Chein-Shan Liu·Bo-Tong Li

Abstract A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function.The R(x)-orthonormal bases in the linear space of boundary functions are constructed,of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x).When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A:= [ai j],ai j = 0 if i + j is odd.Based on the special structure we can prove two new theorems,which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions halflower.Hence,we can sequentially solve the natural frequencies in closed-form owing to the specialty of A.We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.

Keywords Symmetric composite beams · R(x)-orthogonality of second-order derivatives of boundary functions ·R(x)-orthonormal theory·Non-smooth/discontinuous interface·Sequentially closed-from natural frequencies

1 Introduction

In engineering design,beams have a prominent role in different environments,which are subjected to different boundary conditions [1]. The applications of non-uniform/composite beams in engineering have resulted in many studies in the past.Many researchers have proposed various analytical and numerical methods to analyze the free vibration of Euler-Bernoulli beams of varying cross section properties [2-4].Moreover,the free vibrations of the Timoshenko beam with surface effects were also investigated by Zhao et al.[5].

Thecompositebeamswithvariableinterfacehavereceived much attention for their superior structural performance to fit the stress distribution better [6]. The shear deformable thin-walledcompositebeamswithclosedcross-sectionshave been developed by Kim and Choi[7]for the analyses of coupled flexural,torsional,and buckling.

Some micro-electromechanical devices such as the resonators manufactured in the type of non-uniform composite beams[8]or nanotubes[9]are widely adopted.

For the free vibration analysis of composite beams with periodically varying interfaces,Li et al.[10]have proposed a new iterative method based on the weak-form integral equation technique using the sinusoidal test functions.Because of the orthogonality of the sinusoidal functions they find the closed-form expansion coefficients. However, that method requires iteration and is not applicable to the nonsmooth/discontinuous beam.In Ref.[10],we found that for the N-sine wave interfaced composite beam it can achieve the best performance to increase the first-order natural frequency about 7%if we take N =0.5 for the simply supported composite beam. For N = 0.5 the interface function f(x)is symmetric with respect to the middle point x = L/2,i.e.,f(x) = f(L - x). Therefore, we further study the symmetric composite beam with N-wave complex interface and with symmetric support,such as the simply supported beam and the clamped-clamped beam. Other types of symmetric supports can be treated similarly.

In general,finding the closed-form solutions of the natural frequencies is very difficult for a non-uniform/composite beam.In the previous upper bound theory developed in Ref.[11], the formulae to compute the natural frequencies are quite simple. However, when the order of the natural frequency is increased,the upper bound theory gradually loses its accuracy in the prediction of higher-order natural frequencies.

The present paper is arranged as follows.The free vibration of the Euler-Bernoulli beam equation together with the symmetric composite beams with three different interfaces is introduced in Sect. 2. The Rayleigh quotient, the linear space of boundary functions and the upper bound theory are briefly sketched in Sect.3.One of the main contributions is given in Sect.4,where we introduce a new concept of R(x)-orthonormal theory, and then we can transform the natural frequency problem of Rayleigh quotient to the eigenvalue problem with a special coefficient matrix.The special coefficient matrix with aij= 0, if i + j is odd, is investigated in Sect.5 by many examples and the natural frequencies are obtained by the new theory. The comparisons to the upper bound theory are given there.In Sect.6,we carry out the performance study of the first third-order natural frequencies for the symmetric composite beams with three different interfaces. Another major contribution is carried out in Sect. 7,where two new theorems about the R(x)-orthonormal theory are proven.As a consequence,we can sequentially find the closed-form solutions of the higher-order natural frequencies up to any order we desire. However, to save space we terminate at the eighth-order natural frequency. The corresponding free vibration modes are solved in Sect.8.Finally,the conclusions are drawn in the last section.

2 N-wave interfaces

The free vibration of the Euler-Bernoulli beam is governed by

where L is the length of beam, R(x)is the flexural rigidity function of a composite beam, P(x) is the planar inertial function,y(x)is the transverse deflection,and ω is the natural frequency of free vibration.

Fig.1 Symmetric composite beam considered in this work

The configuration of beam is shown in Fig. 1, which is composed of two materials with one material for the upper beam and another for the lower beam,where three interfaces:sine wave,triangular wave,and square wave are considered as

where f0is the amplitude of interface, N is the number of waves, x0= L/N is the wave length,and z = Mod(x,x0)is the modulus function. To generate a symmetric interface function f(x),we only need to do

It is easy to check that f(x) = f(L - x) is a symmetric function.

