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ISSN : 2093-5145(Print)
ISSN : 2288-0232(Online)
Journal of the Korean Society for Advanced Composite Structures Vol.7 No.4 pp.137-144
DOI : https://doi.org/10.11004/kosacs.2016.7.4.137

# The Influence of Natural Frequency for Cantilevered Composite Materials Beam with Different Cross Section

Bong Koo Han1, Jae Chang Jang2
1Professor, Dept. of Civil Engineering, Seoul National University of Science, Seoul, Korea
2Ph.D Candidate, Dept. of Civil Engineering, Seoul National University of Science, Seoul, Korea
Corresponding author: Han, Bong-Koo, Department of Civil Engineering, Seoul National University of Science, 232 Gongneung-ro, Nowon-gu, Seoul 139-743, Korea +82-2-970-6577, +82-2-948-0043, bkhan@seoultech.ac.kr
September 19, 2016 November 21, 2016 December 22, 2016

## Abstract

Theories of advanced composite structures are too difficult for such field engineers and some simple methods are necessary. In this paper, Simple method of vibration analysis is presented. This method presented in this paper is studied self-weight and other loads. The result of the 2~3 times iteration is good enough for field engineering purposes. In the case of cantilevered composite materials beams with different cross section, increase of mass near the support does not significantly affect the vibration characteristics. As a calculations of the simple method of vibration analysis for cantilevered composite materials beams with different cross section, it is noted that the result of the second cycle at the point of free end (actually 5L/6 span) is only 2.2% away from the ‘exact’ solution.

# 변환단면을 갖는 복합신소재 캔틸레버보의 고유진동수에 대한 영향

한 봉구1, 장 재창2
1서울과학기술대학교 건설시스템공학과 교수
2서울과학기술대학교 건설시스템공학과 박사과정

## 초록

Seoul National University of Science and Technology

## 1.INTRODUCTION

The future of construction materials will depend on if and when the conventional construction materials are replaced by advanced composite materials. The field engineers are quite often faced with the problems of designing high rise towers, such as elevated water tanks, transmission as well as observation towers, landmarks, sea berths, offshore oil drilling platforms, and many other miscellaneous types of structures. Unfortunately, they have no access to rigorous and practical methods to handle the vibration problems of such structures.

Most of the structural engineers encountered solving vibration and buckling problems when he designed. Since both buckling of columns and vibration of beams are, mathematically, eigenvalue problems, similar concepts could be applied for both cases. In bucking problems, the deflection is caused by the axial load.

In the case of vibration analysis, the deflection is caused by the flexural rigidity. Calculation of natural frequency and mode shape of structural members with variable cross section was developed for the tower structures.

Simple method of vibration analysis for calculating the natural frequency for cantilevered composite materials beam with different cross section is presented herein. This method consists of determining the mode shape by means of successive approximations of iteration and of calculating the natural period corresponding to this method. Pretlove reported a method of analysis of beams with attached masses using the concept of effective mass. This method, however, is useful only for certain simple types of beams. Such problems can be easily solved by presented method. This method was found to be very simple in calculation, and was proved to be very accurate and efficient. This method may be extended to stability analysis of complex structural elements and proved to be as good as for the one dimensional case.

The originality of this method is open to question and immaterial. This method was found to be the good and reliable one in vibration analysis of laminate plates about vertical load (Kim et al. 1999, Han & Kim, 2011, Han & Suk, 2011).

Eigenvalue problems are involved in seeking such a solution (Ashton, Pagano, Whitney, 1970, Timoshenko, 1989).

## 2.METHOD OF ANALYSIS

### 2.1.Specially Orthotropic Theory

The equilibrium equation for the specially orthotropic plate is :

$D 1 ∂ 4 w ∂ x 4 + 2 D 3 ∂ 4 w ∂ x 2 ∂ y 2 + D 2 ∂ 4 w ∂ y 4 = q x , y$
(1)

where D1 = D11,D2 = D22,D3 = D12 + 2D66

The assumptions needed for this equation are :

1. The transverse shear deformation is neglected.

2. Specially orthotropic layers are arranged so that no coupling terms exist, i.e. Bij = 0, ()16 = ()26 = 0

3. No temperature or hygrothermal terms exist.

The purpose of this paper is to demonstrate, to the practicing engineers, how to apply this equation to the slab systems made of plate girders and cross beams.

