Content uploaded by Agnivo Gosai

Author content

All content in this area was uploaded by Agnivo Gosai on Jun 20, 2016

Content may be subject to copyright.

DETERMINATION OF NATURAL

FREQUENCY OF EULER’S BEAMS

USING ANALYTICAL AND FINITE

ELEMENT METHOD

BACHELOR OF MECHANICAL ENGINEERING

PROJECT SUBMITTED

BY

ARINDAM MUKHERJEE

AGNIVO GOSAI

Under the esteemed guidance of

Dr. S. BHATTACHARYA

READER, JADAVPUR UNIVERSITY

Department of Mechanical Engineering

Jadavpur University, Kolkata-

700032

West Bengal, India

April-2010

2

CONTENTS

TOPIC PAGE

INTRODUCTION 3-12

MATHEMATICAL FORMULATION

(PRISMATIC BEAM)

13-21

RESULTS AND DISCUSSION

(PRISMATIC BEAM)

22-28

MATHEMATICAL FORMULATION

(TAPERED BEAM)

29-35

RESULTS AND DISCUSSION (TAPERED

BEAM)

36-43

CONCLUSION 44

REFERENCES 45-47

APPENDIX 1 48-52

APPENDIX 2 52-74

3

1. INTRODUCTION

A beam is a structural element that is capable of withstanding load primarily by

resisting bending. The use of the cantilever is in fixed-wing aircraft design. Cantilevered beams

are the most ubiquitous structures in the field of micro-electromechanical systems (MEMS).

MEMS cantilevers are commonly fabricated from silicon (Si), silicon nitride (SiN), or polymers.

Without cantilever transducers, atomic force microscopy would not be possible. Cantilevers are

widely found in construction, notably in cantilever bridges and balconies. In cantilever bridges

the cantilevers are usually built as pairs, with each cantilever used to support one end of a central

section. Also simply supported beams are the basic model for bridges all over the world. Beams

of different forms are used in Materials Handling Equipment, Industrial Robotics and Aerospace

Engineering. Also the read/write head of a disk drive (in PCs and laptops) consists of a cantilever

beam. Apart from this, many engineering structures, such as offshore structure piles, oil platform

supports, oil-loading terminals, tower structures and moving arms, can be modeled as beams. In

all these application areas, beams are found in both prismatic forms as well as tapered forms. So

it can be seen that beams as structural elements are widely used in very small as well as large

devices. Hence an analysis of free vibration of prismatic and tapered beams becomes an

important study. The following paragraphs describe the already published research works in the

literature in connection with the analysis of free vibration of beams.

Davis et al. [1] used constant curvature beam finite elements for in-plane vibrations.

The element stiffness and mass matrices are based upon the integration of the exact differential

equations of an infinitesimal element in static equilibrium.. The element which allows shear

deformation and rotary inertia is shown to converge onto frequencies given by a more accurate

finite element analysis providing the correct value of shear coefficient. Prathap and Varadan [2]

analyzed the large amplitude free vibrations of a beam with immovable clamped ends, with the

actual non-linear equilibrium equations and the exact non-linear expression for curvature, and

with no assumption made as to the constancy of axial force, have been determined by a simple

numerically exact successive integration and iterative technique. Rouch and Kao [3] analyzed a

tapered beam finite element for rotor dynamics analysis. The stiffness, mass and gyroscopic

matrices of a rotating beam element were developed and a cubic function being used for the

transverse displacement. Bhashyam and Prathap [4] used Galerkin finite element method for

non-linear beam vibrations. The transverse displacement term alone was used.

4

Sarma and Varadan [5] used Lagrange-type formulation for finite element analysis of

non-linear beam vibrations. A Lagrange-type formulation for finite element analysis of non-

linear vibrations of immovably supported beams was presented. Two equations of motion

coupled in axial and transverse displacements are derived by using Lagrange's equations. Iu et al.

[6] used non-linear vibration analysis of multilayer beams by incremental finite elements. An

incremental variational equation for non-linear motions of multilayer beams composed of n stiff

layers and (n-1) soft cores was derived from the dynamic virtual work equation by an appropriate

integration procedure. To demonstrate its capability, some problems in free non-linear vibrations

of multilayer beams were treated by using the procedure. Mei and Decha [7] used a finite

element method for non-linear forced vibrations of beams. Geometric non-linearities for large

amplitude free and forced vibrations of beam were investigated. Longitudinal displacement and

inertia are included in the formulation. The harmonic force matrix was introduced and derived.

Various out-of-plane and in plane boundary conditions were considered It is concluded that the

effects of longitudinal deformation and inertia are to reduce the non-linearity. Mei [8] discussed

finite element formulations of nonlinear beam vibrations. The Lagrange-type, Galerkin, and Ritz-

type finite element formulations for large amplitude vibrations of immovably supported slender

beams are reexamined. Inconsistency in the definition of frequency or criterion of defining

nonlinearity was discussed, and validity of the frequency solution is examined. Improved finite

element results by including both longitudinal displacement and inertia in the formulation were

presented and compared with available Rayleigh-Ritz continuum solutions.

Heyliger and Reddy [9] used a higher order beam finite element for bending and

vibration problems. The finite element equations for a variationally consistent higher order beam

theory were presented for the static and dynamic behavior of rectangular beams. The higher

order theory correctly accounts for the stress-free conditions on the upper and lower surfaces of

the beam while retaining the parabolic shear strain distribution. The influence of in-plane inertia

and slenderness ratio on the non-linear frequency is examined for beams with a number of

different support conditions. Dumir and Bhaskar [10] showed some erroneous finite element

formulations of non-linear vibrations of beams and plates .The non-linear vibrations of straight

beams and flat plates are governed by differential equations which effectively involve a cubic

non-linearity. This paper brings out the source of errors in some finite element formulations of

these non-linear problems which are based on the introduction of a “linearizing function” in the

5

expression for the strain energy. The magnitude of this error is derived. It was explicitly shown

that the discrepancy between the results of approximate analytical solutions and those of these

finite element solutions is attributable to these errors in the finite element formulations. Khulief

[11] analysed the vibration frequencies of a rotating tapered beam with end mass. The natural

frequencies of vibration of a rotating tapered beam with tip mass were investigated. Explicit

expressions for the finite element mass and stiffness matrices were derived by using a consistent

mass formulation. The beam is assumed to be linearly tapered in two planes. The generalized

eigenvalue problem is defined and numerical solutions are generated for a wide range of

rotational speed and tip mass variations. Both fixed and hinged end conditions were considered.

Noor et al. [12] used mixed finite element models for free vibrations of thin-walled beams.

Simple mixed finite element models were developed for the free vibration analysis of curved

thin-walled beams with arbitrary open cross section. The analytical formulation was based on a

Vlasov's type thin-walled beam theory which includes the effects of flexural-torsional coupling,

and the additional effects of transverse shear deformation and rotary inertia. The fundamental

unknowns consist of seven internal forces and seven generalized displacements of the beam. The

element characteristic arrays were obtained by using a perturbed Lagrangian-mixed variational

principle.

Bangera and Chandrashekhara [13] used nonlinear vibration of moderately thick

laminated beams using finite element method. A finite element model was developed to study

the large-amplitude free vibrations of generally-layered laminated composite beams. The Poisson

effect, which is often neglected, was included in the laminated beam constitutive equation. The

large deformation was accounted for by using von Karman strains and the transverse shear

deformation was incorporated using a higher order theory. The beam element had eight degrees

of freedom with the in-plane displacement, transverse displacement, bending slope and bending

rotation as the variables at each node. Leung and Mao [14] used symplectic integration of an

accurate beam finite element in non-linear vibration. They gave two contributions in this paper:

(1) to form beam finite element matrices within the large deflection and small rotation

assumptions and (2) to integrate the resulting equations by symplectic schemes, The inherent

approximation was introduced by the assumed shape functions only in our finite element

formulation. The induced axial force was not averaged and the stiffness is defined by the first,

second- and third-order matrices. Both free and forced vibrations and damped and undamped

6

vibrations were studied. Dube and. Dumb [15] used tapered thin open section beams on elastic

foundation. The work presents exact solutions for the coupled flexural-torsional vibration of

tapered beams with a thin-walled open section resting on an elastic foundation. A solution was

also obtained by the finite element method using tapered elements with cubic shape functions.

The results for the first five natural frequencies were presented for square, channel and circular

open sections for various boundary conditions at the ends. Babert et al. [16] used a finite element

model for harmonically excited visco-elastic sandwich beams. A finite element model was

presented for the harmonic response of sandwich beams with thin or moderately thick visco-

elastic cores. Nonlinear variation of displacements through the thickness of the core was

assumed. A simple approximation for the response of the core allows all core variables to be

expressed in terms of the face plate displacements. The model used standard beam shape

functions to construct twelve degrees-of-freedom element.

Chaudhari and Maiti [17] made a study of vibration of geometrically segmented

beams with and without crack. In the paper, a method of modeling for transverse vibrations of a

geometrically segmented slender beam, with and without a crack normal to its axis, had been

proposed using Frobenius technique. There are segments of a linearly variable depth. The

thickness is uniform along the whole length. Hua et al. [18] used vibration analysis of

delaminated composite beams and plates using a higher-order finite element. In order to analyze

the vibration response of delaminated composite plates of moderate thickness, a FEM model

based on a simple higher-order plate theory, which can satisfy the zero transverse shear strain

condition on the top and bottom surfaces of plates, had been proposed in the paper. The

influences of delamination on the vibration characteristic of composite laminates had been

investigated. Mazzilli et al. [19] used non-linear normal modes of a simply supported beam for

continuous system and finite-element models. Non-linear normal modes of vibration for a

hinged–hinged beam with fixed ends were evaluated considering both the continuous system and

finite-element models. With regard to the latter ones, two alternative approaches are used namely

the invariant manifold technique and the method of multiple scales. Yaman [20] used finite

element vibration analysis of a partially covered cantilever beam with concentrated tip mass. The

work presented in the paper is the theoretical investigation of the dynamical behavior of a

cantilever beam, partially covered by damping and constraining layers, with concentrated mass at

the free end. A finite element method is used in order to obtain the resonant frequencies and loss

7

factors. The resonant frequencies and loss factors for different physical and geometrical

parameters were determined. The variations of these two parameters are found to be strongly

dependent on the geometrical and physical properties of the constraining layers and the mass

ratio.

Ribeiro [21] used non-linear forced vibrations of thin/thick beams and plates by the

finite element and shooting methods. The shooting, Newton and p-version, hierarchical finite

element methods were applied to study geometrically nonlinear periodic vibrations of elastic and

isotropic, beams and plates. Thin and thick or first-order shear deformation theories are followed.

One of the main goals of the work presented is to demonstrate that the methods suggested are

highly adequate to analyse the periodic, forced non-linear dynamics of beam and plate structures.