According to the geometry of the composite beam, we have

where b is the width of the beam,h1is the upper height of the beam,and h2is the lower height. E1and E2, I1and I2,ρ1and ρ2are,respectively,the Young's modulus,the moment of inertia,and the density of upper and lower beams.

Owing to the symmetry of f(x), the material functions R(x)and P(x)in Eq.(1)are both symmetric,i.e., R(x) =R(L-x)and P(x)= P(L-x).Mathematically speaking,we are going to study the symmetric beam equation and its natural frequency.

3 Rayleigh quotient

For the symmetric composite beam Eq.(1),we consider the following boundary conditions:

Clamped-clamped beam:

First, by multiplying both the sides of Eq. (1) by y(x),integrating it from x = 0 to x = L twice and using the corresponding boundary conditions for each type beam,one can derive the Rayleigh quotient:

which is a useful mathematical tool to study the free vibration of the composite beam.When the vibration mode(eigenfunction) y(x)for each mode can be known we can compute the natural frequency ω for each mode.

The boundary functions are introduced in Ref.[11]by Simple beam: B1= x4-2Lx3+L3x,

which automatically satisfy the boundary conditions for each type beam.

The scalar multiplication and the addition of boundary functions are closure.If Bj(x), j ≥1 is a boundary function then αBj(x), α ∈R, j ≥1 is also a boundary function.If Bj(x),Bk(x) j,k ≥1 are boundary functions then Bj(x)+Bk(x), j,k ≥1 is also a boundary function.Then the span of all boundary functions and the zero element constitutes a linear space,denoted by B.

In the linear space of B,we can define the inner product by

and then the norm of Bj(x)induced by the inner product is

A P(x)-orthogonal system from the boundary functions{Bj(x), j = 1,2,...,n} was constructed by Liu and Li[11]using the orthogonalized technique:

Furthermore,Liu and Li[11]have suggested that the upper bound theory can be used to estimate the lower-order natural frequency by

where we can estimate the j-th order natural frequencies,j = 1,2,...,n. This technique provides the upper bound≥ωjof the j-th order natural frequency.

4 An R(x)-orthonormal theory

By multiplying both the sides of Eq.(1)by φj(x),integrating it by parts from x = 0 to x = L twice and using the corresponding boundary conditions for each type beam,one can derive

where μ:=ω2and

form a P(x)-orthonormal system,i.e.,

Let

be the vibration mode, and insert it into Eq. (17) by taking j =1,2,...,n,and employing Eq.(19),rendering an eigenvalue problem:

where x =(a1,a2,...,an)Tis an n-vector,and the components b jk of the symmetric coefficient matrix B are given by

Upon solving this eigenvalue problem (21), we can obtain the natural frequency by ω j =where μ j is the j-th eigenvalue of Eq. (21). This method to obtain the natural frequency is named the P(x)-orthonormal method.

On the other hand,multiplying Eq.(1)by ψj(x)and integrating by parts twice it follows that

where λ=1/ω2and ψj(x)is given as follows.

In the linear space of B,we can define another inner product by

and then the norm of Bj(x)induced by the inner product is

Then,

form an R(x)-orthonormal system of the second-order derivatives of ~ψj,i.e.,

Let

be the vibration mode, and insert it into Eq. (23) by taking j =1,2,...,n,and employing Eq.(28),rendering another eigenvalue problem:

where x =(x1,x2,...,xn)Tis an n-vector,and the components a jk of the symmetric coefficient matrix A are given by

Upon solving this eigenvalue problem (30), we can obtain the natural frequency by ω j =where λ j is the j-th eigenvalue of Eq. (30). This theory to obtain the natural frequency is named the R(x)-orthonormal theory.Later,we will point out the merits of this approach to find the natural frequency.We may call Eq.(30)an n-dimensional eigenvalue problem,since A is an n×n symmetric matrix.

5 Special matrix with aij =0 if i+j is odd and natural frequencies

5.1 Example 1

In order to compare the upper bound theory, the P(x)-orthonormal method,as well as the R(x)-orthonormal theory on the natural frequencies, let us consider an N-sine wave composite beam with the interface being given in Eqs. (2)and(5),where f0=0.01 m and N =0.5.The material constants are E1=2×1011N/m2and E2=0.5×1011N/m2.The length of beam is L =1 m,and the width is b=0.1 m.The upper height is h1= 0.06 m, and the lower height is h2=0.04 m.The beam is clamped support at two ends.

