TThe theoretical analysis of dynamiche+theoretical+analysis+of+dynamic+response+on+cantilever+beam+of+variable+stiffness

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Vol. 12 No. 2Apr. 2014

The theoretical analysis of dynamic

response on cantilever beam of

variable stiffness

Huo BingyongYi WeijianCollege of Civil EngineeringHunan UniversityChangsha 410082China

AbstractThe paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI

varies continuously along the length of the beam. Dynamic equation yields differential equation with variable co-

efficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients

becomes that with constant coefficients by variable substitution. At lastthe study obtains the solution of dy-

namic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant

and that at clamped end is variedthe dynamic characteristics are analyzed under several cases. The results dem-

onstrate that the natural angular frequency reduces as the flexural rigidity reduces. When the rigidity of clamped

end is higher than that of free endlow-level mode contributes the larger displacement response to the total re-

sponse. On the contrarythe contribution of low-level mode is lesser than that of high-level mode.

Key wordsstiffness functiondifferential equation with variable coefficientscantilever beam

1 Introduction

Vibration of a variable stiffness beam is ana-

lyzed in this paper.Vibration of a continuumor called system with

distributed mass and elasticityhas perfect analytical

solutions in some simple structural systemssuch as

uniform beam. The subject has been presented by

many researches [1-3]. The analytical methods are limi-

ted with respect to the complicated systems in struc-

tures [4]. Such cases have been analyzed by discrete

structure of continuum as systems with a finite num-

ber of degree of freedoms (DOFs) [2]. Some authors

have studied a beam with variable cross-sections

using equivalent representations [5] and analog equa-

tions [6]

respectively.In this paperthe Euler-Bernoulli beam is adop-

ted whose stiffness is variable along the length x of

the beam. It is assumed that in the method presented

here the stiffness function isEI(x)=(A+Bx)4. Then the

differential equation with variable coefficients would

be gottenand with variable substitution which is

simplified to become a differential equation with con-

stant coefficients. So the analytical results can be ob-

tained by method of solving linear ordinary differen-

tial equations with constant coefficients. As an exam-plecantilever beam with variable stiffness is ana-

lyzed of which the 5- order natural frequencies and

modal shapes are calculated. The analysis will not on-

ly help to understand vibration characteristics of the

variable stiffness systembut also advance to build

approximate solution to study the more complex situa-

tion.

2 Euler-Bernoulli beam model

Based on the past researchesfree vibration can

be written as follows

[EI(x)y

]

+m(x)y = 0 (1)whereEI(x) is bending stiffnessm(x) is mass per

unit lengthy(xt) is total lateral deflection.

Assuming

y(x,t)=i=1

i(x)qi(t) (i =1,2,3,) (2)

Substituting Eq. (2) into Eq. (1)we obtain

Received14 October2013

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ENGINEERING SCIENCES

q (t) +2q(t)=0 ( )3

[EI(x)(x)] -2m(x)(x)=0 ( )4

where is natural frequency. When

EI(x)=(A+Bx)4 =H(x) (5)

whereAB are undetermined coefficients. Substitu-

ting Eq. (5) into Eq. (4)we obtain

H(4)+2H +H -2m= 0 (6)

To make

(A+Bx)= ezz =ln(A+Bx) (7)

Substituting Eq. (7) into Eq. (6)we obtain

d4

dz4+2d3

dz3-d2

dz2-2d

dz-

2m

B4 = 0 (8)

If we have the solutions of Eq. (3) and Eq. (8)then

the response of variable stiffness vibration can be an-

swered.

3 Solution of the variable stiffness

vibration equation

Writing directly the solution of Eq. (3)

q(t) = sin(t+ ) (9)

whereare undetermined parameters.

To obtain the solution of Eq. (8)in practical

structurethe coefficients AB and m can be known.

Only the natural frequency is an unknown parame-

ter. There are three steps to solve Eq. (8).

1) Assuming the frequency valve of then sub-

stituting into Eq. (8)we will get 4-order the lin-

ear homogeneous equation with constant coefficients.

The characteristic equation is 4-order algebraic equa-

tions. According to the property of algebraic equa-

tionsthe characteristic equation of Eq. (8) has two

real and a couple of conjugate complex number

rootsor four real roots.

2) Solving the characteristic equation. Assuming

the roots of the characteristic equation is the former

caser1r2r3 r4i then the solution of Eq. (8) is

written as follows

(z)=C1er1z

+C2er2z

+ er3z[C3cos(r4z)+C4sin(r4z)] (10)

Substituting for tthe general solution of Eq. (6) is

expressed as follows(x)=C1(A+Bx)

r1 +C2(A+Bx)

r2 +

C3(A+Bx)r3 cos[r4ln(A+Bx)] +

C4(A+Bx)r3 sin[r4ln(A +Bx)]

(11)

whereunknown constants C1C2C3C4 are de-

termined by the boundary conditions of column.