An additional purpose is to investigate the differences in the predictions of non-linear motions

when thin and thick, either beam or plate theories are followed. Subramanian [22] used dynamic

analysis of laminated composite beams using higher order theories and finite elements. Free

vibration analysis of laminated composite beams was carried out using two higher order

displacement based shear deformation theories and finite elements based on the theories. Both

theories assume a quintic and quartic variation of in-plane and transverse displacements in the

thickness coordinates of the beams respectively and satisfy the zero transverse shear strain/stress

conditions at the top and bottom surfaces of the beams. The difference between the two theories

is that the first theory assumes a non-parabolic variation of transverse shear stress across the

thickness of the beams whereas the second theory assumes a parabolic variation. The equations

of motion are derived using Hamilton’s principle. Fonseca and Ribeiro [23] used a Beam p-

version finite element for geometrically non-linear vibrations in space .A beam p-version,

hierarchical finite element for geometrically non-linear vibrations in space was presented.

Stresses were obtained by adding the effects of traction, torsion and bending. A simplified

version of Green’s strain tensor and the generalized Hooke’s law were used. The mass and

stiffness matrices are obtained by the principle of the virtual work, and the ensuing non-linear

equations of motion are solved by Newmark’s method. Ganesan and Zabihollah [24] carried out

vibration analysis of tapered composite beams using a higher-order finite element. The objective

of the present work was to conduct an investigation of the undamped free vibration response of

such tapered composite beams. A higher-order finite element formulation had been developed

for vibration analysis of tapered composite beams based on classical laminated plate theory.

8

Banerjee et al. [25] made free vibration of rotating tapered beams using the dynamic

stiffness method .The free bending vibration of rotating tapered beams was investigated by using

the dynamic stiffness method. First, the governing differential equation of motion of the rotating

tapered beam in free flap bending vibration was derived for the most general case using

Hamilton’s principle, allowing for the effects of centrifugal stiffening, an arbitrary outboard

force and the hub radius term. Malekzadeh and Karami [26] mixed differential quadrature and

finite element free vibration and buckling analysis of thick beams on two-parameter elastic

foundations. As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE)

method for boundary value structural problems in the context of free vibration and buckling

analysis of thick beams supported on two-parameter elastic foundations was presented. The

formulations were based on the two-dimensional theory of elasticity. The presented formulations

provide an effective analysis tool for beams free of shear locking. Comparisons were made with

results from elasticity solutions as well as higher-order beam theory. Ece et al. [27] used

vibration of a variable cross-section beam. Vibration of an isotropic beam which has a variable

cross-section was investigated. Governing equation was reduced to an ordinary differential

equation in spatial coordinate for a family of cross-section geometries with exponentially varying

width. Analytical solutions of the vibration of the beam were obtained for three different types of

boundary conditions associated with simply supported, clamped and free ends. Natural

frequencies and mode shapes were determined for each set of boundary conditions.

Trindade and Benjeddou [28] used a refined sandwich model for the vibration of

beams with embedded shear piezoelectric actuators and sensors. The work extends a previously

presented refined sandwich beam finite element (FE) model to vibration analysis, including

dynamic piezoelectric actuation and sensing. The mechanical model is a refinement of the

classical sandwich theory (CST), for which the core was modeled with a third-order shear

deformation theory (TSDT). The FE model is developed considering, through the beam length,

electrically: constant voltage for piezoelectric layers and quadratic third-order variable of the

electric potential in the core, while mechanically: linear axial displacement, quadratic bending

rotation of the core and cubic transverse displacement of the sandwich beam. Chen and Hsiao

[29] studied quadruply coupled linear free vibrations of thin-walled beams with a generic open

section. The coupled vibration of thin-walled beams with a generic open section induced by the

boundary conditions was investigated using the finite element method. If the axial displacement

9

of the pin end is restrained at another point rather than the centroid of the asymmetric cross

section, the axial vibration, two bending vibrations, and torsional vibration may be all coupled.

The element developed here has two nodes with seven degrees of freedom per node. The shear

center axis is chosen to be the reference axis and the element nodes are chosen to be located at

the shear centers of the end cross sections of the beam element. Different sets of element nodal

degrees of freedom corresponding to different pin ends are considered here. Alonso and Ribeiro

[30] used a flexural and torsional non-linear free vibrations of beams using a p-version finite

element. A p-version beam finite element with hierarchic basis functions and which may

experience longitudinal, torsional and bending deformations in any plane was employed to

investigate the geometrically non-linear vibrations of beams. Clamped–clamped, isotropic and

elastic beams of circular cross section were analyzed. The geometrical non-linearity was taken

into account by considering a simplified version of Green’s strain tensor. The harmonic balance

method was employed to map the equations of motion to the frequency domain and the resulting

algebraic non-linear system of equations was solved by a continuation method. Assuming a

Fourier series where the constant term and the first three harmonics are considered it was

concluded that internal resonances appear both in bending and torsion.

Piovana et al. [31] made exact solutions for coupled free vibrations of tapered shear-

flexible thin-walled composite beams. A parametric analysis for different taper ratios,

slenderness ratios and stacking sequences was performed. Shavezipur and Hashemi [32] used

free vibration of triply coupled centrifugally stiffened nonuniform beams, using a refined

dynamic finite element method. The application of a Refined Dynamic Finite Element (RDFE)

technique to triply coupled vibration of centrifugally stiffened beams was presented. The

proposed method is a fusion of the Galerkin weighted residual formulation and the Dynamic

Stiffness Matrix (DSM) method, where the basis functions of approximation space were assumed

to be the closed form solutions of the differential equations governing uncoupled bending and

torsional vibrations of the beam. The use of resulting dynamic trigonometric interpolation

(shape) functions leads to a frequency dependent stiffness matrix, representing both mass and

stiffness properties of the beam element. Assembly of the element matrices and the application

of the boundary conditions then leads to a frequency dependent nonlinear eigen problem. The

Wittrick–Williams algorithm was used as a solution technique to compute the natural frequencies

10

and modes of five illustrative example beam configurations, exhibiting doubly and triply coupled

vibrations.

Gupta et al. [33] used a relatively simple finite element formulation for the large

amplitude free vibrations of uniform beams. Large amplitude free vibration analysis of uniform,

slender and isotropic beams is investigated through a relatively simple finite element

formulation, applicable to homogenous cubic nonlinear temporal equation in homogenous

Duffing equation. The finite element formulation begins with the assumption of the simple

harmonic motion and was subsequently corrected using the harmonic balance method and is

general for the type of the nonlinearity mentioned earlier. The nonlinear stiffness matrix derived

in the finite element formulation leads to symmetric stiffness matrix as compared to other recent

formulations. Empirical formulas for the nonlinear to linear radian frequency ratios, for the

boundary conditions considered, were presented using the least square fit from the solutions of

the same obtained for various central amplitude ratios. Ramtekkar [34] used free vibration

analysis of delaminated beams using mixed finite element model. Free vibration analysis of

laminated beams with delamination had been presented. A 2-D plane stress mixed finite element

model developed by the authors had been employed. Two models, namely the unconstrained-

interface model and the contact-interface model had been proposed for the computation of

frequencies and the of delaminated beams with mid-plane delamination as well as off-mid-plane

delamination had been considered. It has been concluded that the contact-interface model

presents a realistic behavior of the dynamics of delaminated beams whereas the unconstrained-

interface model under-predicts the frequencies, particularly at the higher modes.

Das et al. [35] did out-of-plane free vibration analysis of rotating tapered beams in

post-elastic regime. Free vibration dynamic behavior of rotating tapered beams in elastic and

post-elastic regimes was presented in the paper. The entire analysis is carried out in two parts.

First the analysis of the rotating beam under static centrifugal loading is performed and then it is

followed by the dynamic analysis using the solution parameters of the static analysis. The

governing equations were obtained by the application of suitable variational principles. The

displacement fields are assumed using the linear combinations of admissible orthogonal

functions which are generated numerically using Gram–Schmidt schemes. Elastic and post-

elastic dynamic behavior of rotating tapered beams was presented through suitable normalized

parameters of the beam geometries and rotational speeds.

11

Vidal and Polit [36] used vibration of multilayered beams using sinus finite elements

with transverse normal stress. A family of sinus models was presented for the analysis of

laminated beams in the framework of free vibration. A three-node finite element is developed

with a sinus distribution with layer refinement. The transverse shear strain was obtained by using

a cosine function avoiding the use of shear correction factors. This kinematic accounts for the

interlaminar continuity conditions on the interfaces between the layers, and the boundary

conditions on the upper and lower surfaces of the beam. A conforming FE approach was carried

out using Lagrange and Hermite interpolations Zhu and Leung [37] used linear and nonlinear

vibration of non-uniform beams on two-parameter foundations using p-elements for a non-

uniform beam resting on a two-parameter foundation. Legendre orthogonal polynomials were

used as enriching shape functions to avoid the shear-locking problem. With the enriching degrees

of freedom, the accuracy of the computed results and the computational efficiency were greatly

improved. The arc-length iterative method was used to solve the nonlinear eigen value equation.

The computed results of linear and nonlinear vibration analyses show that the convergence of the

proposed element was very fast with respect to the number of Legendre orthogonal polynomials

used. Since the elastic foundation and the axial load applied at both ends of the beam affect the

ratios of linear frequencies associated with the internal resonance, they influence the nonlinear

vibration characteristics of the beam.

In the present work, analytical method has been used to derive the natural frequencies

of prismatic beams subjected to various boundary conditions and subsequently arrive at the mode

shapes for the corresponding natural frequencies. The analysis is done in MATLAB and an

algorithm has been derived to solve the frequency equations resulting from the solution to the

governing differential equations of beams having different boundary conditions. The mode shape

equations for the corresponding configurations are also formulated and utilizing these, the mode

shapes are plotted. The non dimensional natural frequencies and natural frequencies in Hz and

radian per second for various beam geometries can be calculated using the MATLAB code. The

governing equation of tapered beam is formulated and it is seen that there is considerable

difficulty to solve it by well known analytical methods. Hence numerical method (FEM) is used

to solve it.

12

To determine the natural frequency of a tapered beam (tapered in both dimensions,

breadth and thickness), the problem is solved using finite element analysis .The beam of length L

is broken into a finite number of elements, n (here n=50) , and then assembly is done by

programming. After the assembly is done the boundary conditions are applied and the resultant

stiffness and mass matrix are determined. From there simultaneous iteration concept is applied to

determine six eigen vectors and corresponding eigen values. From this eigen values six natural

frequencies are determined. In this project work four type of boundary conditions are discussed

clamped-free, clamped-clamped, clamped-simply supported, simply supported and simply

supported. For the above calculations MATLAB is used.

13

2. MATHEMATICAL FORMULATION (PRISMATIC BEAM)

In this section the mathematical formulation for the entire work has been detailed out.

2.1 GOVERNING DIFFERENTIAL EQUATION

Euler-Bernoulli beam theory (also known as Engineer's beam theory, Classical beam

theory or just beam theory) is a simplification of the linear theory of elasticity which provides a

means of calculating the load-carrying and deflection characteristics of beams. It was first

enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel

Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations,

it quickly became a cornerstone of engineering and an enabler of the Second Industrial

Revolution. Additional analysis tools have been developed such as plate theory and finite

element analysis, but the simplicity of beam theory makes it an important tool in the sciences,

especially structural and mechanical engineering.

