VIBRATION ANALYSIS OF AN ELASTICALLY
RESTRAINED BEAM
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Bachelor of Technology
In
Mechanical Engineering
By
HAREKRISHNA SAHU
Under the guidance of
PROF R.K.BEHRA
Department of Mechanical Engineering
National Institute of Technology
Rourkela2009
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that that the work in this thesis report entitled “Vibration analysis of
elastically restrained beam” submitted by harekrishna sahu in partial fulfillment of
the requirements for the degree of Bachelor of Technology in Mechanical
Engineering Session 2008-2009 , National Institute of Technology Rourkela, is an
authentic work carried out by him under my supervision and guidance.
Date:
Prof. R.K.BEHERA
Department of Mechanical Engineering
National Institute of Technology
Rourkela - 769008
ACKNOWLEDGEMENT
We deem it a privilege to have been the student of Mechanical Engineering
stream in National Institute of Technology, ROURKELA. Our heartfelt thanks to
PROF R.K.BEHRA, my project guide who helped me to bring out this project in
good manner with his precious suggestion and rich experience. We take this
opportunity to express our sincere thanks to our project guide for cooperation in
accomplishing this project a satisfactory conclusion.
HAREKRISHNA SAHU
Roll no: 10503017
Mechanical Engineering
National Institute of Technology
Rourkela - 769008
CONTENTS
Chapter 1
INTRODUCTION…………………………………………………....1
Chapter 2
THEORY……………………………………………………………..4
2.1ALGOR…………………………………………………………....5
2.2Timoshenko Beam………………………………………………..7
2.3 Typical uses of ALGOR………………………………………….7
Chapter 3
3.1 Problem statement……………………………………………….9
3.2 Setup……………………………………………………………..10
Chapter 4
SOLUTION…………………………………………………………..11
4.1 Design Analysis………………………………………………….12
Chapter 5
RESULTS…………………………………………………………….45
Chapter 6
CONCLUSION………………………………………………………53
Chapter 7
References……………………………………………………………55
Abstract
The problem of free vibration of Timoshenko beams with elastically supported ends can be
solved by using a finite element model which can satisfy all the geometric and natural boundary
conditions of an elastically restrained Timoshenko beam. Timoshenko beam takes into account
shear deformation and rotational inertia effects, making it suitable for describing the behavior of
short beams, sandwich composite beams or beams subject to high-frequency excitation when the
wavelength approaches the thickness of the beam.
The effects of the translational and rotational support flexibilities on the natural frequencies of
free vibrations of Timoshenko beams with non-idealized end conditions are investigated in
detail. Results obtained for the Bernoulli-Euler beam, which is a special case of the present
analysis, show excellent agreement with the available results.
In this project, the model is made by using software ANSYS and analysis part is done by using
the software ALGOR.
Chapter 1
INTRODUCTION
1
The problem of free vibrations of beams with elastically supported ends has been discussed by
seveal investigators.
Chun [l] considered a beam hinged at one end by a rotational spring and free at the other end.
Lee [2] obtained the frequency equation for a uniform beam hinged at one end by a rotational
spring and having a mass attached at the other end. MacBain and Genin [3] investigated the
effect of rotational and translational support flexibilities on the fundamental frequency of an
almost clamped-clamped Bernoulli-Euler beam.
The frequency equation for the normal modes of vibration of uniform beams with linear
translational and rotational springs at one end and having a concentrated mass at the other free
end was obtained and solved by Grant [4] using the Newton-Raphson root finding method.
Goel [5] investigated the vibration problem of a beam with an arbitrarily placed concentrated
mass and elastically restrained against rotation at either end by using Laplace transforms.
The deflection of a rotationally restrained beam was obtained by Nassar and Horton [6] by
successively integrating the differential equaton of equilibrium and satisfying the appropriate
boundary conditions, while Maurizi et al. [7] studied the free vibration of a beam hinged at one
end by a rotational spring and subjected to the restraining action of a translational spring at the
other end.
It is clear that elastically restrained beams have been investigated by many authors, but it is seen
that in all these cases the beams were assumed to be of the Bernoulli-Euler type and thus the
effects of rotary inertia and shear deformation were neglected.
In the present analysis, the problem of free vibration of Timoshenko beams with elastically
supported ends is solved, for the first time, by using the unique finite element mode1 developed
2
by Abbas and Thomas [8-lo] which can satisfy all the geometric and natural boundary conditions
of an elastically restrained Timoshenko beam.
The effects of the translational and rotational support flexibilities on the natural frequencies of
free vibrations of Timoshenko beams with non-idealized end conditions are investigated in
detail.
Results are also shown for six limiting (idealized) end conditions of the Timoshenko beam. The
results obtained for the Bernoulli-Euler beam, which is a special case of the present analysis,
show excellent agreement with the available results.
In my current work, the problem of free vibration of Timoshenko beams with elastically
supported ends can be solved by using a finite element model which can satisfy all the geometric
and natural boundary conditions of an elastically restrained Timoshenko beam.
3
Chapter 2
THEORY
4
2.1Timoshenko Beam
A Timoshenko beam takes into account shear deformation and rotational inertia effects, making
it suitable for describing the behaviour of short beams, sandwich composite beams or beams
subject to high-frequency excitation when the wavelength approaches the thickness of the beam.
The resulting equation is of 4th order, but unlike ordinary beam theory - i.e. Bernoulli-Euler
theory - there is also a second order spatial derivative present. Physically, taking into account the
added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is
a larger deflection under a static load and lower predicted eigen frequencies for a given set of
boundary conditions. The latter effect is more noticeable for higher frequencies as the
wavelength becomes shorter, and thus the distance between opposing shear forces decreases.
If the shear modulus of the beam material approaches infinity - and thus the beam becomes rigid
in shear - and if rotational inertia effects are neglected, Timoshenko beam theory converges
towards ordinary beam theory.
This beam theory, allowing for vibrations, may be described with the coupled linear partial
differential equations [1]
where the dependent variables are u, the translational displacement of the beam, and θ, the
angular displacement. Note that unlike the Euler-Bernoulli theory, the angular deflection is
another variable and not approximated by the slope of the deflection. Also,
5