For n = 3, we apply the P(x)-orthonormal method to obtain

Similarly,for n = 3 we apply the R(x)-orthonormal theory to obtain

Upon comparing these two matrices, we can find that the matrix B possesses no special structure.In contrast,the matrix A is a special matrix by neglecting the tiny numbers,for example-2.98971×10-22.Then we have

This encourages us to use the R(x)-orthonormal theory to compute the natural frequencies. Denote the matrix in Eq.(33)by

and by the method of cofactor it follows the following characteristic equation:

whose solutions of λ are obvious. In Table 1, we list the first three orders natural frequencies obtained from the R(x)-orthonormal theory and the upper bound theory. We can conclude that the R(x)-orthonormal theory provides a better estimation of the first-order natural frequency.

Table 1 Comparing the first three orders natural frequencies obtained fromthe R(x)-orthonormaltheoryandtheupperboundtheoryforexample 1

As shown in Table 1,ω2and ω3obtained from the R(x)-orthonormal theory are even equal to or larger than that obtained from the upper bound theory. Of course, we must increase n to obtain more accurate solutions of ω2and ω3.

If we directly solve the eigenvalue problem of A given by Eq. (32), we need to solve a cubic equation, of which we can apply the Cardano formulae to find the closed-form solutions. As compared to the natural frequencies listed in Table 1, we find that the absolute difference of ω1is zero,ω2is 1.09×10-11, and ω3is 2.91×10-11, of which the differences are very small.Hence,we can use the specially structured matrix in Eq.(34),rather than Eq.(32),to find the natural frequencies,which can greatly simplify the work.

5.2 Example 2

In order to further confirm the special structure of the coefficient matrix A, let us consider an N-triangular wave composite beam with the interface given in Eqs.(4)and(5),where f0= 0.01 m and N = 1.5.Other material constants are the same to example 1.The beam is simply supported at two ends.

For n = 4, we apply the R(x)-orthonormal theory to obtain

Again, A is a special matrix by neglecting the tiny numbers:

For this matrix we can obtain the eigenvalues by solving the following characteristic equation:

Therefore,we can derive

We list the first four orders natural frequencies obtained from the R(x)-orthonormal theory and the upper bound theoryinTable2.Theimprovementofthefirsttwoordersnatural frequencies from the R(x)-orthonormal theory can be seen.Similarly,we must increase n to obtain more accurate solutions of ω3and ω4.

5.3 Example 3

Let us consider an N-square wave composite beam with the interface being given in Eqs.(3)and(5),where f0=0.01 m and N = 3. Other material constants are the same to example 1.The beam is simply supported at two ends.

For n = 5 when we apply the R(x)-orthonormal theory as before,we can obtain a symmetric matrix,denoted by

For this matrix we have the eigenvalues solved from the following nonlinear equation:

where

Such that we have

where

In Table 3 we list the first five orders natural frequencies obtained from the R(x)-orthonormal theory and the upperbound theory. The improvement of the first three orders natural frequencies by the R(x)-orthonormal theory can be seen.

Table 2 Comparing the first four orders natural frequencies obtained from the R(x)-orthonormal theory and the upper bound theory for example 2

Table 3 Comparing the first five orders natural frequencies obtained from the R(x)-orthonormal theory and the upper bound theory for example 3

By observing Eqs. (34), (36) and (39) the coefficient matrix A := [aij] for different order eigenvalue problem has a special structure with aij=0, i+j is odd.The coefficient matrix generated from n denoted by Anis a sub-matrix of the coefficient matrix generated from n +1 denoted by An+1,i.e.,

where a is a newly generated n-dimensional vector and an+1,n+1＞0 is a newly generated scalar when n is increased to n+1.

6 Performance study

In order to give a clear picture that the R(x)-orthonormal theory is better than the upper bound theory,we consider a triangular interface symmetric composite beam with varying wave number. Other material constants are the same as that in example 1 and f0= 0.01 m. The composite beam is simply supported at two ends. In the interval of N ∈(0,10] we plot the first three orders of natural frequencies in Fig. 2. As mentioned in Ref. [11], the upper bound theory provides a quite accurate estimation of the first-order natural frequency；hence,as shown in Fig.2a the curve obtained by the R(x)-orthonormal theory, where we take n = 5 and use the closed-form formulae in Eq. (42),is slightly smaller than that obtained by the upper bound theory. However, as shown in Fig. 2b, c the improvements on the second-order and third-order natural frequencies are obvious. It is interesting that the two patterns of the natural frequencies obtained by the upper bound theory and the R(x)-orthonormal theory are similar,in spite of their differences.

