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Determination of Natural Frequency of Euler's Beams Using Analytical and Finite Element Method

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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. 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.
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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 “ 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) (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) (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) (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) (hz) (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:-
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Volume 25, 1972 Journal of Sound and Vibration.
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[10] P.C. Dumir A. Bhaskar Some erroneous finite element formulations of non-linear
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46
[13] K. M. Bangera, K. Chandrashekhara Nonlinear vibration of moderately thick laminated
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and without crack 1998 International Journal of Solids and Structures.
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Structures.
[20] M. Yaman Finite element vibration analysis of a partially covered cantilever beam with
concentrated tip mass 2004 Materials and Design
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order finite element. 2005 Computer and Structures.
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[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.
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with embedded shear piezoelectric actuators and sensors 2007 Computer and Structures.
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beams with a generic open section 2007, Engineering Structures.
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using a p-version finite element 2008 Computer and Structures.
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free vibrations of tapered shear-flexible thin-walled composite beams 2008 Journal of Sound
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[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
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Element Analysis and Design.
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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
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=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
... Simsek and Kocatürk [24] used a third order shear deformation theory to study the natural vibrations of beams. Mukherjee and Gosai [25] determined the natural frequencies of Euler-Bernoulli beams using analytical and finite elements method, and presented solutions for various end support conditions. ...
... Similarly, for Euler-Bernoulli beam with clamped ends, the boundary conditions expressed by Equations (22) and (23) are used to obtain the system of homogeneous equations -Equation (24). For nontrivial solutions, the characteristic frequency equation, obtained from the vanishing of the coefficient matrix is found as Equation (25). Expansion of the determinant gave the characteristic frequency equation as the transcendental equation -Equation (26). ...
Article
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The determination of the natural frequencies of flexural vibrations of Euler-Bernoulli beams is a vital consideration in their analysis and design for dynamic loads. This paper presents the Sumudu transform method for the determination of the natural frequencies of Euler-Bernoulli beams under transverse free harmonic vibration for different boundary conditions. The end support conditions considered are: (a) simply supported at both bends, (b) clamped at both ends, (c) clamped-free ends (d) clamped-simply supported ends, and (e) simply supported-clamped ends. The governing partial differential equation is converted by the Sumudu transformation to an integral equation, which upon evaluation becomes an algebraic equation. The solution gives the dynamic modal displacement shape function in the Sumudu transform space V(u). Inversion gives the dynamic modal displacement function in the physical problem space V(x). The enforcement of boundary conditions for the end supports considered yielded systems of homogeneous equations. The condition for nontrivial solutions is used to determine the characteristic frequency equation for each considered boundary condition. It is found that the characteristic frequency equation has an infinite number of eigenvalues (roots or zeros) corresponding to the continuously distributed parameter model of idealization of the problem. The characteristic frequency equations obtained are solved for the n roots using computational software methods, Symbolic Algebra Software and Mathematica Software to obtain the eigenvalues (zeros or roots) for any (n) vibration mode. The eigenvalues are then used to obtain the eigenfrequencies or natural frequencies of flexural vibration for each considered boundary conditions. It is found that closed form solutions obtained are identical to the solutions in the literature; obtained by classical methods of separation of variables and eigenfunction expansion methods.
... Simsek and Kocatürk [24] used a third order shear deformation theory to study the natural vibrations of beams. Mukherjee and Gosai [25] determined the natural frequencies of Euler-Bernoulli beams using analytical and finite elements method, and presented solutions for various end support conditions. ...
... Similarly, for Euler-Bernoulli beam with clamped ends, the boundary conditions expressed by Equations (22) and (23) are used to obtain the system of homogeneous equations -Equation (24). For nontrivial solutions, the characteristic frequency equation, obtained from the vanishing of the coefficient matrix is found as Equation (25). Expansion of the determinant gave the characteristic frequency equation as the transcendental equation -Equation (26). ...
Article
The determination of the natural frequencies of flexural vibrations of Euler-Bernoulli beams is a vital consideration in their analysis and design for dynamic loads. This paper presents the Sumudu transform method for the determination of the natural frequencies of Euler-Bernoulli beams under transverse free harmonic vibration for different boundary conditions. The end support conditions considered are: (a) simply supported at both bends, (b) clamped at both ends, (c) clamped-free ends (d) clamped-simply supported ends, and (e) simply supported-clamped ends. The governing partial differential equation is converted by the Sumudu transformation to an integral equation, which upon evaluation becomes an algebraic equation. The solution gives the dynamic modal displacement shape function in the Sumudu transform space V(u). Inversion gives the dynamic modal displacement function in the physical problem space V(x). The enforcement of boundary conditions for the end supports considered yielded systems of homogeneous equations. The condition for nontrivial solutions is used to determine the characteristic frequency equation for each considered boundary condition. It is found that the characteristic frequency equation has an infinite number of eigenvalues (roots or zeros) corresponding to the continuously distributed parameter model of idealization of the problem. The characteristic frequency equations obtained are solved for the n roots using computational software methods, Symbolic Algebra Software and Mathematica Software to obtain the eigenvalues (zeros or roots) for any (n) vibration mode. The eigenvalues are then used to obtain the eigenfrequencies or natural frequencies of flexural vibration for each considered boundary conditions. It is found that closed form solutions obtained are identical to the solutions in the literature; obtained by classical methods of separation of variables and eigenfunction expansion methods.