To determine the differential equation for the lateral vibration of beams, consider the

forces and moments acting on an element of the beam shown in the given figure:-

Figure 1

Here, V and M are shear and bending moments, respectively, and p(x) represents the loading per

unit length of the beam.

By summing forces in the y-direction

(1)

By summing moments about any point on the right face of the element,

14

(2)

In the limiting process these equations result in the following important relationships

= V

The first part of the above equations states that the rate of change of shear along the length of the

beam is equal to the loading per unit length and the second states that the rate of change of

moment along the beam is equal to the shear.

Hence we obtain the following,

= = (3)

The bending moment is related to the curvature by the flexural equation which for the

coordinates indicated in the figure is

Substituting this relation into (3) we obtain

For a beam vibrating about its static equilibrium position under its own weight, the load per unit

length is equal to the inertia load due to its mass and acceleration. Since the inertia force is in the

same direction as p(x), as shown in the figure we have by assuming harmonic motion

y

where ρ is the mass per unit length of the beam. Using this relation, the equation for the lateral

vibration of the beam reduces to

15

In the special case where the flexural rigidity EI is a constant as in the case of a prismatic beam

the above equation may be written as

On substituting

we obtain the fourth order differential equation

(4)

for the vibration of a uniform beam.

2.2 SOLUTION OF THE GOVERNING DIFFERENTIAL EQUATION

The general solution of equation (4) can be shown to be

(5)

To arrive at this result, we assume a solution of the form

which will satisfy the differential equation when

Since

the solution in the form of equation (5) is readily established.

The natural frequencies of vibration are found to be

16

ω

n

= β

n2

or

ω

n

= (β

n

l)

2

where the number β

n

depends on the boundary conditions of the problem.

2.3 BOUNDARY CONDITIONS

The beam equation contains a fourth-order derivative in x, hence it mandates at most

four conditions, normally boundary conditions. The boundary conditions usually model supports,

but they can also model point loads, moments, or other effects.

An example is a cantilever beam: a beam that is completely fixed at one end and completely free

at the other. "Completely fixed" means that at the left end both deflection and slope are zero;

"completely free" implies (though it may or may not be obvious) that at the free end, both shear

force and bending moment are zero. Taking the x coordinate of the left end as 0 and the right end

as L (the length of the beam), these statements translate to the following set of boundary

conditions (assume EI is a constant):

Figure 2 (a cantilevered beam)

It is common practice in structural engineering to replace the displacement y by u.

17

Depending on various boundary conditions the first few natural frequencies and the normalized

mode shapes for the following beam configurations were found out:

1> clamped free

2> clamped-clamped

3> clamped-simply supported

4> simply supported-simply supported

BEAM CONFIGURATION B.C. at x = 0 B.C. at x=l

Clamped-free

Clamped-clamped

Clamped-simply supported

Simply supported-simply

supported

18

2.4 SOLUTIONS

For clamped-free beam:

B.C.: At

At

Substituting the BC’s into the general solution we have

(y)

x=0

= A + C = 0 A = C

(

x=0

=

x=0

= 0

(

x=l

=

---(i)

x=l

=

---(ii)

From (i) & (ii) we get,

This is the frequency equation. Now we have to find out different values of for which the

equation is satisfied. Then for this different values of we will get different values of natural

frequency.

19

For clamped-clamped beam:

B.C.: At

At

Substituting the BC’s into the general solution we have

(y)

x=0

= A + C = 0 A = C

(

x=0

=

x=0

= 0

x=l

=

---(i)

x=l

=

---(ii)

From (i) & (ii) we get,

This is the required frequency equation for the concerned beam configuration.

20

For clamped-simply supported:

B.C.: At

At

Substituting the BC’s into the general solution we have

(y)

x=0

= A + C = 0 A = C

(

x=0

=

x=0

= 0

x=l

=

---(i)

(

x=l

=

---(ii)

From (i) & (ii) we get,

This is the required frequency equation for the concerned beam configuration.

21

For simply supported – simply supported:

B.C.: At

At

(y)

x=0

= A + C = 0 A = C

(

x=0

=

Hence A = C =0

x=l

= ---(i)

and

(

x=l

= ---(ii)

As A= C= O from (i) and (ii) we have,

Therefore we get,

This is the required frequency equation for the concerned beam configuration.

22

3. RESULTS AND DISCUSSION (PRISMATIC)

The mathematical formulations are done and then the resulting equations are solved

in MATLAB.A program is coded which would take variable beam data as input and display non

dimensional natural frequencies, natural frequencies in Hz and radian per second and also plot

the normalized mode shapes. The program is enclosed in this report in Appendix 1.

The trigonometric frequency equations are derived analytically and it was decided

to develop an algorithm to solve the frequency equations based on a trial and error method. The

objective was to find out the roots of the equation numerically with the help of the programming

environment offered in MATLAB.

By plotting the frequency equation, a rough estimate of the lower and upper bounds

for an interval containing the first few roots i.e. the natural frequencies can be found out. Then an

incremental search technique can be formulated to find out the roots.

To solve the frequency equations the algorithm is derived based on the change of

sign of a function in the vicinity of a possible root. The MATLAB code is then fed with different

frequency functions to get the non dimensional frequencies corresponding to the relevant beam

geometry. Then depending on the data supplied for beam thickness, breadth, length and material

the natural frequencies in Hz and radian per second are calculated. Then the non dimensional

frequencies are utilized in the mode shape equations to get the mode shape plots.

For generating normalized mode shapes all amplitudes are divided by the maximum

amplitude in a particular case. This is done by choosing the maximum amplitude irrespective of

sign and then dividing all the amplitudes by the same.

The problem with incremental search technique is that if the incremental length is

large and the function is tangential to the x-axis, closely spaced roots can be missed. For the case

of the clamped-clamped beam, the algorithm fails to find out the 5

th

root i.e. the natural

frequency

23

TABLE 1: THE FIRST FIVE NON DIMENSIONAL NATURAL FREQUENCIES FOR

PRISMATIC BEAMS

BEAM

CONFIGURA

TION

FREQUENCY

EQUATION

1

ST

FREQUE

NCY

2

ND

FREQUEN

CY

3

RD

FREQUENC

Y

4

TH

FREQUENC

Y

5

TH

FREQUEN

CY

CLAMPED-

FREE

coshβlcosβl + 1 =

0

1.8753 4.6941 7.8548 10.9955 14.1372

CLAMPED-

CLAMPED

coshβlcosβl - 1 =

0

4.7301 7.8532 10.9956 14.1372

CLAMPED-

SIMPLY

SUPPORTED

sinβl = 0 1.5710 4.7130 7.8540 10.9960 14.1380

SIMPLY

SUPPORTED-

SIMPLY

SUPPORTED

cosβl = 0 3.1420 6.2840 9.4250 12.5670 15.7080

TABLE 2: COMPARISON OF NON DIMENSIONAL NATURAL FREQUENCY

RATIOS FOR VARIOUS BEAM CONFIGURATIONS

Clamped Free Clamped-

Clamped

Clamped-

Simply

supported

Simply

supported -

Simply

supported

1 1 1 1

2.503 1.660 3 1.988

4.188 2.325 4.999 2.999

5.863 2.988 6.999 3.999

7.538 8.999 4.999

24

The analysis for a beam made of mild steel is done. The modulus of elasticity E is 210 GPa and

the material density is 7850 kg/m

3

.The length of the beam is taken as 1 m. The breadth is taken

as 5 cm and the thickness is taken as 3 cm.

TABLE 3: CLAMPED-FREE BEAM

Modes Frequency in rad/sec Frequency in Hz

1 157.5 25.1

2 987 157.1

3 2763.6 439.8

4 5415.5 861.9

5 8952.2 1424.8

TABLE 4: CLAMPED-CLAMPED BEAM

Modes Frequency in rad/sec Frequency in Hz

1 1002.2 159.5

2 2762.5 439.7

3 5415.6 861.9

4 8952.2 1424.8

5

TABLE 5: CLAMPED-SIMPLY SUPPORTED BEAM

Modes Frequency in rad/sec Frequency in Hz

1 110.5 17.6

2 994.9 158.4

3 2763.0 439.8

4 5416.0 862.0

5 8953.3 1425.0

TABLE 6: SIMPLY SUPPORTED-SIMPLY SUPPORTED BEAM

Modes Frequency in rad/sec Frequency in Hz

1 442 70.4

2 1769 281.5

3 3979 633.3

4 7074 1125.9

5 11052 1759.0

25

A separate analysis for the beam made of same material is done with different beam geometry.

The length is fixed at 1 m. But the breadth is changed to 3 cm and the thickness is made 2 cm.

TABLE 7: CLAMPED-FREE BEAM

Modes Frequency in rad/sec Frequency in Hz

1 105.0 16.7146

2 658.0 104.7227

3 1842.4 293.2243

4 3610.3 574.6025

5 5968.1 949.8591

TABLE 8: CLAMPED-CLAMPED BEAM

Modes Frequency in rad/sec Frequency in Hz

1 668.1 106.3328

2 1841.7 293.1084

3 3610.4 574.6095

4 5968.1 949.8587

5

TABLE 9: CLAMPED-SIMPLY SUPPORTED BEAM

Modes Frequency in rad/sec Frequency in Hz

1 73.7 11.7297

2 663.3 105.5672

3 1842.0 293.1677

4 3610.6 574.6505

5 5968.8 949.9708

TABLE 10: SIMPLY SUPPORTED-SIMPLY SUPPORTED BEAM

Modes Frequency in rad/sec Frequency in Hz

1 294.8 46.9

2 1179.2 187.7

3 2652.6 422.2

4 4716.0 750.6

5 7368.1 1172.7

26

TABLE 11: THE MODE SHAPE EQUATIONS FOR VARIOUS BEAM

CONFIGURATIONS ARE TABULATED

BEAM CONFIGURATION MODE SHAPE EQUATIONS

clamped free Y=(coshβx-cosβx)-( )*(sinhβx-sinβx)

clamped- clamped Y=(sinβx-sinhβx)-( )*(cosβx-coshβx)

clamped -simply supported Y=(coshβx-cosβx)-( )*(sinhβx-sinβx)

simply supported-simply supported Y=sinβx

N.B. The coefficients from the general solution were suitably adjusted to give the normalized

mode shape plots.

27

Figure 3: Cantilever-Free beam

Figure 4: Cantilever-Cantilever beam

28

Figure 5: Cantilever-Simply supported

Figure 6: Simply supported-Simply supported beam

29

4.

MATHEMATICAL FORMULATION (TAPERED BEAM)

A linear tapered beam has been considered here, for determining the natural frequency

using finite element method (fem) the following steps that are taken. (1) Discretization of the

continuum. (2) Selection of interpolation functions. (3) Finding element matrices. (4) Assembly

of element matrices to obtain the system equations. (5) Solving the system equations. Here the

beam of length L is broken into a finite number of elements, n (here n=50).

