Nomenclature
u,                                        displacement;
E,                                        Young’s modulus;
I,                                         inertial moment;
ρ,                                        density;
A,                                        area;
p,                                        exciting force;
x,                                        coordinate;
f₁(x),                                  dimensionless bending stiffness;
f₂(x),                                   dimensionless linear density;
L,                                        length of beam;
β,                                        dimensionless frequency;
ω,                                        cycle frequency;
D,                                        Wronski determinant;
c,                                         integral constant;
C,                                        constant;
a,                                        scale factor;
i,                                         imaginary unit;
t,                                         time. 1 Introduction

The vibration of non-uniform beams is a subject of considerable scientific and practical interest that has been studied extensively,and is still receiving attention in literatures. Many papers have been published on transverse vibrations of non-uniform beams due to their relevance to aeronautical,mechanical,and civil engineering. However,analytical solutions are available only for a few particular types of variable cross-sections. These solutions are important since “it is difficult to draw general conclusions about the behavior of a system using only numerical methods”^[1] .

A few exact solutions available in terms of elementary functions have been reported. Gottlieb ^{[2, 3]} presented exact solutions of seven classes of non-uniform beams. It has also been found that the motion of non-uniform beams with exponentially varying width could be transformed to an ordinary differential equation with constant coefficients ^{[4, 5]} . When the crosssectional area and the inertia moment of a non-uniform beam vary together with the position x along the beam as (1 + ax)⁴,the equation of motion could be transformed into the equation of uniform beams^[6] . Lateral vibration of a beam with constant depth and linearly varying breadth was reported by Naguleswaran^{[7, 8]} ,and it was shown that the solution of the mode shape differential equation of the beam consisted of three independent polynomials and a fourth function containing logarithmic terms. Some analytical solutions in terms of special functions including orthogonal polynomials,Bessel functions,hypergeometric series,and power series by the Frobenius method have been reported in literatures. Cranch and Adler^[9] considered the case of non-uniform Euler-Bernoulli beams with rectangular cross-section. They found that the motion could be exactly described in terms of Bessel functions for three special cases^{[10, 11, 12, 13, 14, 15]} . Closed form solutions of Jacobi beams with orthogonal Jacobi polynomials were obtained by Caruntu^[16]. Recently,in terms of hypergeometric functions,Caruntu^[1] studied the rectangular cross-section with parabolic thickness variation and the cantilevers of circular cross-section with parabolic radius variation. Stori and Aboelnaga^[17] studied bending vibrations of a class of rotating beams with hypergeometric function. Transverse vibration of a beam of linearly variable depth with edge crack was analyzed by the Frobenius method^[18] . Naguleswaran^{[19, 20]} studied a class of non-uniform beams with varying width xs in terms of the Frobenius method, where s is positive. Wright^[21] also studied a non-uniform rotating beam with the Frobenius method.

In this paper,a method is proposed to solve the free and steady state transverse vibration of non-uniform beams,which results in a series solution. Its convergence is verified by convergence tests. In Section 2,the governing equations and the proposed method are introduced. Section 3 investigates the uniform beams using the proposed method. Section 4 deals with the case of non-uniform beams. The frequency equation and mode function of free vibration are obtained in Section 5. An example using the proposed method is presented in Section 6. Conclusions are drawn in Section 7. 2 Governing equation and main results

The concerned non-uniform beam is a single span Euler-Bernoulli beam without intermediate support. Young’s modulus E,the inertial moment I,the density ρ,and the cross-sectional area A vary with E(x)I(x) = E₀I₀f₁(x) and ρ(x)A(x) = ρ₀A₀f₂(x). If Young’s modulus E(x) or the density ρ(x) varies with the longitudinal coordinate x,and the inertia moment I(x) and cross-sectional area A(x) are constant,it is the so called axial functionally graded material. If Young’s modulus and the density keep constant,and the cross-sectional area and the inertial moments vary along the longitudinal direction,it is a non-uniform beam. The two cases both could be described by the two functions f₁(x) and f₂(x).

The general governing equation of transverse vibrations reads

where A₀,I₀,ρ₀,and E₀ are the nominal area,the nominal inertia moment,the nominal density, and the nominal Young’s modulus,respectively,u is the lateral displacement,the exciting force p(x,t) imposed on the beam is assumed as harmonic p(x,t) = p(x)e^{± iωt},and the displacement is assumed as u(x,t) = u(x)e^{± iωt}. the governing equation (1) can be rewritten as

where

The functions f₁(x) and f₂(x) are assumed to be positive,continuous,and bounded. is piecewise continuous and bounded. The function p(x) is bounded,

Although these assumptions can be weaken in mathematics,they are consistent with the common case in mechanics. If the exciting force p(x) vanishes,the problem is reduced to the free vibration of non-uniform beams. The solution form of Eq. (2) is assumed as

Substituting Eq. (4) into Eq. (2) and vanishing the coefficients of β yield the recursive equations as follows:

The general solutions of u₀(x) and u_n(x) in Eq. (5) can be expressed as

and

where n = 1,2,3,· · · ,and ξ,c_n0,c_n1,c_n2,and c_n3 are arbitrary free selected constants. For simplicity,vanishing these constants leads to the following equations:

and

Comparing Eqs. (8) and (9) with Eq. (4),one can obtain

where c₀,c₁,c₂,and c₃ are determined by boundary conditions,and U_i(x) (i = 1,2,3,4,5) are shown as

where n = 1,2,3,· · ·,and

The convergence of the functions U_i(x) is discussed in Section 4,and the results are as follows:

(i) The five series U_i(x) (i = 1,2,· · · ,5) are convergent under the given assumptions.