In case of an orthotropic plate with boundary conditions other than Navier or Levy solution type, or with irregular cross section, or with nonuniform mass including point masses, analytical solution is very difficult to obtain. Numerical methods for eigenvalue problems are also very much involved in seeking such a solution. Finite difference method is used in this paper. The resulting linear algebraic equations can be used for any cases with minor modifications at the boundaries, and so on.

The problem of deteriorating infrastructures is very serious all over the world. Before making any decision on repair work, reliable non-destructive evaluation is necessary. One of the dependable methods is to evaluate the in-situ stiffness of the structure by means of obtaining the natural frequency. By comparing the in-situ stiffness with the one obtained at the design stage, the degree of damage can be estimated rather accurately.

The basic concept of the Rayleigh method, the most popular analytical method for vibration analysis of a single degree of freedom system, is the principle of conservation of energy ; the energy in a free vibrating system must remain constant if no damping forces act to absorb it. In case of a beam, which has an infinite number of degree of freedom, it is necessary to assume a shape function in order to reduce the beam to a single degree of freedom system [Clough 1995]. The frequency of vibration can be found by equating the maximum strain energy developed during the motion to the maximum kinetic energy. This method, however, yields the solution either equal to or larger than the real one. Recall that Rayleigh's quotient ≥1 [Kim, 1995]. For a complex beam, assuming a correct shape function is not possible. In such cases, the solution obtained is larger than the real one.

### 2.2.Simple method of vibration analysis

Structural engineers need to calculate the natural frequencies of such element but obtaining exact solution to such problems is very much difficult. This method, however, is useful only for certain simple types of beams. Such problems can be easily solved by presented method.

Simple method of vibration analysis for calculating the natural frequency is presented in this paper. A simple but exact method of calculating the natural frequency corresponding to the first mode of vibration of beam and tower structures with irregular cross sections and attached mass/masses is presented. This method consists of determining the deflected mode shape of the member due to the inertia force under resonance condition. Beginning with initially “guessed” mode shape, “exact” mode shape is obtained by the process similar to iteration. Recently, this method was extended to two dimensional problems including composite laminates, and has been applied to composite plates with various boundary conditions with/without shear deformation effects. This method is used for vibration analysis in this paper.

A natural frequency of structure is the frequency under which the mode shape. If the mode shape is accurate, then the relative deflections will remain unchanged. A natural frequency of a structure is the frequency under which the deflected mode shape corresponding to this frequency begins to diverge under the resonance condition. From the deflection caused by the free vibration, the force required to make this deflection can be found, and from this force, resulting deflection can be obtained. If the mode shape as determined by the series of this process is sufficiently accurate, then the relative deflections (maximum) of both the converged and the previous one should remain unchanged under the inertia force related with this natural frequency. Vibration of a structure is a harmonic motion and the amplitude may contain a part expressed by a trigonometric function.

In this paper, Simple method of vibration analysis is a simple method and can be calculated natural frequencies for the composite materials beam.

Considering only the first mode as a start, the deflection shape of a structural member can be expressed as

$w = W x , y F t = W x , y sin ω t$
(2)

where

W : maximum amplitude

ω : circular frequency of vibration

t : time

By Newton's second law, the dynamic force of the vibrating mass, m, is

$F = m ∂ 2 w ∂ t 2$
(3)

Substituting (2) into this,

(4)

In this expression, ω and W are unknowns. In order to obtain the natural circular frequency ω, the following process is taken.

The magnitudes of the maximum deflection at a certain number of points are arbitrarily given as

$w i , j 1 = W i , j 1$
(5)

where (i,j) denotes the point under consideration. This is absolutely arbitrary but educated guessing is good for accel erating convergence. The dynamic force corresponding to this (maximum) amplitude is

$F i , j 1 = m i , j ω i , j 1 2 w i , j 1$
(6)

The “new” deflection caused by this force is a function of F and can be expressed as

$w i , j 2 = f m k , l ω i , j 1 2 w k , l 1 = Σ k , l Δ i , j , k , l m k , l ω i , j 1 2 w k , l 1$
(7)

where Δ is the deflection influence surface. The relative (maximum) deflections at each point under consideration of a structural member under resonance condition, w(i,j)(1) and w(i,j)(2), have to remain unchanged and the following condition has to be held

$w i , j 1 / w i , j 2 = 1$
(8)

From this equation, w(i,j)(1) at each point of (i,j) can be obtained, but they are not equal in most cases. Since the natural frequency of a structural member has to be equal at all points of the member, i.e., w(i,j)should be equal for all (i,j), this step is repeated until sufficient equal magnitude of w(i,j)is obtained at all (i,j) points.