ρ is the density of the beam material (but not the linear density).

A is the cross section area.

E is the elastic modulus.

G is the shear modulus.

I is the second moment of area.

κ, called the Timoshenko shear coefficient, depends on the geometry. Normally, κ = 5 / 6
for a rectangular section.

w is a distributed load (force per length).
These parameters are not necessarily constants.
Determining the shear coefficient is not straightforward (nor are the determined values widely
accepted, i.e. there's more than one answer), generally it must satisfy:
For a static beam, the equations can be decoupled:
and it is readily seen that the Timeoshenko beam theory for this static case is equivalent to the
Euler-Bernoulli theory when the last term above is neglected, an approximation that is valid
when
where L is the length of the beam and H is the maximum deflection.
6
2.2 ALGOR
ALGOR is a general-purpose multiphysics finite element analysis software package developed
by ALGOR Incorporated for use on the Microsoft Windows and Linux computer operating
systems. It is distributed in a number of different core packages to cater to specifics
applications, such as mechanical event simulation and computational fluid dynamics.
ALGOR is used by many scientists and engineers worldwide. It has found application in
aerospace and it has received many favorable reviews.
2.3 Typical uses of ALGOR

Bending

Mechanical contact

Thermal (conduction, convection, radiation)

Fluid dynamics

Coupled and uncoupled multiphysics
7
Chapter 3
PROBLEM STATEMENT
8
3.1 Problem statement
To find out the natural frequency and mode shape of a elastically restrained beam using a
software ALGOR.The problem consists of a beam of length 100 cm made up of steel material
and four springs attached to it in the position described in the fig . Below .The material used
for the spring are mild steel.
Data:
 Beam

Material : steel

Length : 100 cm
 Breadth : 2cm
 Modulus of elasticity : 0 .724 x10 ^11 n/m^2
 Poisson’s ratio : .334
 Density :2713 kg/m^3
 Springs
 material : mild steel
 diameter: 6mm
 length :30 cm
 modulus of elasticity : 2.1 x10^11 N/M^2
 Poisson ratio : .3
 density:7860kg/m^3
9
3.2 Setup
The setup for the spring and beam
10
Chapter 4
SOLUTION
11
4.1 Design Analysis-:
Summary
Model Information
Analysis Type - Natural Frequenc y (Modal)
Units - Custom - (N , mm, s, deg C, K, V , ohm, A , J)
Model location - D:\ajay \beam- fillet \ Product2
12
Analysis Parameters Information
Process Information Table
Number o f Frequencies To Calculate
5
Cutoff Frequency
0 cycles / s
Frequency Shift
0 cycles / s
Expected Rigid Body Modes
0
Maximum Number of Iterations
32
Number o f Vectors in Subspace
Iteration
Orthogonality Check Print out
0
Convergence Value for Eigen value
1e -00 5
Avoid Stern Sequence Check
No
Avoid Band width Minimization
No
Stop After Stiffness Calculations
No
Attempt to Run Despite Errors
No
Do Not Save Restart Files
No
Displacement Data in Output File
No
Equation Numbers Data in Output
File
Matrices in Output File
No
Element Input Data in Output File
No
Nodal Input Data in Output File
No
None
No
13
Part Information
Element Properties used for :