(ii) The four functions U_i(x) (i = 1,2,3,4) are linearly independent. 3 Transverse vibrations of uniform beams

In the case of free vibration of uniform beams,we have

Substituting them into Eqs. (8)-(9) and Eqs. (11)-(12) yields

or

Then,the exact solution reads

where

For the case of steady state vibration of uniform beams,p(x) ≠0,and f₁(x) = f₂(x) = 1, we can get

Then,

Substituting Eqs. (14) and (16) into Eq. (10) yields the exact solution

4 Transverse vibrations of non-uniform beams 4.1 Linear independence of functions U_i (x)

The Wronski determinant of U_i (x) is

where

Then,D_U (0) reads

By use of Liouville’s formula or Abel’s identity,the Wronski determinant D_U (x) can be calculated as

It is easy to obtain the following equation from Eqs. (11) and (12):

Substituting it into Eq. (20) reads

Then,the four functions U_i (x) (i = 1,2,3,4) are linearly independent. 4.2 Convergence of series U_i (x)

Let us consider the convergence of the series. The series is convergent if the series is con- vergent absolutely. From Eq. (11),one can find an evaluation of U_in (x),i.e.,

Substituting U₁₀ (x) = 1 into Eq. (23) and calculating that yield

Then,one obtains an evaluation of the series of U₁ (x),

where

That is to say,U₁ (x) is convergent absolutely. Thus,the series is convergent. For the case of other series,one can analyze it similarly.

Then,

Hence,U₂(x),U₃(x),U₄(x),and U₅(x) are also convergent. 5 Free vibrations of non-uniform beams 5.1 Mode functions and frequency equations

Let the exciting force p(x) vanish in Eq.(10). The free vibration solution reads

where c₀,c₁,c₂,and c₃ are determined by boundary conditions,and U_i(x) (i = 1,2,3,4) are defined in Eqs.(11) and (12). The functions U_i(x) are linear independent and construct the basic set of solutions. In the present study,the ends of the beam are considered to be simply supported,and clamped or free. The boundary conditions associated with both ends being simply supported (SS),both ends being clamped (CC),and the left end being clamped while the right end being free (CF) can be written in the same order as follows:

Case I SS

Considering the SS boundary conditions (26) and the solution form (25),one obtains

where

Substituting the results of (21) and U₂⁽²⁾(0) = 0 into Eq.(29) yields

Then,the corresponding mode function u(x) and the frequency equation are

Case II CC

Considering the clamped boundary conditions (27) and the solution form (25),one obtains

Substituting the results (21) into Eq. (33) yields

Then,the corresponding mode function u(x) and the frequency equation are

Case III CF

Considering the boundary conditions (28) and the solution form (25),one obtains

Substituting the results (21) into Eq. (37) yields

Then,the corresponding mode function u(x) and the frequency eq

5.2 Further discussion

Although the functions U_i (x) (i = 1,2,3,4) are linearly independent and convergent,there are other basic sets of solutions. Considering the general solution expressed by Eqs. (6) and (7) and vanishing the excitation p(x) and the constant ξ yield the the following equations:

and

or

Note that Eqs. (41) and (43) are slightly different with Eqs. (8) and (9),one can obtain another basic set of the solution,

where

The corresponding recursive equations of U_in(x) are

or

From Eq. (45),one can obtain

With these obtained identities and related properties of infinite series,U_i(x) can be linearly expressed by U_i(x) as follows:

Using the definition ,Eq. (47) can be rewritten as

where T_ij can be found in Appendix A,|T_ij | = f₁²(0)D_U(0),and the transformation matrix T_ij is nonsingular if and only if f₁(0) ≠ 0 and D_U(0) ≠ 0,where D_U(0) is the value of the Wronski determinant D_U(x) at x = 0. Note the discussion in Subsection 4.1,the condition D_U(0) ≠ 0 implies the functions U_i(x) are linearly independent.

If Ui0(x) ≠ U_i0(x),one can obtain another basic set with similar process,which can be linearly expressed by the basic set U_i(x),

where the transformation matrix (k_ij) is non-singular and can be found in Appendix A. The concerned terms can be iterated as

or

Obviously, are linearly independent if and only if the matrix (k_ij) is non-singular. In order to ensure the non-singularity of the matrix (k_ij) and the simplicity of the functions , the constants must be carefully selected. 6 Example

In this section,the proposed method is verified by an example. Considering the free vibration of a non-uniform beam,one obtains the functions f₁(x) and f₂(x) in Eq. (2) as follows:

Then, (i = 1,2,3,4) can be calculated based on Eqs. (49) and (50). From Eq. (12),we can obtain Ui0(x) as follows:

The functions are selected as

Using Eq. (14) and in Eq. (53),one can obtain

Using the obtained identities in Eq. (14),the series in Eq. (54) can be simplified as

Then,the exact solution of free lateral vibrations of conical beams can be expressed as

or

where

The solution (57) is the same as the exact solution obtained by Abrate^[6] . 7 Conclusions

An exact solution of free and steady state forced transverse vibrations of non-uniform Euler- Bernoulli beams is obtained,which is a convergent series solution. The frequency equation and mode function of free vibration are obtained under three boundary conditions. Finally,the proposed method is verified by an example. The advantage of the method is that it can deal with arbitrary non-uniform Euler-Bernoulli beams in principle. Another feature is that one can solve both the uniform and non-uniform cases using the same method.