the assumed constant stress state and the actual state.The preceding studies show that the shear correction factor has a great effect on the accuracy of the FSDBT solution.The HSDBT in which the shear correction factor is not used,has been suggested to eliminate the deficiencies of CBT and FSDBT.It includes either the higher-order variation of axial displacement or the axial and transverse displacements along the thickness of the beam.Regarding the higher-order shear deformation theory,some models using different shear strain shape functions have been proposed.

Recently,some researchers studied the vibration characteristics of LCB by using the refined higher-order shear deformation theories (HSDTs) [11–21].However,the HSDTs have some disadvantages that they demand more computation time than the FSDT and include complex formulations and BCs which are not easily solved [22].For this reason,the free vibration analysis of LCB needs a selection of the appropriate beam theory.

From the literature review,we came to the conclusion that the FSDT using the proper shear correction factor is capable of predicting the global behaviors of moderately thick LCB.In addition to the aforementioned theories,other analytical and numerical methods have been developed for the free vibration analysis of laminated beam.The examples are the meshless method [23–25],the closed-form solution [26–29],transfer matrix method [30,31],dynamic stiffness method[32–34],differential quadrature method [35–38]and finite element method [39–43].

As mentioned above,the vibration problem of laminated beam has been widely studied,but there are still many boundary valued problems unsolved.The HWDM is used to obtain the numerical solution of the generalized eigenvalue problem.The Haar wavelet series owing to its advantage in treating singularities,has been utilized to study the vibration of functionally graded plate in Ref [44]and evaluate damages of plates in Ref [45].Majak et al.[46]developed this method and applied it to solve problems in solid mechanics.In Ref [47],Majak discussed the strong and weak formulations,and pointed out that the weak formulation based on the HWDM is more efficient and stable for larger number of collocation points.

Recently,with the help of the Haar wavelet series,Hein and Feklistova [48,49]solved the vibration problem of a nonuniform and functionally graded beam with various BCs.R.Talebitooti et al.[50]investigated the free vibration of the FGM doubly-curved shells of revolution by using the Haar wavelet discretization approach.Jin GY et al.[51]performed the free vibration analysis of a functionally graded cylindrical shell by using HWDM,and Xie et al.[52–54]proposed a numerical solution method based on the HWDM for free vibration study of the laminated composite conical,cylindrical shell and annular plate with various classical BCs.

From the literature review on the vibration study of LCB using HWDM,the research has been carried out for structures with various classical BCs,but the study of laminated structures with generalized elastic boundaries is rarely found.Therefore,in this work,we use HWDM to investigate the free vibration of LCB with generalized elastic BCs.Through the free vibration study on the cross-ply LCB and angle-ply LCB,the accuracy,convergency,performance of our method is demonstrated.

Numerical results are shown graphically for different sizes and parameters of LCB,and they verify the accuracy of solution.

2.Governing equation

Consider an LCB made of several laminas as shown in Fig.1.The LCB has widthb,lengthLand thicknessh.The Cartesian coordinate system (x,y,z) is introduced to describe free vibration of the beam and its origin is set at the center of the left side of the beam.For the processing of the beam’s BCs,two translating spring groups (ku0,kw0andkuL,kwL) and one rotating spring group (kϕ0,kϕL) are adopted to the beam as shown in the figure.The BCs of the beam vary with the stiffness of the springs.zk,zk+1are the intervals between they-axis and the bottom as well as top surfaces of the (k+ 1)th lamina,respectively.

The Timoshenko beam theory is applied to obtain the displacements of LCB and they can be written as

whereU,VandWare the displacements in axial,transverse and vertical directions respectively.Meanwhileu,v,wstands for the translational displacements inx,y,zaxes,andϕ,θare the rotational displacements inx-zplane andy-zplane,respectively.

In this work,since we discuss the transverse vibration of the beam,the tensile displacements inx,zaxes as well as the transverse bending displacement are only considered.Therefore,the relationships between strain and displacement are given by

Fig.1.Geometry of LCB.

The relationship for thekth layer of LCB is written as

strain components and shear strain components of thekth layer,respectively.is the transformed elastic coefficient obtained byHereare the lamina elastic coefficient and the orientational angle of thekth layer fiber.For thekth layer where the principal material direction is placed,the coefficientis written as

By integrating the stresses over the cross-section,we obtain the force and moment in the following form.