However, in most cases, the difference between the maximum and the minimum values of w(i,j)obtained by the first cycle of calculation is sufficiently negligible for engineering purposes. The accuracy can be improved by simply taking the average of the maximum and the minimum, or by taking the value of w(i,j)(2) where the deflection is the maximum. For the second cycle, w(i,j)(3) in the absolute numerics of w(i,j)(2) can be used for convenience.

$w i , j 3 = f m i , j ω i , j 2 2 w i , j 2$
(9)

In case of a structural member with irregular section including composite one, and non-uniformly distributed load, regardless of the boundary conditions, it is convenient to consider the member as divided by finite number of elements. The accuracy of the result is proportional to the accuracy of the deflection calculation.

For practical design purposes, it is desirable to simplify the vibration analysis procedure. One of the methods is to neglect the weight of the structural element. The effect of neglecting the weight (thus mass) of the plate is studied as follow. If a weightless plate is acted upon by a concentrated load, P = Nqab, the critical circular frequency of this plate is

$ω n = g δ st$
(10)

where δst is the static deflection.

For a massless simple supported beam loaded at the center

$w = 48 EIg PL 3$
(11)

For a massless cantilevered beam loaded at the free end,

$w = 2 EIg PL 3$
(12)

Replacing P is Nm, N is gradually increased and Eq.(11) and Eq.(12) are calculated for each of N.

Similar result can be obtained by the use of Eqs. (7) and (8).

$w i , j 2 = 1 Δ i , j , i , j ⋅ P i , j g$
(13)

where,

$P i , j = N ⋅ q ⋅ a ⋅ b$
(14)

In case of the plate with more than one concentrated loads,

$ω i , j 2 = 1 Σ k . l Δ i , j , k , l ⋅ P k , l g$
(15)

If we consider the mass of the plate as well as the concentrated loads,

(16)

where (m,n) is the location of the concentrated loads. The effect of neglecting the weight of the plate can be found by simply comparing Eqs. (15) and (16).

Since no reliable analytical method is available for the subject problem, F.D.M. is applied to the governing equation of the special orthotropic plates.

The number of the pivotal points required in the case of the order of error Δ2, where Δ is the mesh size, is five for the central differences of the fourth order single derivative terms. This makes the procedure at the boundaries complicated. In order to solve such problem, the three simultaneous partial differential equations of equilibrium with three dependent variables, w, Mx, and My, are used instead of Eq.(1) for the bending of the specially orthotropic plate.

$D 11 ∂ 2 Mx ∂ x 2 + 4 D 66 ∂ 4 w ∂ x 2 ∂ y 2 + D 22 ∂ 2 My ∂ y 2 = − q x , y + kw x , y$
(17)
$M x = − D 11 ∂ 2 w ∂ x 2 − D 12 ∂ 2 w ∂ y 2$
(18)
$M y = − D 12 ∂ 2 w ∂ x 2 − D 22 ∂ 2 w ∂ y 2$
(19)

If F.D.M. is applied to these equations, the resulting matrix equation is very large in sizes, but the tridiagonal matrix calculation scheme is very efficient to solve such equations. (Kim, 1974, 1993)

In order to confirm the accuracy of the Simple method of vibration analysis, [A/B/A]r type laminate with aspect ratio of a/b=1m/1m=1 is considered. The material properties are :

E1 = 67.36 GPa

E1 = 8.12 GPa

G12 = 3.0217 GPa

ν12 = 0.272

ν21 = 0.0328

The thickness of a ply is 0.005m. As the r increases, B16, B26, D16, and D26 decrease and the equations for special orthotropic plates can be used. For simplicity, it is assumed that A = 0°, B = 90° and r=1. Then D22=18492 N-m.

Since one of the few efficient analytical solutions of the special orthotropic plate is Navier solution, and this is good for the case of the four edges simple supported, Simple method of vibration analysis is used to solve this problem and the result is compared with the Navier solution.

The mesh size is Δx=a/10=0.1m, Δy=b/10=0.1m. The deflection at (x, y), under the uniform load of 100N/ ㎡, the origin of the coordinates being at the corner of the plate, is obtained, and the ratio of the Navier solution to the Simple method of vibration analysis solution is 1.005~1.00028.

For simplify the vibration analysis procedure, in this paper, Simple method of vibration analysis is presented. The Influence of neglect the self-weight for cantilevered composite materials beam loaded at the free end is studied in this paper.