Figure 7. Tapered Beam

Here ‘h-refinement’ meshing is done. From there for each two node beam element the

shape functions are determined. For each element of length (l), for calculating the stiffness and

mass matrix , the element is considered to be of constant cross section with dimensions having

mean at the ends as shown in figure 2.

Figure 8. Element discretized

30

4.1

DERIVATION OF THE GOVERNING EQUATION

Now considering a 2-D beam with uniform load (q), if we consider our attention on a

piece of the beam then it will look like this.

Figure 9. Element formulation

Where V is the shear force acting on it and M is the bending moment it is experiencing.

So in this element we can write that the total force in y direction it is having is

(1)

from this equation we can deduce

as we know

Thus we can write

Substituting the above deductions in equation (1), we can write

(2)

31

4.2 SOLUTION OF THE EQUATION

Considering the part of the beam as a two node 2-D beam where v’s represent the transverse

displacement and Ѳ the corresponding slope angle, we can write the displacement of any point of

the beam as

(3)

Figure 10. A two node 2-D Beam

Where N’s are the corresponding shape functions,

Then application of weighted residuals method in which the weight functions when multiplied

with the residual value of an approximate solution and is then integrated over the domain yields

zero. In Galerkin’s method the weight functions are the corresponding shape functions. Using

this method in equation (2)

32

i=1,4.

(4)

Evaluating for the first term of the above equation we get,

=ρA = [ (5)

So [ = ρA is the consistent mass matrix of one element.

Similarly the second term

placing equation 3

we get

(6)

which is referred as the stiffness matrix. Here the nodal force and moments are taken as zero, as

there are no external loadings, q=0.

Thus we can write

(7)

where

For this equation as is function of both time and place we can replace it with

Where is the amplitude of vibration changing with time (t), assuming the system vibrates

harmonically under inertia forces.

Substituting this value in equation (7) we get,

33

(8)

After determining the assembled stiffness and mass matrices, boundary conditions are applied

and modified stiffness and mass matrices are generated.

4.3 ASSEMBLY OF ELEMENTS

For “n” number of elements two (2n+2)x(2n+2) null global matrices of stiffness(K) and

mass(M) of zero values are generated .

Then the tapered angles are calculated as

in breadth and,

in thickness direction

.

Then for the first stiffness (4x4) and mass (4x4) matrix for the first element of length l

(l=L/n), are

Mean breadth for the element,

where

Mean thickness for the element

where

This stiffness and mass matrix are added to the leftmost top part of the corresponding

global stiffness(K) and mass(M) matrix.

In the next iteration,

B=b1 and T=t1, and b1 and t1 for the second element values are calculated accordingly.

34

As the values of b and t are changed, and so the values of stiffness and mass matrices are

also changed. The new mass and stiffness matrices are added to the global one by shifting two

values in both row and column as shown.

Figure 11. Process of Assembly

After the assembled stiffness and mass matrices are determined, boundary conditions are

applied to get the working stiffness [K] and mass [M] matrix. To get the natural frequencies and

eigen vectors simultaneous iteration concept is utilized.

4.4 SIMULTANEOUS ITERATION

In an iteration process if any normal vector is multiplied to the respective matrix, it

eventually gives the eigen vector of the matrix, thus from it we can get the eigen values and thus

natural frequencies.Here, to apply simultaneous iteration concept we first have to triangularise

the stiffness matrix(K) such that

In the next step we have to take a identity matrix U such that

[U]=[1 0 0 0 ….0;0 1 0 0..0;0 0 1 0 0…..: ;….;….0 0 0 1].

now considering another vector [X] such that

then making [Y] such that

then, multiplying Y with inverse transpose of L

35

then, finding D as

now [D] is triangularised , where

now making U such that,

this [U] is used for next iteration.

After few iterations [U] becomes eigen matrix where each column are eigen vectors of

corresponding eigen values(λ).

Here we have used the inverse iteration process to get the first

few natural frequencies , that is largest λ for smallest natural frequency(ω), as .

The mode shapes U1 are found from the vector U as

The programs of clamped-free, clamped-clamped, clamped-simply supported and simply

supported- simply supported beams are shown in appendix 2.

36

5. RESULTS AND DISCUSSION (TAPERED)

From the stiffness and mass matrix, simultaneous iteration method is employed to get the eigen

vectors. Then the eigen values are developed and natural frequency are obtained. Changing the

boundary conditions the natural frequencies of the clamped-free, clamped-clamped, clamped-

simply supported and simply supported-simply supported beams are obtained. Here “ω “denotes

the frequency in radian per second, “f” denotes the frequency in hertz, and “Ω” in non

dimensional form. The non dimensional value is obtained as [15].

.

Where ρ (7860 kg/cubic metres) is the material density, E ( GPa) is the modulus of

elasticity, A is the cross sectional area and I is the moment of inertia.

The results have been generated with the following geometric parameters, L=1m,

B=5cm, Bn=3cm, T=3cm, Tn=2cm.

In table 1 the first five natural frequencies of a prismatic beam obtained from the

present analysis is compared with the exact values. [27]

Table 12 Comparison of Natural Frequency of Prismatic Beam Found and Exact values

Natural Frequency obtained Exact Natural frequency

1

st

mode

156.26 156.25

2

nd

mode

979.28 979.28

3

rd

mode

2742.0 2472.02

4

th

mode

5373.3 5373.26

5

th

mode

8882.5 8882.38

37

It is seen that the error occurred is nearly zero. Thus if the program validates for a

prismatic beam where the tapered value is made zero, then for a linearly tapered beam the results

should also be correct.

In tables 13, 14, 15 and 16 the natural frequencies for first six modes of clamped-free,

clamped-clamped, clamped-simply supported, and simply supported- simply supported tapered

beams are shown in radian per seconds, hertz and in non dimensional values.

Table 13 Natural Frequency of a Clamped-Free Beam

Table 14 Natural Frequency of a Clamped-Clamped Beam

Frequencies

ω (rad/sec) f (Hz) Ω (N.D.)

1

st

mode 189.74 30.2 4.2294

2

nd

mode 914.90 145.61 20.5849

3

rd

mode 2370.3 377.24 53.3310

4

th

mode 4542.5 722.96 102.2060

5

th

mode 7437.7 1183.75 167.3489

6

th

mode 11057 1759.78 248.7827

Frequencies

ω (rad/sec) f (Hz) Ω (N.D.)

1

st

mode 616.28 98.08 13.8664

2

nd

mode 1885.3 300.05 42.4190

3

rd

mode 3877.2 617.07 87.2370

4

th

mode 6591.8 1049.12 148.3152

5

th

mode 10030 1596.32 225.6750

6

th

mode 14196 2259.36 319.4101

38

Table 15 Natural Frequency of a Clamped-Simply supported Beam

Frequencies

ω (rad/sec) f (Hz) Ω (N.D.)

1

st

mode 823.21 131.02 18.5222

2

nd

mode 2265.1 360.5 50.9648

3

rd

mode 4437.1 706.19 99.8348

4

th

mode 7332.1 1166.94 164.9719

5

th

mode 10952 1743.06 246.4202

6

th

mode 15300 2435.07 344.2494

Table 16 Natural Frequency of a Simply supported -Simply supported Beam

Frequencies

ω (rad/sec) Ω (hz) Ω (N.D.)

1

st

mode 356.8041 56.79 8.0280

2

nd

mode 1459.2 231.44 32.7197

3

rd

mode 3266.7 519.91 73.5007

4

th

mode 5801.2 923.29 130.5269

5

th

mode 9058.9 1441.77 203.8253

6

th

mode 13042 2075.70 293.4449

A comparison has been made in table 6 to show the variation of the natural frequencies

with change in number of elements used to model the clamped-free beam. The variation of few

radians is observed as the tapered value in both directions is very small.

39

Table 17 Change of Natural Frequency of a Clamped-Free Beam with change in number of

Elements (n)

Frequencies

n=10 n=50 n=100

n=200

% diff. of (n=200) and (n=10)

1

st

mode 188.9798

189.74 189.7678

189.7738

0.4184

2

nd

mode 910.7328

914..90

915.0335

915.0667

0.4736

3

rd

mode 2359.8 2370.3 2370.7 2370.7 0.4598

4

th

mode 4525.7 4542..5

4543.2 4543.3 0.3874

5

th

mode 7422.8 7437.7 7438.8 7439 0.2178

6

th

mode 11067 11057 11058 11058 -0.0814

40

Table 18: Change in frequency between the two beams

Natural Frequency of tapered

beam

Exact frequency of prismatic beam

1

st

mode

189.74 156.25

2

nd

mode

914..90 979.28

3

rd

mode

2370.3 2472.02

4

th

mode

4542..5 5373.26

5

th

mode

7437.7 8882.38

41

Table 19: Comparison of Natural frequency Ratios for Clamped Free, Clamped-Clamped,

Clamped-Simply supported and Simply supported -Simply supported beams

Clamped

Free

Clamped-Clamped

Clamped-Simply

supported

Simply supported -

Simply supported

1 1 1 1

4.82 2.75 3.06 4.08

12.49 5.39 6.29 9.15

23.94 8.91 10.70 16.26

39.20 13.30 16.27 25.39

58.27 14.00 23.03 36.55

The plot of mode shapes of Clamped-Free, Clamped-Clamped and Clamped- Simply

supported and Simply supported- Simply supported beams are shown in figures 12, 13, 14 and

15.

42

0 0.1 0.2 0.3 0.4 0.5 0. 6 0.7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

length

displac ement

1

2

3

4

5

6

Figure 12 Mode shapes of a Clamped-Free Beam

0 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

length

disp lacem ent

1

2

3

4

5

6

Figure 13 Mode shapes of a Clamped-Clamped Beam

43

0 0.1 0.2 0.3 0.4 0. 5 0.6 0.7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

length

displac ement

1

2

3

4

5

6

Figure 14 Mode shapes of a Clamped - Simply Supported Beam

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0. 8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

length

displacement

1

2

3

4

5

6

Figure 15 Mode shapes of a Simply supported - Simply Supported Beam

44

6. CONCLUSION

The comparison of mode shapes and natural frequencies of various beam configurations

have been found out. The results clearly indicate the variation in these quantities. Also two

separate beam geometries were chosen and the tapered beam geometry takes these geometries as

its bounding values. Hence a fair comparison of the two analyses is possible.

The present study deals with the natural frequency of tapered beams under different

boundary conditions. Variation of width and thickness of the beam was chosen linearly. In the

above different boundary conditions it is seen that when the tapered value was made zero the

natural frequencies of the prismatic beam corresponds to that of exact case [27]. This study can

be extended to other structural members i.e. plates and shells or to other materials composites or

functionally graded materials.

45

References:-

[1] R. Davis, R.D. Henshell Constant curvature beam finite elements for in-plane vibration

Volume 25, 1972 Journal of Sound and Vibration.

[2] G. Prathap, , T.K. Varadan The large amplitude vibration of tapered clamped beams Volume

58, 1978 Journal of Sound and Vibration.