Fig.2 The first three orders of natural frequencies of the simply supported composite beam with f0 =0.01 m and triangular interfaces

InTables1,2and3,thenaturalfrequenciesobtainedbythe R(x)-orthonormal theory are acceptable if they are smaller than those obtained by the upper bound theory.For example,ω3and ω4obtained by the R(x)-orthonormal theory with n = 4 in Table 2 are not accurate. However, ω1and ω2are accurate.In principle,the first n-2 natural frequencies obtained by the R(x)-orthonormal theory with n are accurate,and the R(x)-orthonormal theory with n+1 leads to the same values. In order to demonstrate this concept, in Table 4 we list the first four orders of natural frequencies obtained fromthe R(x)-orthonormal theory with n =4 and that with n =5 under the same values of parameters in example 2.

Table 4 For example 2,comparing the first four orders of natural frequencies obtained from the R(x)-orthonormal theory with n =4 and n =5

Fig.3 The first three orders of natural frequencies for three interfaces of simply supported composite beam with f0 =0.01 m

For n = 4, ω1and ω2are accurate, but for n = 5, ω1,ω2and ω3are accurate.It can be seen that ω1= 1542.675 obtained from n =5 is slightly smaller than ω1=1542.677 obtained from n = 4, of which the difference can be neglected.Therefore,we can conclude that the first two ω1and ω2have the same values,no matter n =4 or n =5.One reason for this phenomenon is that our R(x)-orthonormal theory reduces the problem of finding the natural frequencies to solve the eigenvalue problem in Eq.(30).By the same token, if we want to obtain accurate natural frequencies of the first three orders,n =5 is enough,and we do not need to take n =6,which leads to the same values of the first three orders of natural frequencies.

Fig.4 The first three orders of natural frequencies for three interfaces of simply supported composite beam with f0 =0.02 m

Now we are ready to apply the R(x)-orthonormal theory with n =5 to investigate the influence of N and f0on the natural frequencies of the first three orders.We first consider the simply supported symmetric composite beam with three different interfaces.In Fig.3,we compute the first three orders of natural frequencies in the range of N ∈(0,10]where the amplitude of the N-wave is f0= 0.01 m. With N ≤1,the natural frequencies of the square wave are constants,because the interface is a horizontal line and the material properties are constants. Before a certain value of N, the natural frequencies of the square wave are larger than that of the sine wave and triangular wave. It is interesting that when the second-order natural frequencies of different interfaces are tending to the same steady value after N ＞3.5,the other orders of natural frequencies are still varying and different. Basically, the curves obtained from larger amplitudes f0= 0.02 m in Fig. 4 and f0= 0.03 m in Fig. 5 have the same patterns with that of f0= 0.01 m, but with larger values of the first three orders of natural frequencies.Larger amplitude leads to larger natural frequency,and with N ＞5 the influences of different interfaces disappear gradually.

Fig.5 The first three orders of natural frequencies for three interfaces of simply supported composite beam with f0 =0.03 m

7 Higher-order natural frequencies

For n =6,the characteristic equation is

As shown in Table 4,ω2and ω4obtained from n =4 and n = 5 are the same. Similarly, we can expect that ω1, ω3and ω5obtained from n = 5 and n = 6 are the same.From Eqs.(39)and(40),we know that ω1,ω3and ω5are solved from the characteristic equation of the following sub-matrix:

Therefore,it is a factor of Eq.(45).Indeed through a lengthy derivation,we can find that

The decomposition of a six-order characteristic equation into the product of two three-order characteristic equations greatly simplifies the solutions of eigenvalues. Hence, we have

where p,q and α were defined by Eq.(43),and the new p1,q1and α1are given by

Table 5 Comparing the natural frequencies obtained from the R(x)-orthonormal theory with n =5,n =6,and n =7

In order to show the improvement by using the results from n = 6 we consider a clamped composite beam with a triangular wave interface, where we take f0= 0.01 m and N = 5. In Table 5, we list the first five orders of natural frequencies obtained from the R(x)-orthonormal theory with n = 5 and that with n = 6. The value of ω2= 9673.345 obtained from n = 6 is slightly smaller than ω2= 9676.815 obtained from n = 5. The improvement is very limited. It can be seen that n = 5 and n = 6 give the same values of ω1, ω3, and ω5, and meanwhile n = 6 gives more accurate value of ω4. In contrast to that for ω2, the improvement of ω4is significant.