## 3.NUMERICAL EXAMPLES

### 3.1.Simple method of vibration analysis

As a numerical example of the simple method of vibration analysis. A cantilevered composite materials beam is considered. The uniform load of 500 kg/m is treated as five concentrated loads as shown in Fig. 1.

The length of the cantilevered composite materials beam is 10 m. If necessary, the number of points under consideration may be change and the two adjacent points may be made closer, i.e. nonuniform spacing, when the geometry of mode shape requires us to do so. The influence coefficients for cantilevered composite materials beam are shown in Table 1.

The initially maximum amplitude is guessed as follows

W(1)(1) = 5, W(2)(1) = 15, W(3)(1) = 50, W(4)(1) = 80, W(5)(1) = 100

From Eq.(5), Eq.(6) and Eq.(7), the deflection is obtained:

w(1)(2) = 829m(1)[ω(1)(1)]2/EI

w(2)(2) = 6809m(2)[ω(2)(1)]2/EI

w(3)(2) = 16895m(3)[ω(3)(1)]2/EI

w(4)(2) = 29235m(3)[ω(3)(1)]2/EI

w(5)(2) = 45845m(3)[ω(3)(1)]2/EI

Letting w(i)(1) / w(i)(2) = 1, we get

w(1)(1) = 0.0770A(1), w(2)(1) = 0.469A(2)

w(3)(1) = 0.0544A(3), w(4)(1) = 0.052A(4)

w(5)(1) = 0.0468A(5)

After the second step of calculation.

w(1)(2) = 0.0494A(1)

w(2)(2) = 0.0490A(2)

w(3)(2) = 0.0494A(3)

w(3)(2) = 0.0490A(4)

w(5)(2) = 0.0507A(5)

From the exact theory is

w = 0.0496A

The calculation result of second step is good enough for the field engineering. It is noted that the result of the second cycle at the point of free end (actually 5L/6 span), w(5)(2), is only 2.2% difference from the exact theory. In the case of the simply supported composite materials beam is only 0.77% difference from the exact theory.

### 3.2.Influence of different cross section on natural frequencies

This method is study the influence of different cross section on natural frequencies for cantilevered composite materials beam. The calculation results are shown in Fig.2. In the case of cantilevered composite materials beam, increase moment of inertia near the end has a similar influence in Fig. 2.

Increase of moment of inertia near the support influences the natural frequencies quite profoundly in Fig.2 and Fig.3

Replacing P is Nm, N is gradually increased and Eq. (11) and Eq. (12) are calculated for each of N.

It is noted that does not directly indicate the ratio of the weight of the concentrated load to the total weight of a uniform load. For example, N=10 indicates that the ratio is (10-1)/3=3, i.e. the weight of which is three times that of the beam, the critical frequency different between the correct value, obtained by considering the self-weight of the beam, and the approximate one, obtained by neglect of self-weight, is 2.30%. In the case of a fixed beam with similar condition, the difference is 0.68%. In the case of cantilevered composite materials beam loaded at the end (actually 5L/6 span) is only 0.92% difference.

In this paper, the relation between the different cross section and the natural frequency of vibration for some structural elements is presented. This method is a simple but exact method of calculating natural frequencies for cantilevered composite materials beam with different cross sections.

Because of complexity of the geometry or loading condition, no 'exact' theory is available for vibration analysis of most of the practical design problems. Such problems can be solved with relative ease by this method. If the accuracy of the influence coefficients is in question because of the complexity of the geometry and the mass distribution, the number of finite elements to obtain these coefficients can be changed and by Schwarz's inequality the range in which the exact solution lies can be found.

## 4.CONCLUSION

Simple method of vibration analysis for calculating the natural frequency is present in this paper. This method is a simple method and can be calculated natural frequencies of the cantilevered composite materials beam with different cross section. This method presented in this paper is studied self-weight and other loads. The result of the 2~3 times iteration is good enough for engineering purposes. In the case of cantilevered composite materials beams, increase of mass near the support does not significantly affect the vibration characteristics.

The influence of difference cross section for the cantilevered composite materials beam as the amount of attached loads are thoroughly studied.

As a calculations of the simple method of vibration analysis for cantilevered composite materials beam, it is noted that the result of the second cycle at the point of free end (actually 5L/6 span) is only 2.2% away from the ‘exact’ solution.

## ACKNOWLEDGMENT

This study was partially supported by Seoul National University of Science and Technology.

## Figure

cantilevered composite materials beam

Cantilevered composite materials beam with different cross section

Cantilevered composite materials beam with different cross section

## Table

Influence coefficients for cantilevered composite materials beam

## Reference

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