[3] K. E. Rouch, J. S. Kao A tapered beam finite element for rotor dynamics analysis Volume 66,

1979 Journal of Sound and Vibration.

[4] G.R. Bhashyam, G. Prathap Galerkin finite element method for non-linear beam vibrations,

Volume 72, 1980, Journal of Sound and Vibration.

[5] B.S. Sarma, T.K. Varadan Lagrange-type formulation for finite element analysis of non-

linear beam vibrations Volume 86, 1983, Journal of Sound and Vibration.

[6] V. P. Iu, Y. K. Cheung, S. L. Lau Non-linear vibration analysis of multilayer beams by

incremental finite elements, Part I: Theory and numerical formulation Volume 100, 1985,

Journal of Sound and Vibration.

[7] C. Mei, K. Decha-Umphai A finite element method for non-linear forced vibrations of beams

Volume 102, 1985, Journal of Sound and Vibration.

[8] C. Mei Discussion of finite element formulations of nonlinear beam vibrations Volume 22,

1986, Computer and Structures.

[9] P.R. Heyliger J.N. Reddy A higher order beam finite element for bending and vibration

problems Volume 126, 1988, Journal of Sound and Vibration.

[10] P.C. Dumir A. Bhaskar Some erroneous finite element formulations of non-linear

vibrations of beams and plates Volume 123, 1988, Journal of Sound and Vibration.

[11] Y.A. Khulief Vibration frequencies of a rotating tapered beam with end mass Volume 134,

1989, Journal of Sound and Vibration.

[12] A. K. Noor Jeanne M. Peters, Byung-Jin Min Mixed finite element models for free

vibrations of thin-walled beams Volume 5, 1989, Finite Element Analysis and Design.

46

[13] K. M. Bangera, K. Chandrashekhara Nonlinear vibration of moderately thick laminated

beams using finite element method Volume 9, 1991, Finite Element Analysis and Design.

[14] A. Y. T. Leung S. G. Mao Symplectic integration of an accurate beam finite element in

non-linear vibration 1993 Computer and Structures.

[15] G. P. Dube ,P. C. Dumb tapered thin open section beams on elastic foundation-ii. vibration

analysis Vol. 61. 1996 Computer and Structures.

[16] T. T. Babert , R. A. Maddox and C. E. Orozco A finite element model for harmonically

excited viscoelastic sandwich beams 1997 Computer and Structures.

[17] T.D. Chaudhari , S.K. Maiti A study of vibration of geometrically segmented beams with

and without crack 1998 International Journal of Solids and Structures.

[18] N. Hua ,H. Fukunagab, M. Kameyamab, Y. Aramakib, F.K. Changc Vibration analysis of

delaminated composite beams and plates using a higher-order finite element 2002 International

Journal of Mechanical Sciences.

[19] E.N. Mazzilli Ma´rio E.S. Soares, Odulpho G.P. B.. N. Carlos Non-linear normal modes of

a simply supported beam: continuous system and finite-element models 2004 Computer and

Structures.

[20] M. Yaman Finite element vibration analysis of a partially covered cantilever beam with

concentrated tip mass 2004 Materials and Design

[21] P. Ribeiro Non-linear forced vibrations of thin/thick beams and plates by the finite element

and shooting methods May 2004 Computer and Structures.

[22] P. Subramanian Dynamic analysis of laminated composite beams using higher order

theories and finite elements 2005 Computer and Structures.

[23] J.R. Fonseca , P. Ribeiro Beam p-version finite element for geometrically non-linear

vibrations in space 2005 Computer Methods in Applied Mechanics and Engineering.

[24] R. Ganesan , A. Zabihollah Vibration analysis of tapered composite beams using a higher-

order finite element. 2005 Computer and Structures.

[25] J.R. Banerjee H. Su, D.R. Jackson Free vibration of rotating tapered beams using the

dynamic stiffness method 2006 Journal of Sound and Vibration.

47

[26] P. Malekzadeh G. Karami A mixed differential quadrature and finite element free

vibration and buckling analysis of thick beams on two-parameter elastic foundations 2007

Applied Mathematical Modeling.

[27] M. C. Ece, M.Aydogdu . V. Taskin Vibration of a variable cross-section beam 2006

Mechanics Research Communications.

[28] Marcelo A. Trindade A. Benjeddou Refined sandwich model for the vibration of beams

with embedded shear piezoelectric actuators and sensors 2007 Computer and Structures.

[29] Hong Hu Chen ,K. M. Hsiao Quadruply coupled linear free vibrations of thin-walled

beams with a generic open section 2007, Engineering Structures.

[30] R. Lopes Alonso, P. Ribeiro Flexural and torsional non-linear free vibrations of beams

using a p-version finite element 2008 Computer and Structures.

[31] Marcelo T. Piovana Carlos P. Filipicha,, Vı´ctor H. Cortı´neza Exact solutions for coupled

free vibrations of tapered shear-flexible thin-walled composite beams 2008 Journal of Sound

and Vibration.

[32] M. Shavezipur S.M. Hashemi Free vibration of triply coupled centrifugally stiffened

nonuniform beams, using a refined dynamic finite element method 2008 Aerospace Science

and Technology.

[33] R.K. Gupta , G.J. Babua, G.R.Janardhanb, G.VenkateswaraRao Relatively simple finite

element formulation for the large amplitude free vibrations of uniform beams 2009 Finite

Element Analysis and Design.

[34] G.S. Ramtekkar Free vibration analysis of delaminated beams using mixed finite element

model August 2009 Journal of Sound and Vibration.

[35] D. Das P. Sahoo , K. Saha Out-of-plane free vibration analysis of rotating tapered beams in

post-elastic regime 2009 Materials and Design.

[36] P. Vidal O. Polit Vibration of multilayered beams using sinus finite elements with

transverse normal stress 2009 Computer and Structures.

[37] B. Zhu, A.Y.T. Leung Linear and nonlinear vibration of non-uniform beams on two-

parameter foundations using p-elements 2009 Computers and Geotechnics.

48

APPENDIX 1

%THIS PROGRAM WILL CALCULATE THE FIRST FEW NATURAL FREQUENCIES AND PLOT THE

CORRESPONDING MODE SHAPES

%OF A PRISMATIC BEAM BASED ON DIFFERENT BOUNDARY CONDITIONS.

%MATERIAL PROPERTIES FOR MILD STEEL ARE:-E=200 GPa;Density=7850 kg/m^3

clear all %clears pre determined variable values

%declaring material property

E = input('ENTER MATERIAL MODULUS OF ELASTICITY IN Pa\n');

density = input('ENTER MATERIAL DENSITY IN kg/m^3\n');

breadth = input('ENTER BREADTH OF THE BEAM IN m\n');

thickness = input('ENTER THICKNESS OF THE BEAM IN m\n');

length = input('ENTER LENGTH OF THE BEAM IN m\n');

area = breadth*thickness;

I = breadth*thickness^3/12;

cal = sqrt((E*I)/(density*area*length^4));

%solving the natural frequency equation based on boundary conditions

zz = input('FOR SOLUTION ENTER AS STATED:CF=1;CC=2;CS=3;SS=4\n');

p=0.001;

sol=zeros(1,1); %array of zeros to store solution

q=0.00001;

f=0; %array position locator

tolerance=p;

if zz==1

for x=1:p:15

value=cf(x); %calling clamped free beam function

newvalue=cf(x+p) ; %giving an increment

if ((value<=0) && (newvalue>0)) || ((newvalue<=0) && (value>0)) %checking for change in sign of the function

f=f+1; %new postion in the array

for j=0:q:1

xx=x+p*j; %refining the search

new=cf(xx);

if abs(new)<=tolerance

sol(f,1)=xx; % putting the root into the array

end

end

end

if value==tolerance

sol(f,1)=x; f=f+1; % putting the root into the array

end

end

non_di_natfreq = sol %print out the first few roots in the MATLAB window

natfreq_rad = zeros(1,1);

natfreq_Hz = zeros(1,1);

nn = 0;

for i=1:1:5

mm = sol(i,:)^2*cal;

nn = i;

natfreq_rad(nn,1) = mm;

gg = mm /(2*pi);

natfreq_Hz(nn,1) = gg;

end

natfreq_rad %calculating natural frequency in rad per s

natfreq_Hz %calculating frequency in Hz

for i=1:1:5 %plotting mode shape

49

m = sol (i,1);

zeta = linspace(0,1,5000);

mdash = cosh(m*zeta)-cos(m*zeta);

ndash = cos(m)+cosh(m);

odash = sin(m)+sinh(m);

pdash = sinh(m*zeta)-sin(m*zeta);

X = (mdash - (ndash/odash)*pdash);

C = max(abs(X));

X1 = ((mdash - (ndash/odash)*pdash))/C;

plot(zeta,X1,zeta,0,'r')

xlabel('Non dimensional length','FontSize',16)

ylabel('Amplitude','FontSize',16)

title('Mode shape plots','FontSize',16)

hold on

end

elseif zz==2

for x=1:p:20

value=cc(x); %calling clamped clamped beam function

newvalue=cc(x+p) ; %giving an increment

if ((value<=0) && (newvalue>0)) || ((newvalue<=0) && (value>0)) %checking for change in sign of the function

f=f+1; %new postion in the array

for j=0:q:1

xx=x+p*j; %refining the search

new=cc(xx);

if abs(new)<=tolerance

sol(f,1)=xx; % putting the root into the array

end

end

end

if value==tolerance

sol(f,1)=x; f=f+1; % putting the root into the array

end

end

non_di_natfreq = sol %print out the first few roots in the MATLAB window

natfreq_rad = zeros(1,1);

natfreq_Hz = zeros(1,1);

nn = 0;

for i=1:1:4

mm = sol(i,:)^2*cal;

nn = i;

natfreq_rad(nn,1) = mm;

gg = mm /(2*pi);

natfreq_Hz(nn,1) = gg;

end

natfreq_rad %calculating natural frequency in rad per s

natfreq_Hz %calculating frequency in Hz

for i=1:1:4 %plotting mode shape

m = sol (i,1);

zeta = linspace(0,1,5000);

mdash = cosh(m*zeta)-cos(m*zeta);

ndash = cosh(m)-cos(m);

odash = sinh(m)-sin(m);

pdash = sinh(m*zeta)-sin(m*zeta);

X = (mdash - (ndash/odash)*pdash);

50

C = max(abs(X));