The constitutive equations of lamination based on the classical lamination theory are written as

whereNx,Ny,Nxyare the in-plane forces,Mx,My,Mxyare the bending and twisting moments,andQx,Qyare the shear forces respectively.κstands for the shear correction factor.Since the beam has a rectangular crosssection,κequals to 5/6.The laminate stiffness coefficientsare defined by

Since the beam only has axial displacement inx-axis,vertical displacement inz-axis and bending displacement on the middle plane,the constitutive equation is

where

The following Hamilton’s principle is utilized to obtain governing equations and BCs.

The whole elastic strain energy of LCB,UV,is

From Eq.(1),the total kinetic energyTis

where

The potential energy stored in the boundary spring is expressed as

wherekt,0(t=u,w,ϕ)andkt,Lstand for the boundary spring stiffness at the both ends of LCB,respectively.

The variational operations result in the following governing equations of motion.

The general BCs for the LCB are

3.Implementation of the HWDM

For the help of understanding,we first describe some aspects of the Haar wavelet.Let us suppose that the Haar wavelet is defined ashi(x),and its orthogonal group is a set of square waves whose magnitudes are ±1 in some intervals as well as 0 elsewhere.

In the interval [0,1],thehi(x) is written as [50-54]

It is well known that with the help of Haar wavelet,any square integrable functiony(x) is able to be expanded in the interval [0,1]and its series expansion includes infinite terms.In case thaty(x) is piecewise constant itself,or may be approximated as piecewise constant within each subinterval,y(x)is truncated with finite terms,which is

whereai(i=1,...,2M) is unknown wavelet coefficient.The interval [0,1]is equally divided into 2Msubintervals(Δx=1/2M) and the collocation points are

The Haar coefficient matrixHis represented asH(i,l)=hi(ξl).The following integrals are used to solve then-th order PDE.[55]

α=1,2,…,n,i=1,2,…,2MForα=0,it corresponds to the functionhi(t).The integrals can be solved analytically.Fori=1,pα,i(ξ)=ξα/α!,and Fori＞1,the integrals are [48]

wheni=1,

wheni＞1,

For solving boundary value problems,it is necessary to determine the valuespα,i(0) andpα,i(1).

Substituting the collocation points in Eq.(22) into Eq.(26),

whereP(α)(i,l) is a 2M×2Mmatrix.It should be mentioned that the matricesH(i,l) andP(α)(i,l) must be calculated only once.The HWDM is used to discretize the governing equations in terms of translational and rotational displacements as well as boundary conditions.Since the Haar wavelet series is defined in the interval [0,1],in order to solve the finite domain problem by using the Haar wavelet method,we should convert the displacement domain into the unit interval [0,1].For this,the following normalized variable is used.

The principle of the HWD method is to expand the highest order derivatives of the functions with the help of Haar wavelet series and then determine the lower order derivatives as well as the function itself by integrating the expansions.As the highest order derivatives of the displacement and rotation functions in Eq.(13) are of order two,the second order derivatives of the unknown functions can be written by means of the Haar wavelet series in the following forms

whereai,biandciare the unknown Haar wavelets coefficients.Integrating Eq.(29) two times and considering Eq.(23),the following expressions are obtained.

The displacement and rotation functions are written in the following discrete matrix forms.

where H,P1and P2which denote the Haar matrices,are respectively

The notations in Eq.(31) are defined as

The BCs are used to determine the integral constantsf,gandm.In the Eq.(18),u,wandϕare of first order derivatives at most.Consequently,substitutingξ=0 andξ=1 into Eq.(31),we can obtain the values ofu,wandϕat the boundaries.Finally,using the same method as in Eq.(31),and applying the collocation points,the rotation and displacement components in Eq.(18) are written in the following discrete matrices.

where

Fig.2.Dependence of Ω on the truncation number of Haar wavelet series J for LCB with C-C BC.

The HWD method is then employed to discretize the Eqs.(17) and (18).By assembling the two equations,the following linear algebraic equations are obtained,that is

whereais the Haar wavelet coefficient andbis the integral constant.Eqs.(17.a -c) are represented by Kdb,Kdd,Mdband Mdd.In addition,Eq.(18) is represented by Kbband Kbd.In order to eliminate ℜbfrom Eq.(37),some algebraic manipulations are carried out.As a result,we obtain the following standard characteristic equation.