Part 2

Part 3

Part 4

Part 5
14
Material Information
AISI 1006 Steel cold drawn, –Brick
AISI 1005 Steel –Brick
Load and Constraint Information
Part ID Part Name Element Type Material Name
1 Part 1 Brick AISI 1006 Steel cold drawn
2 Part 2 Brick AISI 1006 Steel cold drawn
3 Part 3 Brick AISI 1005 Steel
4 Part 4 Brick AISI 1006 Steel cold drawn
5 Part 5 Brick AISI 1006 Steel cold drawn
Element Type
Element Type
Brick
Compatibility
Not Enforced
Integration Order
Stress Free Reference Temperature
2nd Order
0 °C
15
Material Information
AISI 1006 Steel, cold drawn –Brick
16
Load and Constraint Information
AISI 1005 Ste el -Brick
17
Constraints
Constraint Set 1: Unnamed
Surface Boundary Conditions
Surface Boundary Conditions
18
Processor Output
Processor Summary
ALGOR (R) Natural Frequency (Modal)
Version 19.36 -WIN 25 -JAN -2007
Copyright (c) 2000 -2007, ALGOR, Inc. All rights reserved.
**** Memory Dynamically Allocated = 518710 KB
------------------------------------------------DATE: MAY 8, 2009
TIME: 11:47 PM
INPUT MODEL: D:\ajay\beam -fillet\Product2
PROGRAM VERSION: 19360009
alg.dll VERSION: 19300071
agsdb_ar.dll VERSION: 18000000
algconfig.dll VERSION: 19200097
algsolve.exe VERSION: 19300071
amgsolve.exe VERSION: 03300000
------------------------------------------------Structural
1**** CONTROL INFORMATION
number of node points (NUMNP) = 38081
19
number of element types (NELTYP) = 5
number of load cases (LL) = 1
number of frequencies (NF) = 5
analysis type code (NDYN) = 1
equations per block (KEQB) = 0
gravitational constant (GRAV) = 9.8146E+03
number of equations (NEQ) = 114081
**** PRINT OF NODAL DATA SUPPRESSED
**** PRINT OF EQUATION NUMBERS SUPPRESSED
**** EQUATION PARAMETERS
total number of equations = 114081
bandwidth = 114030
number of equations in a block = 272
number of blocks = 420
blocking memory (kilobytes) = 531159
available memory (kilobytes) = 531159
**** Hard disk file size information for processor
Available hard disk space on current drive = 14502.098 megabytes
**** Invoking PVSS Sparse Eigen solver ...
**** Symbolic Assembling Using the Row -Hits Matrix Profile ...
**** Assembled in One Block.
**** Real Sparse Matrix Assembly ...
1**** STIFFNESS MATRIX PARAMETERS
20
minimum non -zero diagonal element = 8.8306E+04
maximum diagonal element = 1.2452E+08
maximum/minimum = 1.4101E+03
average diagonal element = 1.0679E+06
the minimum is found at equation 51889: node=17351 Tx
the maximum is found at equation 17012: node=5725 Ty
in the upper off -diagonal matrix:
number of entries in the profile = 11766150
number of symbolic nonzero entries= 3647658
number of real nonzero entries = 3647627
**** Sparse Matrix Assembled in One Block
**** SOLUTION SOUGHT FOR FOLLOWING EIGENPROBLEM
number of equations = 114081
number of eigen values required = 5
rigid body modes ( 0 for none) = 0
Requested working vectors = 15
**** Sparse Matrix Solving ...
""C:\PROGRA1\ALGOR\algsolve.exe""D:\ajay\beamfillet\Product2.dat