... The volume and surface area of panel are LWT and 2ðLW þ LT þ WTÞ, respectively. The detailed description of FE model are given as [61] and Euler [62] beam theory with respect to the FE analysis for varying dimensions are shown in Figure 10. The expression of normalized frequency for homogeneous cantilever panel ...
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This paper communicates an analytical study on computing the natural frequencies and in-plane deflections caused by static forces in the panel walls using Euler-Bernoulli, Timoshenko, Timoshenko and Goodier, Couple-stress, and Micropolar-Cosserat theory. The study highlights the formulation of the transfer matrix via the state-space method in the spatial domain; from coupled governing equations of motion that arises from the Micropolar-Cosserat theory. This theory captures the novel curvature of edges and moments of the panels at energy density level due to its unique feature of asymmetric shear stresses; that emphasizes the loss of ellipticity of governing equations. The analytical solution of the Micropolar-Cosserat theory yield appropriate results compared to plane-stress simulation of the panels using finite element analysis. KEYWORDS: Couple-stress theory; Micropolar-Cosserat panel; size-dependent behavior; eigenvalue problems.
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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, has been proposed in this paper. To set up a C0-type FEM model, two artificial variables have been introduced in the displacement field to avoid the higher-order derivatives in the higher-order plate theory. The corresponding constraint conditions from the two artificial variables have been enforced effectively through the penalty function method using the reduced integration scheme within the element area. Furthermore, the implementation of displacement continuity conditions at the delamination front has been described using the present FEM theory. Various examples studied in many previous researches have been employed to verify the justification, accuracy and efficiency of the present FEM model. The influences of delamination on the vibration characteristic of composite laminates have been investigated. Especially the variation of ‘curvature of vibration mode’ (i.e., the second-order differential of deflections in vibration mode) caused by delamination has been studied in detail to provide valuable information for the possible identification of delamination. Furthermore, two approaches have been investigated to detect a delamination in laminates by employing this information.
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In this paper, a method of modelling for transverse vibrations of a geometrically segmented slender beam, with and without a crack normal to its axis, has been proposed using the Frobenius technique. There are two segments; one segment is uniform in depth and the other segment has a linearly variable depth. The thickness is uniform along the whole length. In the presence of a crack, the crack section is represented by a rotational spring. Thereby, it is possible to solve both the forward and inverse problems. In the forward problem, the frequencies can be determined by giving the rotational spring stiffness as an input. In the inverse problem, the method can be employed to detect the location and size of a crack by providing the natural frequencies as an input. A number of numerical examples are presented to demonstrate the accuracy of the method. Wherever possible, results have been compared with analytical solutions available in the literature. In the remaining cases, the results are found to be in very good agreement with finite element solutions. In the inverse problems, the error in prediction of crack location is less than 3% and that in size is around 25%.
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A finite element model is developed to study the large-amplitude free vibrations of generally-layered laminated composite beams. The Poisson effect, which is often neglected, is included in the laminated beam constitutive equation. The large deformation is accounted for by using von Karman strains and the transverse shear deformation is incorporated using a higher order theory. The beam element has eight degrees of freedom with the inplane displacement, transverse displacement, bending slope and bending rotation as the variables at each node. The direct iteration method is used to solve the nonlinear equations which are evaluated at the point of reversal of motion. The influence of boundary conditions, beam geometries, Poisson effect, and ply orientations on the nonlinear frequencies and mode shapes are demonstrated.
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A finite element model is presented for the harmonic response of sandwich beams with thin or moderately thick viscoelastic cores. Nonlinear variation of displacements through the thickness of the core is 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 uses standard beam shape functions to construct a 12 d.f. element, SANDWICH12. Driving point impedence predicted by the model agrees closely with experimental data, even at high frequencies. The model also predicts displacements at “resonant” driving frequencies that agree closely with available closed form solutions.
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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 expression for the strain energy. The magnitude of this error is derived. It is 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.
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Geometric non-linearities for large amplitude free and forced vibrations of beam are investigated. Longitudinal displacement and inertia are included in the formulation. The finite element method is used. The harmonic force matrix is introduced and derived. Various out-of-plane and inplane boundary conditions are considered. Results showing the dependence of the amplitude on the frequency ratio and on the strain are presented for different boundary conditions and loads. It is concluded that the effects of longitudinal deformation and inertia are to reduce the non-linearity.
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We make 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 is introduced by the assumed shape functions only in our finite element formulation. The induced axial force is not averaged and the stiffness is defined by the first-, second- and third-order matrices. In the solution stage we use symplectic integration which does not require the linearization of stiffness. All necessary conservative laws during numerical integration are observed. Both free and forced vibrations and damped and undamped vibrations are studied.
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Free vibration analysis of laminated composite beams is 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. Further two-node C1 finite elements of eight degrees of freedom per node, based on the theories, are presented for the free vibration analysis of the beams in this paper. Numerical results have been computed for various length to thickness ratios and for various boundary conditions of the beams and compared with the results of other theories and finite elements available in literature. The comparison study shows that the theories and the finite elements predict the natural frequencies of the laminated composite beams better than the other theories and the finite elements considered.