Solving Eq.(38),the natural frequencies and the corresponding modes are decided.It is noted that since most elements in Eq.(32a) are zero,there appear sparse matrices in Eq.(38),which leads to the increase in convergence rate and the decrease in computation time.Finally,the HWD method is employed to solve the Eq.(38) approximately.

Fig.3.Dependence of Ω on the stiffness values of boundary springs:(a) ku,(b) k w and (c) kϕ.

Table1 Stiffness of boundary springs.

Table2 Comparison of the natural frequencies for LCB.

Table3 Non-dimensional frequency parameters for an LCB of different layouts and BCs.

Table4 Non-dimensional frequency parameters for cross-ply LCBs of different stacking structures.

Table5 Non −dimensional frequency parameters for LCBs of different stacking structures with classical BCs.

Table6 Non-dimensional frequency parameters for LCBs of different stacking structures with elastic BCs.

4.Results and discussions

For simplicity of the BC description,in this paper,we introduce alphabetic strings.For example,C-S means the clamped BC at the left end and the simply-supported BC at the right end.The non-dimensional frequency parameterΩfor LCB is

For the consideration of free vibration for LCB,the Eq.(38) which contains Haar wavelet should be solved.The accuracy of solution is dependent upon the number of truncation number of the Haar wavelet series.Therefore,we first do simulations to determine truncation number.

Fig.2 shows the dependence ofΩon the truncation numberJfor LCB with C-C BC.The material properties and size of LCB areE1=144.8 GPa,E2=9.65 GPa,G12=G13=4.14 GPa,G23=3.45 GPa,ν12=0.3,ρ=1389.23 kg/m3,ϕfiber=[90 °/0 °/90 °/0 °],L=0.381 m,h=L/15 m.

From Fig.2,it is seen that the value ofΩfor LCB converge rapidly along with the increase ofJ.It can be also seen thatΩis almost unchanged whenJis greater than a certain value (J=7).In other words,it means that the accuracy of solution is not proportional to the value ofJ.Consequently,in this work,the value ofJis set to 7.

Fig.4.Variation of the non-dimensional frequency parameter versus lamina number k for [30 °/60 °]k laminated beams.(a);C-C,(b);S-C,(c);S-S,(d);G 1 -G 1,(e);G 2 -G 2,(f);G 3 -G 3.

As mentioned earlier,in our model,the BCs of LCB are determined by the stiffness of boundary springs.Thus,we then study the dependence ofΩon the stiffness of boundary springs.Fig.3 depicts the dependence ofΩon the stiffness of boundary springs:(a)ku,(b)kwand (c)kϕ.Here the material properties and size of LCB are the same as in Fig.2,and the only difference is that the laminating order is [90 °/0 °/90 °/0°].

In this work,it is assumed that the right boundary of LCB is clamped and the left boundary is elastically fixed by three springs.Only one spring in the left boundary restrained by three elastic springs has a variable stiffness.The other springs are supposed to have infinite stiffness.From Fig.3,we can see that the values ofkuandkwhave a great effect on theΩin the range from 106to 1011,but the value ofkϕhas no influence on theΩin the whole range from 104to 1018.Table1.shows the stiffness of boundary springs corresponding to different BCs.

The above study shows that our method has a fast convergence.The accuracy and reliability of the proposed method are then demonstrated through the comparison with the results in previous works.The comparison ofΩfor LCB with different classical BCs is shown in Table2.We can see that our results agree well with the ones in the reference.Here the material properties and size of LCB are the same as in Ref.[20].

Fig.5.Variation of Ω versus E 1 / E 2 for different stacking structures.(a)- [0 45 0 45],(b)- [0/45/45/0],(c)- [45/0/0/45],(d)- [45/-45/45/-45],(e)- [45/-45/-45/45],(f)- [-45/45/45/-45].

The natural frequencies for LCB in Table2 are in good agreement with those in the previous literature.As mentioned above,LCBs have a wide range of applications.To help engineers’ designs for LCB structure,some examples are given with the parameter study.