X1 = X/C;

plot(zeta,X1,zeta,0,'r')

xlabel('Non dimensional length','FontSize',16)

ylabel('Amplitude','FontSize',16)

title('Mode shape plots','FontSize',16)

hold on

end

elseif zz==3

for x=1:p:15

value=cos(x); %clamped simply supported beam function

newvalue=cos(x+p) ; %giving an increment

if ((value<=0) && (newvalue>0)) || ((newvalue<=0) && (value>0)) %checking for change in sign of the function

f=f+1; %new postion in the array

for j=0:q:1

xx=x+p*j; %refining the search

new=cos(xx);

if abs(new)<=tolerance

sol(f,1)=xx; % putting the root into the array

end

end

end

if value==tolerance

sol(f,1)=x; f=f+1; % putting the root into the array

end

end

non_di_natfreq = sol %print out the first few roots in the MATLAB window

natfreq_rad = zeros(1,1);

natfreq_Hz = zeros(1,1);

nn = 0;

for i=1:1:5

mm = sol(i,:)^2*cal;

nn = i;

natfreq_rad(nn,1) = mm;

gg = mm /(2*pi);

natfreq_Hz(nn,1) = gg;

end

natfreq_rad %calculating natural frequency in rad per s

natfreq_Hz %calculating frequency in Hz

for i=1:1:5 %plotting mode shape

m = sol (i,1);

zeta = linspace(0,1,5000);

mdash = cosh(m*zeta)-cos(m*zeta);

ndash = sinh(m)-sin(m);

odash = cosh(m)-cos(m);

pdash = sin(m*zeta)-sinh(m*zeta);

X = (mdash + (odash/ndash)*pdash);

C = max(abs(X));

X1 = X/C;

plot(zeta,X1,zeta,0,'r')

xlabel('Non dimensional length','FontSize',16)

ylabel('Amplitude','FontSize',16)

title('Mode shape plots','FontSize',16)

hold on

51

end

else

for x=1:p:18

value=sin(x); %simply supported beam function

newvalue=sin(x+p) ; %giving an increment

f=f+1; %new postion in the array

for j=0:q:1

xx=x+p*j; %refining the search

new=sin(xx);

if abs(new)<=tolerance

sol(f,1)=xx; % putting the root into the array

end

end

end

if value==tolerance

sol(f,1)=x; f=f+1; % putting the root into the array

end

end

non_di_natfreq = sol %print out the first few roots in the MATLAB window

natfreq_rad = zeros(1,1);

natfreq_Hz = zeros(1,1);

nn = 0;

for i=1:1:5

mm = sol(i,:)^2*cal;

nn = i;

natfreq_rad(nn,1) = mm;

gg = mm /(2*pi);

natfreq_Hz(nn,1) = gg;

end

natfreq_rad %calculating natural frequency in rad per s

natfreq_Hz %calculating frequency in Hz

for i=1:1:5 %plotting mode shape

m = sol (i,1);

zeta = linspace(0,1,5000);

X = sin(m*zeta);

plot(zeta,X,zeta,0,'r')

xlabel('Non dimensional length','FontSize',16)

ylabel('Amplitude','FontSize',16)

title('Mode shape plots','FontSize',16)

hold on

end

end

The function files for 2 of the beam configurations are to be written separately.

Cantilever - free:

function [r] = cc (t)

r=cos(t)*cosh(t)-1;

52

Cantilever- cantilever:

function [r] = cf (t)

r=cos(t)*cosh(t)+1;

Functions for rest of the beam configurations came in built with the MATLAB software.

APPENDIX 2

Program for clamped-free beam:

%This code calculates the free vibration first 6 mode shapes of a clamped

%free beam by Finite Element Analysis Method

%The material is mild steel and the beam geometry is fixed

clear all;

% beam geometry

E=207*(10^9); % modulus of elasticity of mild steel

rho=7860; % density of mild steel

B=0.05; %initial breadth

Bn=0.03; %tapered final breadth

T=0.03; % initial thickness

Tn=0.02;% tapered final thickness

L=1;%length of the beam

% element shape function formulation

n=50;%the number of 2D 2DOF elements

l=L/n;%element length

tb=(B-Bn)/(2*L);

tt=(T-Tn)/(2*L);

k1=[12 (6*l) -12 (6*l);(6*l) (4*l*l) -(6*l) (2*l*l);-12 -(6*l) 12 -(6*l);(6*l) (2*l*l) -(6*l) (4*l*l)];

k=k1*E/(l^3);%initial element stiffness matrix excluding moment of inertia

m1=[156 (22*l) 54 -(13*l);(22*l) (4*l*l) (13*l) -(3*l*l);54 (13*l) 156 -(22*l);-(13*l) -(3*l*l) -(22*l) (4*l*l)];

m=m1*rho*l/420;%initial element mass matrix

b=B;

t=T;

p=0;

%--------------------------------------------------------------------------

% ASSEMBLY OF MATRICES

K=zeros(((2*n)+2),((2*n)+2));%initial global stiffness matrix

M=zeros(((2*n)+2),((2*n)+2));%initial global mass matrix

for q=1:n

b1=b-(2*l*tb);

t1=t-(2*l*tt);

I=((b+b1)/2)*(((t+t1)/2)^3)/12;%mean area moment of inertia

A=((b+b1)/2)*((t+t1)/2);%mean element area

k2=k*I;

m2=m*A;

for i=1:4

for j=1:4

K((p+i),(p+j))=K((p+i),(p+j))+k2(i,j);

M((p+i),(p+j))=M((p+i),(p+j))+m2(i,j);

end

end

%going to the next element in the initial for loop

53

p=p+2;%shifting the rows and columns by two positions

b=b1;

t=t1;

end

K1=K(:,2:((2*n)+2));

K2=K1(2:((2*n)+2),:);

K3=K2(:,2:((2*n)+1));

K4=K3(2:((2*n)+1),:);

M1=M(:,2:((2*n)+2));

M2=M1(2:((2*n)+2),:);

M3=M2(:,2:((2*n)+1));

M4=M3(2:((2*n)+1),:);

%--------------------------------------------------------------------------

% TRIANGULARISATION OF STIFFNESS MATRIX

L=zeros((2*n),(2*n));

l1=sqrt(K4(1,1));

L(1,1)=L(1,1)+l1;

s1=0;

s3=0;

for i=2:(2*n)

l2=K4(i,1)/l1;

L(i,1)=L(i,1)+l2;

end

for i=2:(2*n)

for j=i:(2*n)

if (i==j)

for s=1:(i-1)

s1=s1+((L(i,s))^2);

end

l3=sqrt((K4(i,i))-s1);

L(i,i)=L(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(L(j,s2)*L(i,s2));

end

l4=((K4(i,j))-s3)/L(i,i);

L(j,i)=L(j,i)+l4;

s3=0;

end

end

end

%--------------------------------------------------------------------------

% SIMULTANEOUS ITERATION PROCESS

U=zeros((2*n),6);

U(1,1)=1;

U(2,2)=1;

U(3,3)=1;

U(4,4)=1;

U(5,5)=1;

U(6,6)=1;

for i=1:10

X=inv(L')*U;

54

Y=M4*X;

V=inv(L)*Y;

D=(V')*V;

J=zeros(6,6);

l1=sqrt(D(1,1));

J(1,1)=J(1,1)+l1;

s1=0;

s3=0;

for i=2:6

l2=D(i,1)/l1;

J(i,1)=J(i,1)+l2;

end

for i=2:6

for j=i:6

if (i==j)

for s=1:(i-1)

s1=s1+((J(i,s))^2);

end

l3=sqrt((D(i,i))-s1);

J(i,i)=J(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(J(j,s2)*J(i,s2));

end

l4=((D(i,j))-s3)/J(i,i);

J(j,i)=J(j,i)+l4;

s3=0;

end

end

end

U=V*inv(J');

end

% the codes below calculate the eigen value matrix B1 (6 X 6) and extracts

% the eigen values from B1. The square root of the reciprocals of the eigen

% values gives the natural frequency in radian / s

A1=inv(L)*M4*inv(L');

B1=(U')*A1*U;

w1=sqrt(1/B1(1,1));

w2=sqrt(1/B1(2,2));

w3=sqrt(1/B1(3,3));

w4=sqrt(1/B1(4,4));

w5=sqrt(1/B1(5,5));

w6=sqrt(1/B1(6,6));

% the codes below calculate the mode shapes by forming the matrix U1 by

% multiplying U with the inverse of the transpose of L and extracting the eigen

% vectors

55

U1=inv(L')*U;

e1v1=U1(:,1);

e1v2=U1(:,2);

e1v3=U1(:,3);

e1v4=U1(:,4);

e1v5=U1(:,5);

e1v6=U1(:,6);

%--------------------------------------------------------------------------

A2=inv(K4)*M4;

[V,D] = eigs(A2);

f1=sqrt(1/D(1,1));

f2=sqrt(1/D(2,2));

f3=sqrt(1/D(3,3));

f4=sqrt(1/D(4,4));

f5=sqrt(1/D(5,5));

f6=sqrt(1/D(6,6));

ev1=V(:,1);

ev2=V(:,2);

ev3=V(:,3);

ev4=V(:,4);

ev5=V(:,5);

ev6=V(:,6);

%--------------------------------------------------------------------------

% PLOTTING OF MODE SHAPES

S=zeros(n,6);

ev11=S(:,1);

ev22=S(:,1);

ev33=S(:,1);

ev44=S(:,1);

ev55=S(:,1);

ev66=S(:,1);

p=1;

for i=1:2:(2*n)

ev11(p)=ev11(p)+e1v1(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev11(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev111=ev11/a;

x=linspace(0,1,n);

plot(x,ev111,'r');

hold on

56

p=1;

for i=1:2:(2*n)

ev22(p)=ev22(p)-e1v2(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev22(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev222=ev22/a;

x=linspace(0,1,n);

plot(x,ev222);

p=1;

for i=1:2:(2*n)

ev33(p)=ev33(p)-e1v3(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev33(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev333=ev33/a;

x=linspace(0,1,n);

plot(x,ev333,'k');

p=1;

for i=1:2:(2*n)

ev44(p)=ev44(p)+e1v4(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev44(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev444=ev44/a;

x=linspace(0,1,n);

plot(x,ev444,'g');

p=1;

57

for i=1:2:(2*n)

ev55(p)=ev55(p)+e1v5(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev55(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev555=ev55/a;

x=linspace(0,1,n);

plot(x,ev555,'m');

p=1;

for i=1:2:(2*n)

ev66(p)=ev66(p)-ev6(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev66(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev666=ev66/a;

x=linspace(0,1,n);

plot(x,ev666,'y');

hold off

Program for cantilever-cantilever beam:

% Program for cantilever-cantilever beam:

clear all;

% beam geometry

E=207*(10^9); % modulus of elasticity of mild steel

rho=7860; % density of mild steel

B=0.05; %initial breadth

Bn=0.03; %tapered final breadth

T=0.03; % initial thickness

Tn=0.02;% tapered final thickness

L=1;%length of the beam

%element dicretization

n=50;%number of elements

58

l=L/n;%element length

tb=(B-Bn)/(2*L);

tt=(T-Tn)/(2*L);

k1=[12 (6*l) -12 (6*l);(6*l) (4*l*l) -(6*l) (2*l*l);-12 -(6*l) 12 -(6*l);(6*l) (2*l*l) -(6*l) (4*l*l)];

k=k1*E/(l^3);%initial element stiffness matrix excluding area moment of inertia

m1=[156 (22*l) 54 -(13*l);(22*l) (4*l*l) (13*l) -(3*l*l);54 (13*l) 156 -(22*l);-(13*l) -(3*l*l) -(22*l) (4*l*l)];

m=m1*rho*l/420;%inital element mass matrix

b=B;

t=T;

p=0;