**** End Sparse Matrix Solution
**** Participation Factor Calculation
**** Participation factors and modal masses are calculated
for an assumed base acceleration unit loading in each
of the three global directions. Only masses or moments
of inertia on unfixed degrees of freedom are considered.
21
X Direction
Mass available in X direction: 8.2907E -03
Weight available in X direction: 8.1369E+01
Percent used in this direction: 95.38%
Modes used in this direction: 5
**** Modal Effective Mass & Participation Factor X Direction
Mode
Frequency
(HZ)
Modal eff. mass Cumulative mass Participat.
(weight) ( %)
(weight) (%)
Factor
1
2.46257E+1
2.0756E-01 0.26 2.0756E-01
0.26
-4.5987E-03
2
2.85595E+1
7.7093E+01 94.75 7.7301E+01 95.00
8.8628E-02
3
3.16276E+1
2.7466E-01 0.34 7.7576E+01
95.34
5.2901E-03
4
4.89933E+1
1.4513E-03 0.00 7.7577E+01
95.34 -3.8454E-04
5
5.25569E+1
3.3439E-02 0.04 7.7610E+01
95.38 1.8458E-03
Y Direction
Mass available in Y direction: 8.2907E-03
Weight available in Y direction: 8.1369E+01
Percent used in this direction: 97.79
Modes used in this direction: 5
**** Modal Effective Mass & Participation Factor Y Direction
22
Mode Frequency
(HZ)
1 2.46257E+1
2 2.85595E+1
3 3.16276E+1
4 4.89933E+1
5 5.25569E+1
Modal eff.
(weight)
1.0420E-01
3.0222E-01
7.8869E+01
2.4320E-01
5.6002E-02
mass Cumulative mass Participat.
(%)
( weight) (%) Factor
0.13 1.0420E-01 0.13 3.2583E-03
0.37 4.0642E-01 0.50 5.5492E-03
96.93 7.9276E+01 97.43 -8.9644E-02
0.30 7.9519E+01 97.73 4.9779E-03
0.07 7.9575E+01 97.79 2.3887E-03
Z Direction
Mass available in Z direction: 8.2907E -03
Weight available in Z direction: 8.1369E+01
Percent used in this direction: 95.38%
Modes used in this direction: 5
**** Modal Effective Mass & Participation Factor Z Direction
Mode
1
2
3
4
4
5
Frequency
(HZ)
2.46257E+1
2.85595E+1
3.16276E+1
4.89933E+1
4.89933E+1
5.25569E+1
Modal eff. Mass Cumulative
(weight)
(%)
(weight)
7.7140E+01
94.80 7.7140E+01
2.1609E-01
0.27 7.7356E+01
1. 0520E-01 0.13 7.7462E+01
1.3203E-01
0.16 7.7594E+01
1.3203E-01
0.16 7.7594E+01
1.4004E-02
0.02 7.7608E+01
mass
(%)
94.80
95.07
95.20
95.36
95.36
95.38
g = *****
Mode Frequency Modal effective mass (weight)
no.
(HZ)
X -dir.
Y -dir.
Z -dir.
1 2.463E+01 2.0756E -01 1.0420E-01 7.7140E+01
2 2.856E+01 7.7093E+01 3.0222E-01 2.1609E-01
3 3.163E+01 2.7466E-01 7.8869E+01 1.0520E-01
4 4.899E+01 1.4513E-03 2.4320E-01 1.3203E-01
5 5.256E+01 3.3439E-02 5.6002E-02 1.4004E-02
Total mass used 7.7610E+01 7.9575E+01 7.7608E+01
Mass available 8.1369E+01 8.1369E+01 8.1369E+01
23
Participat.
Factor
-8.8655E-02
-4.6923E-03
-3.2739E-03
3.6678E-03
3.6678E-03
1.1945E-03
Percent used 95.38 97.79 95.38
Mode
no.
1
2
3
4
5
Frequency
(HZ)
2.463E+01
2.856E+01
3.163E+01
4.899E+01
5.256E+01
Modal mass (%)
X -dir. Y -dir.
0.26
0.13
94.75
0.37
0.34
96.93
0.00
0.30
0.04
0.07
Mode
no.
1
2
3
4
5
Frequency
Modal Participation Factors
(HZ)
X -dir.
Y -dir.
Z -dir.
2.463E+01 -4.5987E-03 3.2583E-03 -8.8655E-02
2.856E+01 8.8628E-02 5.5492E-03 -4.6923E-03