In Table3,the first four values ofΩfor LCBs with different BCs are shown according to some fiber direction angles.The parameters for LCB areE1=37.41 GPa,E2=13.67 GPa,G12=G13=5.478 GPa,G23=6.03 GPa,ν12=0.3,ρ=1938.9 kg/m3,L=0.11179 m,h=0.00338 m.

As seen in Table3,the value ofΩfor the same fiber direction angle in each layer dereases regardless of the BC.

Table4 shows values ofΩfor cross-ply LCBs of different lamina structures.The parameters are the same as in Fig.1 and the size isL=0.5 m,h=L/10 m.

From Table3,we can see that in the three-layer laminate structure,the values ofΩfor 90 °/0 °/90 ° are larger compared to those for 90 °/0 °/90 ° Here 90 °/0 °/90 ° means that the fiber orientation angles for three layers of the three-layer laminate structure are 90 °,0 ° and 90 ° respectively.The same phenomenon is found in the four-layer laminate structure.In other words,the values ofΩfor 0 °/90 °/90 °/0 ° are bigger than the ones for 90 °/0 °/0 °/90 ° For the same stacking structure,it can be also seen that the BC greatly affects theΩof LCB.

The variation ofΩversusL/hfor [30 °/60 °]klaminated beams,for different BCs is shown in Fig.4.Herekdenotes the lamina number of composite beams.k=1 stands for the two-layered laminations [30 °/60 °],k=2 denotes four-layered[30 °/60 °/30 °/60 °]un-symmetric schemes,and the number of layers fork=3 andk=4 increases in the same rule.It can be seen from the Fig.4 that the value ofΩgets larger as theL/hincreases for all cases.In addition,we also see that the value ofΩfork=4,are much larger than those fork=2,and fork≧ 4,the variation ofΩis not large.Specially,in case of the elastic BC,fork≧ 4 the value ofΩof LCB are nearly unchanged.

Tables 5 and 6 show the values ofΩfor LCBs of stacked structures of type [ϕfiber/-ϕfiber/ϕfiber]and [ϕfiber/-ϕfiber/-ϕfiber/ϕfiber]with classical and elastic BCs,respectively.Even in this case,it is apparent that the frequency parameters for the LCB stacked in four layers are larger than those for the three-layered structure in all BCs.However,it is seen that the values ofΩfor the two stacking types are the same only when the fiber direction angle equals to 90 ° This means that when the fiber direction angle is 90 °,the sign " +,-" specifying the fiber direction of composite material is meaningless.

Fig.5 shows the variation ofΩversusE1/E2for different stacking structures.The material properties and size of LCB are;E1=150 GPa,G12=G13=5 GPa,G23=6 GPa,ν12=0.25,ρ=1500 kg/m3,L=1 m,h=L/20 m.

From Fig.5,it is concluded that all curves have the same tendency,that is,the value ofΩdecrease as the value ofE1/E2increases.

5.Conclusion

In the present work,one method for the analysis of free vibration of LCB with generalized BCs have been systematized based on the Haar wavelet discretization.From the Timoshenko beam theory and Hamilton principle,the governing equation has been derived to evaluate the vibration of LCB.Then,the displacements in the equation have been discretized by using Haar wavelet to construct the differential equation of vibration.From the simulation calculations,the following results are obtained.

Firstly,the orientation angles of fibers in every lamina of LCB greatly effect on the LCB’s natural frequency.Secondly,not only the orientation angles of fibers,but also the number of laminas and the laminated structure (e.g.,in case that the orientation angles of fibers are not similar to each other,though the numbers of laminas for two different LCBs are the same) have an infulence on the LCB’s natural frequency.Thirdly,as the displacement in width direction for LCB is ignored,the rationE1/E2has no effect on the LCB’s natural frequency.From the above results,the orientation angles of fibers and the laminated structure should be considered in the design of LCB in reality.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

CRediT authorship contribution statement

Sung-Ryol So:Conceptualization,Methodology,Formal analysis,Resources,Writing - review &editing.Hoyong Yun:Investigation,Data curation,Writing - review &editing.Yongho Ri:Validation,Data curation.Ryongsik O:Validation,Writing - original draft.Yong-Il Yun:Investigation.