%--------------------------------------------------------------------------

% ASSEMBLY OF MATRICES

K=zeros(((2*n)+2),((2*n)+2));%initial global stiffness matrix

M=zeros(((2*n)+2),((2*n)+2));%initial global mass matrix

for q=1:n

b1=b-(2*l*tb);

t1=t-(2*l*tt);

I=((b+b1)/2)*(((t+t1)/2)^3)/12;%mean area moment of inertia

A=((b+b1)/2)*((t+t1)/2);%mean area of an element

k2=k*I;

m2=m*A;

for i=1:4

for j=1:4

K((p+i),(p+j))=K((p+i),(p+j))+k2(i,j);

M((p+i),(p+j))=M((p+i),(p+j))+m2(i,j);

end

end

%going to the next element in the initial for loop

p=p+2;%shifting the rows and columns by two positions

b=b1;

t=t1;

end

K1=K(:,3:((2*n)+2));

K2=K1(3:((2*n)+2),:);

K3=K2(:,1:((2*n)-2));

K4=K3(1:((2*n)-2),:);

M1=M(:,3:((2*n)+2));

M2=M1(3:((2*n)+2),:);

M3=M2(:,1:((2*n)-2));

M4=M3(1:((2*n)-2),:);

%--------------------------------------------------------------------------

% TRIANGULARISATION OF STIFFNESS MATRIX

L=zeros((2*n)-2,(2*n)-2);

l1=sqrt(K4(1,1));

L(1,1)=L(1,1)+l1;

s1=0;

s3=0;

for i=2:(2*n)-2

l2=K4(i,1)/l1;

L(i,1)=L(i,1)+l2;

end

for i=2:(2*n)-2

for j=i:(2*n)-2

if (i==j)

for s=1:(i-1)

59

s1=s1+((L(i,s))^2);

end

l3=sqrt((K4(i,i))-s1);

L(i,i)=L(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(L(j,s2)*L(i,s2));

end

l4=((K4(i,j))-s3)/L(i,i);

L(j,i)=L(j,i)+l4;

s3=0;

end

end

end

%--------------------------------------------------------------------------

% SIMULTANEOUS ITERATION PROCESS

U=zeros((2*n)-2,6);

U(1,1)=1;

U(2,2)=1;

U(3,3)=1;

U(4,4)=1;

U(5,5)=1;

U(6,6)=1;

for i=1:10

X=inv(L')*U;

Y=M4*X;

V=inv(L)*Y;

D=(V')*V;

J=zeros(6,6);

l1=sqrt(D(1,1));

J(1,1)=J(1,1)+l1;

s1=0;

s3=0;

for i=2:6

l2=D(i,1)/l1;

J(i,1)=J(i,1)+l2;

end

for i=2:6

for j=i:6

if (i==j)

for s=1:(i-1)

s1=s1+((J(i,s))^2);

end

l3=sqrt((D(i,i))-s1);

J(i,i)=J(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(J(j,s2)*J(i,s2));

60

end

l4=((D(i,j))-s3)/J(i,i);

J(j,i)=J(j,i)+l4;

s3=0;

end

end

end

U=V*inv(J');

end

% the codes below calculate the eigen value matrix B1 (6 X 6) and extracts

% the eigen values from B1. The square root of the reciprocals of the eigen

% values gives the natural frequency in radian / s

A1=inv(L)*M4*inv(L');

B1=(U')*A1*U;

w1=sqrt(1/B1(1,1));

w2=sqrt(1/B1(2,2));

w3=sqrt(1/B1(3,3));

w4=sqrt(1/B1(4,4));

w5=sqrt(1/B1(5,5));

w6=sqrt(1/B1(6,6));

% the codes below calculate the mode shapes by forming the matrix U1 by

% multiplying U with the inverse of the transpose of L and extracting the eigen

% vectors

U1=inv(L')*U;

e1v1=U1(:,1);

e1v2=U1(:,2);

e1v3=U1(:,3);

e1v4=U1(:,4);

e1v5=U1(:,5);

e1v6=U1(:,6);

%--------------------------------------------------------------------------

A2=inv(K4)*M4;

[V,D] = eigs(A2);

f1=sqrt(1/D(1,1));

f2=sqrt(1/D(2,2));

f3=sqrt(1/D(3,3));

f4=sqrt(1/D(4,4));

f5=sqrt(1/D(5,5));

f6=sqrt(1/D(6,6));

ev1=V(:,1);

ev2=V(:,2);

ev3=V(:,3);

ev4=V(:,4);

ev5=V(:,5);

ev6=V(:,6);

%--------------------------------------------------------------------------

% PLOTTING OF MODE SHAPES

61

S=zeros(n,6);

ev11=S(:,1);

ev22=S(:,1);

ev33=S(:,1);

ev44=S(:,1);

ev55=S(:,1);

ev66=S(:,1);

p=1;

for i=1:2:(2*n)-2

ev11(p)=ev11(p)+e1v1(i);

p=p+1;

end

ev11(n)=0;

a=0;

for i=1:n

g=((ev11(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev111=ev11/a;

x=linspace(0,1,n);

plot(x,ev111,'r');

hold on

p=1;

for i=1:2:(2*n)-2

ev22(p)=ev22(p)-e1v2(i);

p=p+1;

end

ev22(n)=0;

a=0;

for i=1:n

g=((ev22(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev222=ev22/a;

x=linspace(0,1,n);

plot(x,ev222);

p=1;

for i=1:2:(2*n)-2

ev33(p)=ev33(p)+e1v3(i);

p=p+1;

end

ev44(n)=0;

62

a=0;

for i=1:n

g=((ev33(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev333=ev33/a;

x=linspace(0,1,n);

plot(x,ev333,'k');

p=1;

for i=1:2:(2*n)-2

ev44(p)=ev44(p)+e1v4(i);

p=p+1;

end

ev55(n)=0;

a=0;

for i=1:n

g=((ev44(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev444=ev44/a;

x=linspace(0,1,n);

plot(x,ev444,'g');

p=1;

for i=1:2:(2*n)-2

ev55(p)=ev55(p)+e1v5(i);

p=p+1;

end

ev33(n)=0;

a=0;

for i=1:n

g=((ev55(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev555=ev55/a;

x=linspace(0,1,n);

plot(x,ev555,'m');

p=1;

for i=1:2:(2*n)-2

ev66(p)=ev66(p)+e1v6(i);

p=p+1;

63

end

ev66(n)=0;

a=0;

for i=1:n

g=((ev66(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev666=ev66/a;

x=linspace(0,1,n);

plot(x,ev666,'y');

hold off

Program for clamped-simply supported beam:

% Program for clamped-simply supported beam; All are in standard S.I.

% units

clear all;

% beam geometry

E=207*(10^9); % modulus of elasticity of mild steel

rho=7860; % density of mild steel

B=0.05; %initial breadth

Bn=0.03; %tapered final breadth

T=0.03; % initial thickness

Tn=0.02;% tapered final thickness

L=1;%length of the beam

% element shape function formulation

n=50; % the number of 2-D 2-DOF elements

l=L/n; % element length

tb=(B-Bn)/(2*L);% tan theta of breadth

tt=(T-Tn)/(2*L);% tan phi of thickness

k1=[12 (6*l) -12 (6*l);(6*l) (4*l*l) -(6*l) (2*l*l);-12 -(6*l) 12 -(6*l);(6*l) (2*l*l) -(6*l) (4*l*l)]; % calculation of

stiffness shape functions

k=k1*E/(l^3); % initial element stiffness matrix excluding area moment of inertia as it is variable over element

length

m1=[156 (22*l) 54 -(13*l);(22*l) (4*l*l) (13*l) -(3*l*l);54 (13*l) 156 -(22*l);-(13*l) -(3*l*l) -(22*l) (4*l*l)]; %

calculation of mass shape functions

m=m1*rho*l/420; % initial element mass matrix

b=B;

t=T;

p=0;

%--------------------------------------------------------------------------

% ASSEMBLY OF MATRICES

K=zeros(((2*n)+2),((2*n)+2)); % initial global stiffness matrix

M=zeros(((2*n)+2),((2*n)+2)); % initial global mass matrix

for q=1:n

b1=b-(2*l*tb);

64

t1=t-(2*l*tt);

I=((b+b1)/2)*(((t+t1)/2)^3)/12; % mean area moment of inertia

A=((b+b1)/2)*((t+t1)/2); % mean area

k2=k*I;

m2=m*A;

for i=1:4

for j=1:4

K((p+i),(p+j))=K((p+i),(p+j))+k2(i,j);

M((p+i),(p+j))=M((p+i),(p+j))+m2(i,j);

end

end

% going to the next element in the inital for loop

p=p+2; % shifting the rows and columns by two positions

% changing geometry of individual element at element boundary

b=b1;

t=t1;

end

K1=K(:,3:((2*n)+2));

K4=K1(3:((2*n)+2),:);

K4(:,((2*n)-1))=[];

K4(((2*n)-1),:)=[];

M1=M(:,3:((2*n)+2));

M4=M1(3:((2*n)+2),:);

M4(:,((2*n)-1))=[];

M4(((2*n)-1),:)=[];

%-----------------------------------------------------------------------

% TRIANGULARISATION OF STIFFNESS MATRIX

L=zeros((2*n)-1,(2*n)-1);

l1=sqrt(K4(1,1));

L(1,1)=L(1,1)+l1;

s1=0;

s3=0;

for i=2:(2*n)-1

l2=K4(i,1)/l1;

L(i,1)=L(i,1)+l2;

end

for i=2:(2*n)-1

for j=i:(2*n)-1

if (i==j)

for s=1:(i-1)

s1=s1+((L(i,s))^2);

end

l3=sqrt((K4(i,i))-s1);

L(i,i)=L(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(L(j,s2)*L(i,s2));

end

l4=((K4(i,j))-s3)/L(i,i);

L(j,i)=L(j,i)+l4;

s3=0;

end

end

65

end

%--------------------------------------------------------------------------

% SIMULTANEOUS ITERATION PROCESS

U=zeros((2*n)-1,6);

U(1,1)=1;

U(2,2)=1;

U(3,3)=1;

U(4,4)=1;

U(5,5)=1;

U(6,6)=1;

for i=1:10

X=inv(L')*U;

Y=M4*X;

V=inv(L)*Y;

D=(V')*V;

J=zeros(6,6);

l1=sqrt(D(1,1));

J(1,1)=J(1,1)+l1;

s1=0;

s3=0;

for i=2:6

l2=D(i,1)/l1;

J(i,1)=J(i,1)+l2;

end

for i=2:6

for j=i:6

if (i==j)

for s=1:(i-1)

s1=s1+((J(i,s))^2);

end

l3=sqrt((D(i,i))-s1);

J(i,i)=J(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(J(j,s2)*J(i,s2));

end

l4=((D(i,j))-s3)/J(i,i);