3.163E+01 5.2901E-03 -8.9644E-02 -3.2739E-03
4.899E+01 -3.8454E-04 4.9779E-03
3.6678E-03
5.256E+01 1.8458E-03 2.3887E-03
1.1945E-03
Z -dir.
94.80
0.27
0.13
0.16
0.02
Cumulative mass (%)
X -dir. Y -dir. Z -dir.
0.26
0.13
94.80
95.00
0.50
95.07
95.34
97.43
95.20
95.34
97.73
95.36
95.38
97.79
95.38
1**** PRINT OF NATURAL FREQUENCIES
mode
number
circular
frequency
frequency
period
(rad/sec)
(Hertz)
(sec)
------ ----------- ----------- ----------1
1.5473E+02
2.4626E+01
4.0608E-02
2
1.7944E+02
2.8560E+01
3.5015E-02
3
1.9872E+02
3.1628E+01
3.1618E-02
4
3.0783E+02
4.8993E+01
2.0411E-02
5
3.3022E+02
5.2557E+01
1.9027E-02
1**** TEMPORARY FILE STORAGE (MEGABYTES)
---------------------------------UNIT NO. 7 : 4.361
UNIT NO. 8 : 1.743
UNIT NO. 9 : 0.875
UNIT NO. 10 : 0.000
UNIT NO. 11 : 0.000
24
UNIT NO. 12 : 0.000
UNIT NO. 14 : 4.713
UNIT NO. 15 : 0.000
UNIT NO. 51 : 5.387
UNIT NO. 52 : 141.741
UNIT NO. 54 : 0.435
UNIT NO. 55 : 13.479
UNIT NO. 56 : 26.959
UNIT NO. 58 : 0.870
TOTAL : 200.565 Megabytes
Processor Log
ALGOR (R) Natural Frequency (Modal)
Version 19.36 -WIN 25 -JAN -2007
Copyright (c) 2000 -2007, ALGOR, Inc. All rights reserved.
Structural
38081 5 1 5 0 1
**** Dynamic linear modal analysis
**** Memory Dynamically Allocated = 518710 KB
Options executed are:
SUPELM
SUPNOD
SUPTRN
25
processing ...
**** OPENING TEMPORARY FILES
NDYN = 1
DATE: MAY 8, 2009
TIME: 11:47 PM
INPUT MODEL: D:\ajay\beam -fillet\Product2
PROGRAM VERSION: 19360009
alg.dll VERSION: 19300071
algconfig.dll VERSION: 19200097
algsolve.exe VERSION: 19300071
amgsolve.exe VERSION: 03300000
**** BEGIN NODAL DATA INPUT
38081 NODES
27000 (71.%) nodes
18000 (47.%) nodes
9000 (24.%) nodes
0 ( 0.%) nodes
114081 DOFS
**** END NODAL DATA INPUT
**** BEGIN TYPE -8 data INPUT
PART 1 CONTAINING 8198 ELEMENTS
26
6000 (73.%) elements
5000 (61.%) elements
4000 (49.%) elements
3000 (36.%) elements
2000 (24.%) elements
1000 (12.%) elements
0 ( 0.%) elements
**** end TYPE -8 data INPUT
**** BEGIN TYPE -8 data INPUT
PART 2 CONTAINING 9013 ELEMENTS
7000 (77.%) elements
6000 (66.%) elements
5000 (55.%) elements
4000 (44.%) elements
3000 (33.%) elements
2000 (22.%) elements
1000 (11.%) elements
0 ( 0.%) elements
**** end TYPE -8 data INPUT
**** begin type -8 data INPUT
PART 3 CONTAINING 26483 ELEMENTS
24000 (90.%) elements
22000 (83.%) elements
27
20000 (75.%) elements
18000 (68.%) elements
16000 (60.%) elements
14000 (53.%) elements
12000 (45.%) elements
10000 (38.%) elements
8000 (30.%) elements
6000 (23.%) elements
4000 (15.%) elements
2000 ( 8.%) elements
0 ( 0.%) elements
**** end TYPE -8 data INPUT
**** BEGIN TYPE -8 data INPUT
PART 4 CONTAINING 215 ELEMENTS
**** end TYPE -8 data INPUT
**** BEGIN TYPE -8 data INPUT
PART 5 CONTAINING 220 ELEMENTS
**** end TYPE -8 data INPUT
**** EQUATION PARAMETERS
total number of equations = 114081
bandwidth = 114030
number of equations in a block = 272
number of blocks = 420
28
blocking memory (kilobytes) = 531159