Simple boundary has four equations at two

endstwo at each end of the beam. For examplethe

boundary conditions of cantilever beam can be ex-

pressed as follows

(0)=0(0)=0(l)= 0(l) = 0 (12)

wherelis the length of cantilever beam.

Rewriting Eq. (11) as

(x) =C11(x)+C22(x) +C33(x)+C44(x) (13)

Associating Eq. (12) and Eq. (13)we obtain

1(0) 2(0) 3(0) 4(0)

1(0)

2(0)

3(0)

4(0)

1(l)

2(l)

3(l)

4(l)

1(l)

2(l)

3(l)

4(l)

C1C2C3C4

=0 (14)

Abbreviating Eq. (14) to

C=0 (15)

Since C1C2C3C4 cannot be zeroso

||=0 (16)

3) The Eq. (16) can be satisfied by selecting

then i (i = 123...) can be found. Substituting i

into Eq. (8) to obtain the roots of Eq. (8). ThusEq. (14) is specific algebraic equations. Now remove

directly the forth row of the matrix because of ||=0 .

Application of Cramers Rule provides the values

of C1C2C3 with C4=1. The natural vibration mode

corresponding to i is obtained by substituting

C1C2C3C4 in Eq. (11). Hencedynamic equa-

tion of variable stiffness is solved completely.

4 Cantilever beam as an example

The process above can solve a class of vibration

problem of variable stiffness. As an example dy-

namic characteristics of cantilever beam are analyzed.

Five cases are presented in Table 1. Properties of the

beam are listed asbeam length l=1.5 mmass per

unit length m=112.5 kg/mthe bending stiffnesses of

the free ends are all equal EI= 9.45 106 Nm2the

bending stiffnesses of the clamped ends which are dif-

ferentthe coefficients AB corresponding and the

bending stiffness function (A+Bx)4 are shown in

Table 1. Bending stiffness EI(x) of the working condi-

tions is shown in Fig. 1.

Table 1 Coefficients of the beam at working conditions

Working

conditions

Case 1

Case 2

Case 3

Case 4

Case 5

Bending stiffness of

clamped end

0.9EI

0.95EI

EI

1.05EI

1.1EI

A

54.003 1

54.738 0

55.444 4

56.124 9

56.781 4

B

0.960 9

0.471 0

0

-0.453 6

-0.891 3

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ENGINEERING SCIENCES

In the cases of variable stiffnessthe stiffness

values are constant at free end and variable from large

to small at clamped end. The results show that each-

level natural frequency all declines when stiffness of

clamped end declines. Low- level frequency reduces

more than that of high-level. Meanwhilethe low-level mode has smaller contribution to the displace-

ment when stiffness of clamped end was less than

free end. On the contraryit has larger contribution.

5 Conclusions

An analytical approach has been performed on a

class of vibration problem of variable stiffness. This

method limits the form of stiffness functionEI(x)=(A+

Bx)4. For practical structurethe stiffness of one end

is larger than the other end. This behavior is likely to

find more applications for cantilever beam model.

The stiffness of clamped end is often designed stron-

ger than that of free end by increasing reinforcement

ratio. Howeverthe stiffness of the actual project is

more complex. The method and analysis is not suffi-

cient enough to solve variety service structure.

Through this methodfive cases of variable stiffness

are analyzed. The results show that natural frequency

declines as stiffness of clamped end decreases. If stiff-

ness of clamped end is cut down by just 10 %funda-

mental frequency will drop by 4 %. And modal contri-

bution to the displacement response can yield change.

References[1] Chopra A K. Dynamic of StructuresTheory and Applications to

Earthquake Engineering [M]. 3rd ed. New JerseyPrentice-Hall

2007.

[2] Meirovitch L. Fundamentals of Vibrations [M]. New YorkMc-

Graw-Hill2001.

[3] Tang Y. Numerical evaluation of uniform beam modes [J]. Journal

of Engineering Mechanics ASCE2003129 (12)1475-1477.

[4] Gimena F NGonzaga PGimena L. 3D-curved beam element

with varying cross-sectional area under generalized loads [J]. En-

gineering Structures200830(2)404-411.

[5] Zheng T XJi T J. Equivalent representations of beams with peri-

odically variable cross-sections [J]. Engineering Structures

201133(3)706-719.

[6] Katsikadelis J TTsiatas G C. Non-linear dynamic analysis of

beams with variable stiffness [J]. Journal of Sound and Vibration

2004270 (4/5)847-863.

Author

Huo Bingyongreceived his PhD from Hunan University. His current research is structural vibration tes-

tingsystem identification of structuresnew theoretical formulation and numerical techniques,etc. He can be

reached by [email protected]

Foundation itemNational Natural Science Foundation of China (No. 51178175)

96

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TThe theoretical analysis of dynamiche+theoretical+analysis+of+dynamic+response+on+cantilever+beam+of+variable+stiffness