J(j,i)=J(j,i)+l4;

s3=0;

end

end

end

U=V*inv(J');

end

% the codes below calculate the eigen value matrix B1 (6 X 6) and extracts

% the eigen values from B1. The square root of the reciprocals of the eigen

% values gives the natural frequency in radian / s

66

A1=inv(L)*M4*inv(L');

B1=(U')*A1*U;

w1=sqrt(1/B1(1,1))

w2=sqrt(1/B1(2,2))

w3=sqrt(1/B1(3,3))

w4=sqrt(1/B1(4,4))

w5=sqrt(1/B1(5,5))

w6=sqrt(1/B1(6,6))

% the codes below calculate the mode shapes by forming the matrix U1 by

% multiplying U with the inverse of the transpose of L and extracting the eigen

% vectors

U1=inv(L')*U;

e1v1=U1(:,1);

e1v2=U1(:,2);

e1v3=U1(:,3);

e1v4=U1(:,4);

e1v5=U1(:,5);

e1v6=U1(:,6);

%--------------------------------------------------------------------------

A2=inv(K4)*M4;

[V,D] = eigs(A2);

f1=sqrt(1/D(1,1));

f2=sqrt(1/D(2,2));

f3=sqrt(1/D(3,3));

f4=sqrt(1/D(4,4));

f5=sqrt(1/D(5,5));

f6=sqrt(1/D(6,6));

ev1=V(:,1);

ev2=V(:,2);

ev3=V(:,3);

ev4=V(:,4);

ev5=V(:,5);

ev6=V(:,6);

%--------------------------------------------------------------------------

% PLOTTING OF MODE SHAPES

S=zeros(n,6);

ev11=S(:,1);

ev22=S(:,1);

ev33=S(:,1);

ev44=S(:,1);

ev55=S(:,1);

ev66=S(:,1);

p=1;

for i=1:2:(2*n)-2

ev11(p)=ev11(p)+e1v1(i);

p=p+1;

end

67

ev11(n)=0;

a=0;

for i=1:n

g=((ev11(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev111=ev11/a;

x=linspace(0,1,n);

plot(x,ev111,'r');

hold on

p=1;

for i=1:2:(2*n)-2

ev22(p)=ev22(p)+e1v2(i);

p=p+1;

end

ev22(n)=0;

a=0;

for i=1:n

g=((ev22(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev222=ev22/a;

x=linspace(0,1,n);

plot(x,ev222);

p=1;

for i=1:2:(2*n)-2

ev33(p)=ev33(p)+e1v3(i);

p=p+1;

end

ev33(n)=0;

a=0;

for i=1:n

g=((ev33(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev333=ev33/a;

x=linspace(0,1,n);

plot(x,ev333,'k');

68

p=1;

for i=1:2:(2*n)-2

ev44(p)=ev44(p)+e1v4(i);

p=p+1;

end

ev44(n)=0;

a=0;

for i=1:n

g=((ev44(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev444=ev44/a;

x=linspace(0,1,n);

plot(x,ev444,'g');

p=1;

for i=1:2:(2*n)-2

ev55(p)=ev55(p)+e1v5(i);

p=p+1;

end

ev55(n)=0;

a=0;

for i=1:n

g=((ev55(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev555=ev55/a;

x=linspace(0,1,n);

plot(x,ev555,'m');

p=1;

for i=1:2:(2*n)-2

ev66(p)=ev66(p)+e1v6(i);

p=p+1;

end

ev66(n)=0;

a=0;

for i=1:n

g=((ev66(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev666=ev66/a;

69

x=linspace(0,1,n);

plot(x,ev666,'y');

hold off

Program for simply supported-simply supported beam:

%Program for simply supported-simply supported beam:

clear all;

% beam geometry

E=207*(10^9); % modulus of elasticity of mild steel

rho=7860; % density of mild steel

B=0.05; %initial breadth

Bn=0.03; %tapered final breadth

T=0.03; % initial thickness

Tn=0.02;% tapered final thickness

L=1;%length of the beam

%element discretization

n=50;%number of 2D 2DOF elements

l=L/n;%element length

tb=(B-Bn)/(2*L);

tt=(T-Tn)/(2*L);

k1=[12 (6*l) -12 (6*l);(6*l) (4*l*l) -(6*l) (2*l*l);-12 -(6*l) 12 -(6*l);(6*l) (2*l*l) -(6*l) (4*l*l)];

k=k1*E/(l^3);%initial element stiffness matrix excluding area moment of inertia

m1=[156 (22*l) 54 -(13*l);(22*l) (4*l*l) (13*l) -(3*l*l);54 (13*l) 156 -(22*l);-(13*l) -(3*l*l) -(22*l) (4*l*l)];

m=m1*rho*l/420;%initial mass matrix

b=B;

t=T;

p=0;

%--------------------------------------------------------------------------

% ASSEMBLY OF MATRICES

K=zeros(((2*n)+2),((2*n)+2));%initial global stiffness matrix

M=zeros(((2*n)+2),((2*n)+2));%initial global mass matrix

for q=1:n

b1=b-(2*l*tb);

t1=t-(2*l*tt);

I=((b+b1)/2)*(((t+t1)/2)^3)/12;%mean area moment of inertia

A=((b+b1)/2)*((t+t1)/2);%mean area of an element

k2=k*I;%element stiffness matrix

m2=m*A;%element mass matrix

for i=1:4

for j=1:4

K((p+i),(p+j))=K((p+i),(p+j))+k2(i,j);

M((p+i),(p+j))=M((p+i),(p+j))+m2(i,j);

end

end

%going to the next element in the initial for loop

p=p+2;%shifting the rows and columns by two positions

b=b1;

70

t=t1;

end

K1=K(:,2:((2*n)+2));

K4=K1(2:((2*n)+2),:);

K4(:,((2*n)))=[];

K4(((2*n)),:)=[];

M1=M(:,2:((2*n)+2));

M4=M1(2:((2*n)+2),:);

M4(:,((2*n)))=[];

M4(((2*n)),:)=[];

%---------------------------------------------------------------------

% TRIANGULARISATION OF STIFFNESS MATRIX

L=zeros((2*n),(2*n));

l1=sqrt(K4(1,1));

L(1,1)=L(1,1)+l1;

s1=0;

s3=0;

for i=2:(2*n)

l2=K4(i,1)/l1;

L(i,1)=L(i,1)+l2;

end

for i=2:(2*n)

for j=i:(2*n)

if (i==j)

for s=1:(i-1)

s1=s1+((L(i,s))^2);

end

l3=sqrt((K4(i,i))-s1);

L(i,i)=L(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(L(j,s2)*L(i,s2));

end

l4=((K4(i,j))-s3)/L(i,i);

L(j,i)=L(j,i)+l4;

s3=0;

end

end

end

%--------------------------------------------------------------------------

%SIMULTANEOUS ITERATION PROCESS

U=zeros((2*n),6);

U(1,1)=1;

U(2,2)=1;

U(3,3)=1;

U(4,4)=1;

U(5,5)=1;

U(6,6)=1;

for i=1:10

X=inv(L')*U;

Y=M4*X;

71

V=inv(L)*Y;

D=(V')*V;

J=zeros(6,6);

l1=sqrt(D(1,1));

J(1,1)=J(1,1)+l1;

s1=0;

s3=0;

for i=2:6

l2=D(i,1)/l1;

J(i,1)=J(i,1)+l2;

end

for i=2:6

for j=i:6

if (i==j)

for s=1:(i-1)

s1=s1+((J(i,s))^2);

end

l3=sqrt((D(i,i))-s1);

J(i,i)=J(i,i)+l3;

s1=0;

end

if (i<j)

for s2=1:(i-1)

s3=s3+(J(j,s2)*J(i,s2));

end

l4=((D(i,j))-s3)/J(i,i);

J(j,i)=J(j,i)+l4;

s3=0;

end

end

end

U=V*inv(J');

end

% the codes below calculate the eigen value matrix B1 (6 X 6) and extracts

% the eigen values from B1. The square root of the reciprocals of the eigen

% values gives the natural frequency in radian / s

A1=inv(L)*M4*inv(L');

B1=(U')*A1*U;

w1=sqrt(1/B1(1,1));

w2=sqrt(1/B1(2,2));

w3=sqrt(1/B1(3,3));

w4=sqrt(1/B1(4,4));

w5=sqrt(1/B1(5,5));

w6=sqrt(1/B1(6,6));

% the codes below calculate the mode shapes by forming the matrix U1 by

% multiplying U with the inverse of the transpose of L and extracting the eigen

% vectors

U1=inv(L')*U;

72

e1v1=U1(:,1);

e1v2=U1(:,2);

e1v3=U1(:,3);

e1v4=U1(:,4);

e1v5=U1(:,5);

e1v6=U1(:,6);

%--------------------------------------------------------------------------

A2=inv(K4)*M4;

[V,D] = eigs(A2);

f1=sqrt(1/D(1,1));

f2=sqrt(1/D(2,2));

f3=sqrt(1/D(3,3));

f4=sqrt(1/D(4,4));

f5=sqrt(1/D(5,5));

f6=sqrt(1/D(6,6));

ev1=V(:,1);

ev2=V(:,2);

ev3=V(:,3);

ev4=V(:,4);

ev5=V(:,5);

ev6=V(:,6);

%--------------------------------------------------------------------------

% PLOTTING OF MODE SHAPES

S=zeros(n+1,6);

ev11=S(:,1);

ev22=S(:,1);

ev33=S(:,1);

ev44=S(:,1);

ev55=S(:,1);

ev66=S(:,1);

p=2;

for i=2:2:(2*n)-1

ev11(p)=ev11(p)+e1v1(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev11(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev111=ev11/a;

x=linspace(0,1,n+1);

plot(x,ev111,'r');

hold on

p=2;

for i=2:2:(2*n)-1

73

ev22(p)=ev22(p)-e1v2(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev22(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev222=ev22/a;

x=linspace(0,1,n+1);

plot(x,ev222);

p=2;

for i=2:2:(2*n)-1

ev33(p)=ev33(p)+e1v3(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev33(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev333=ev33/a;

x=linspace(0,1,n+1);

plot(x,ev333,'k');

p=2;

for i=2:2:(2*n)-1

ev44(p)=ev44(p)-e1v4(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev44(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev444=ev44/a;

x=linspace(0,1,n+1);

plot(x,ev444,'g');

p=2;

for i=2:2:(2*n)-1

74

ev55(p)=ev55(p)+e1v5(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev55(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev555=ev55/a;

x=linspace(0,1,n+1);

plot(x,ev555,'m');

p=2;

for i=2:2:(2*n)-1

ev66(p)=ev66(p)-e1v6(i);

p=p+1;

end

a=0;

for i=1:n

g=((ev66(i))^2);

h=sqrt(g);

if (a<=h)

a=h;

end

end

ev666=ev66/a;

x=linspace(0,1,n+1);

plot(x,ev666,'y');

hold off