When n is increased,finding the closed-form solutions of higher-order natural frequencies becomes tedious.For example,for n =7 we have

While the first characteristic equation gives ω2,ω4,and ω6,thesecondcharacteristicequationgivesω1,ω3,ω5,andω7.In order to obtain ω1,ω3,ω5,and ω7,we face a problem to solve a quartic equation. However, the first and third eigenvalues λ1and λ3are the same with that obtained from n = 6,and thus we can decompose the second eigenvalue problem in Eq.(50)into

where

which are obtained by comparing the coefficients preceding λ3and λ0on both the sides of Eq.(51).

Now instead of solving the quartic equation,we can solve the quadratic equation λ2+ a0λ + b0= 0 to obtain the fifth and seventh eigenvalues λ5and λ7.We list those seven natural frequencies in Table 5, from which we can see that the natural frequency ω5obtained from n = 7 is largely corrected from 55,453.6 obtained from n = 6 to 46,881.9.Continuing this line,we can sequentially find more accurate and more high-order natural frequencies.

As a summary of the results displayed in Tables 1,2,3,4 and 5 we can sketch a special structure of the natural frequencies obtained from the R(x)-orthonormal theory.For n =3,the first-order natural frequency ω1is already very accurate；hence,we start from n = 3 and terminate at n = 7 to view the structure:

The“?”means that the natural frequency obtained from n is equaltothesameordernaturalfrequencyobtainedfromn+1.For example, in n = 3 and n = 4 the natural frequencies ω1and ω3are equal. As compared to that in n = 3, two new natural frequencies ω2and ω4are obtained, where ω2is more accurate than that in n =3.Similarly,as comparing the natural frequencies obtained from n = 4 and n = 5,ω2and ω4are equal, while ω1and ω3are more accurate than that in n =4.When n is increased the first n-2 orders natural frequencies obtained are more accurate.However,the diferences between the first n-3 orders natural frequencies with the previous one are either zero or very small,of which the difference can be neglected. For example, the natural frequencies ω1,ω2,ω3,and ω4obtained from n =7 are the same as that obtained from n =6 as shown in Table 5.

Now we are ready to prove two new theorems；however,before that we give two definitions.

Definition 1A sub-matrix Bois said to be an odd-positioned matrix,if all its elements are taken from the coefficient matrix A at the i j-th entry with both i and j being odd integer.Because i + j is an even integer, the elements of Boare non-zero.

Definition 2Asub-matrix Beissaidtobeaneven-positioned matrix,if all its elements are taken from the coefficient matrix A at the i j-th entry with both i and j being even integer.Because i + j is an even integer, the elements of Beare non-zero.

Theorem 1For the coefficient matrix A = [aij],aij= 0 if

i + j is an odd integer,the characteristic equation of A can be decomposed into two factors:

where m1= m2= n/2 if n is an even integer,m1= (n+1)/2,m2=(n-1)/2 if n is an odd integer,and m1+m2=n.

Proof For n =3,4,5,6,7,we have shown the product equations previously.For larger n,tedious work can complete the proof.

Theorem 1 is significant, which greatly simplifies the works to find the natural frequencies of symmetric composite beam. The number of natural frequencies is either odd or even,like that ω1and ω2.The odd numbered natural frequencies are solved from

while the even numbered natural frequencies are solved from

By using the structure in Eq. (53) we can summarize the following result.

Theorem 2If n is an odd integer,its odd numbered natural frequencies are succeeded by the next ones in the natural frequencies solved from the(n+1)-dimensional eigenvalue problem. Conversely, if n is an even integer, its even numbered natural frequencies are succeeded by the next ones in the natural frequencies solved from the(n+1)-dimensional eigenvalue problem.

Moreover, by using the structure in Eq. (53) and Theorem 2, we can find more high-order natural frequencies in closed-from by employing the results from the previous one.For example,for n =8,

is already solved in Eqs.(51)and(52),and the natural frequencies ω1,ω3,ω5and ω7are the same.To obtain the natural frequency ω6and a new one ω8we can apply a similar technique:

where λ2and λ4are already known from that in n =7,and

In Table 6,we list the natural frequencies obtained from the R(x)-orthonormal theory with n = 7 and n = 8. The results support the above arguments. The sixth natural frequency ω6= 65,948.07 obtained from n = 8 is much improved than that ω6=84,174.92 obtained from n =7.

From Table 6, we can conclude that the first six orders natural frequencies obtained from n = 8 are accurate. By using the closed-form formulae we plot the first six orders of natural frequencies of the clamped symmetric composite beam for a triangular interface with f0= 0.02 m in Fig.6.It can be seen that ω2and ω4tend to steady values fast when N ＞6,and the amplitudes of the variations for other natural frequencies are gradually decreased and then they also tend to steady values after certain value of N.

Table 6 Comparing the natural frequencies obtained from the R(x)-orthonormal theory with n =7 and n =8

Fig.6 The first six orders of natural frequencies of a clamped-clamped composite beam with a triangular interface f0 =0.02 m

8 Vibration modes

When the eigenvalues(natural frequencies)are available,we can find the corresponding eigenvectors x in Eq. (30) and then by Eq. (29) the corresponding vibration modes of the symmetric composite beam.

Let m =n-1 and B be an m×m sub-matrix of A,which is obtained from A by deleting the last column and last row.Let ˉx be the sub-vector of x by deleting the last element xn,and c j := -xnajn, j = 1,2,...,m are the components of an m-vector c. Then we can recast the first m equations of Eq.(30)to

Because the eigenvector is not unique,we can fix it by taking xn=1,and other components in=(x1,x2,...,xm)Tare solved from the given linear equations. We can apply any linear solver to solve Eq. (58), for example, the Gaussian elimination method.Finally,we normalize x to a unit vector.

In Fig. 7, we plot the first three vibration modes for a clamped symmetric composite beam with a triangular interface where f0=0.01 m and N =5.In Fig.8,we plot the first four vibration modes for a simply supported symmetric composite beam with a sine wave interface where f0= 0.02 m and N = 0.5. The vibration modes are either symmetric or skew-symmetric with respect to the middle point of the composite beam.

Fig.7 The first three vibration modes of a clamped-clamped composite beam with a triangular interface f0 =0.01 m and N =5

Fig.8 The first four vibration modes of a simply supported composite beam with a sine wave interface f0 =0.02 m and N =0.5

9 Conclusions

Based on the Rayleigh quotient for the Euler-Bernoulli beam equation we have derived an R(x)-orthonormal system of bases in the linear space of boundary functions.The new theory is asking the second-order derivatives of the bases being orthonormal with respect to the weight function R(x).Using thenew R(x)-orthonormal theorydevelopedinthis paper and taking the advantage of the symmetry of beam and the R(x)-orthonormality, the problem to find the natural frequencies andvibrationmodesistransformedtotheeigenvalueproblem with a special structure of the coefficient matrix: A:=[aij],aij=0 if i + j is odd.The new concepts of odd-positioned sub-matrix and even-positioned sub-matrix of A were introduced.We have proven that the characteristic equation of A can be decomposed into two factors, which are the characteristic equations for the odd-positioned sub-matrix and the even-positioned sub-matrix.Therefore,the order of the nonlinear characteristic equation is reduced to one half. Also,we proved that some of the natural frequencies were succeeded by the next eigenvalue problem when n is increased by one. As a consequence, all the natural frequencies were sequentially solved in closed-form owing to the specialty of A. In each n-dimensional eigenvalue problem, there exist four actions:succeeding,improving,creating and transmitting. For example, for the eigenvalue problem with n = 4,it succeeds the natural frequencies ω1and ω3from n = 3,improves the accuracy of ω2,creates a new natural frequency ω4,and then transmits ω2and ω4to the eigenvalue problem with n =5.

To confirm the improvements made by the closed-form natural frequencies derived from the R(x)-orthonormal theory, we have compared them with that obtained from the upper bound theory.Many examples for the varying N-wave interfaces of symmetric composite beams of three types were carried out to analyze the free vibration performance and investigatethefreevibrationmodes.Thesignificantimprovements of the presented theory for analyzing the vibration behavior of the symmetric composite beams are no-iteration,sequentially closed-form natural frequencies to any order,being applicable to non-smooth/discontinuous symmetric composite beams and highly accurate solutions of the free vibration modes.

AcknowledgementsThe Thousand Talents Plan of China (Grant A1211010)and Fundamental Research Funds for the Central Universities(Grant 2017B05714)for the financial support to the first author are highlyappreciated.Dr.LiuisgratefultothetwelfthGuanghuaEngineering Science and Technology Prize.The work of Botong Li is supported by the Fundamental Research Funds for the Central Universities(Grant FRF-TP-17-020A1).