available memory (kilobytes) = 531159
**** Hard disk file size information for processor:
Available hard disk space on current drive = 14502.098 megabytes
**** BEGIN MASS INPUT
**** END MASS INPUT
**** Invoking PVSS Sparse Eigensolver ...
**** Symbolic Assembling Using the Row -Hits Matrix Profile ...
**** Assembled in One Block.
**** Real Sparse Matrix Assembly ...
in the upper off -diagonal matrix:
number of entries in the profile = 11766150
number of symbolic nonzero entries= 3647658
number of real nonzero entries = 3647627
number of real nonzero entries = 3647627
**** Sparse Matrix Assembled in One Block
**** Sparse Matrix Solving ...
**** End Sparse Matrix Solution
**** PRINT OF DISPLACEMENT OUTPUT SUPPRESSED
**** END DISPLACEMENT OUTPUT
**** Participation Factor Calculation
Weight available in X direction: 8.1369E+01
Percent used in this direction: 95.38%
Modes used in this direction: 5
29
**** Modal Effective Mass & Participation Factor X Direction
Mode
Frequency
(HZ)
2.46257E+1
2.85595E+1
3.16276E+1
4.89933E+1
5.25569E+1
1
2
3
4
5
Modal eff. mass
(weight)
2.0756E-01
7.7093E+01
2.7466E-01
1.4513E-03
3.3439E-02
(%)
0.26
94.75
0.34
0.00
0.04
Cumulative
(weight)
2.0756E-01
7.7301E+01
7.7576E+01
7.7577E+01
7.7610E+01
mass Participat.
(%)
Factor
0.26 -4.5987E-03
95.00 8.8628E-02
95.34 5.2901E-03
95.34 -3.8454E-04
95.38 1.8458E-03
Weight available in Y direction: 8.1369E+01
Percent used in this direction: 97.79%
Modes used in this direction: 5
**** Modal Effective Mass & Participation Factor Y Direction
Mode
1
2
3
4
5
Frequency Modal eff.
(HZ)
(weight)
2.46257E+1 1.0420E-01
2.85595E+1 3.0222E-01
3.16276E+1 7.8869E+01
4.89933E+1 2.4320E-01
5.25569E+1 5.6002E-02
mass
(%)
0.13
0.37
96.93
0.30
0.07
Cumulative mass Participat.
(weight) (%) Factor
1.0420E-01 0.13 3.2583E-03
4.0642E-01 0.50 5.5492E-03
7.9276E+01 97.43 -8.9644E-02
7.9519E+01 97.73 4.9779E-03
7.9575E+01 97.79 2.3887E-03
Weight available in Z direction: 8.1369E+01
Percent used in this direction: 95.38%
Modes used in this direction: 5
**** Modal Effective Mass & Participation Factor Z Direction
Mode
1
2
3
4
5
Frequency Modal eff.
(HZ)
(weight)
2.46257E+1 7.7140E+01
2.85595E+1 2.1609E-01
3.16276E+1 1.0520E-01
4.89933E+1 1.3203E-01
5.25569E+1 1.4004E-02
mass
(%)
94.80
0.27
0.13
0.16
0.02
30
Cumulative
(weight)
7.7140E+01
7.7356E+01
7.7462E+01
7.7594E+01
7.7608E+01
mass Participat.
(%) Factor
94.80 -8.8655E-02
95.07 -4.6923E-03
95.20 -3.2739E-03
95.36 3.6678E-03
95.38 1.1945E-03
Frequencies
mode
number
circular
frequency
frequency
(rad/sec)
(Hertz)
------ ----------- --------- -----1
1.5473E+02
2.4626E+01
2
1.7944E+02
2.8560E+01
3
1.9872E+02
3.1628E+01
4
3.0783E+02
4.8993E+01
5
3.3022E+02
5.2557E+01
Product2.t7 = 4465.828 kilobytes
Product2.t8 = 1785.090 kilobytes
Product2.t9 = 895.781 kilobytes
Product2.t10 = 0.000 kilobytes
Product2.t11 = 0.312 kilobytes
Product2.t12 = 0.070 kilobytes
Product2.t13 = 0.000 kilobytes
Product2.t14 = 4826.609 kilobytes
Product2.t15 = 0.000 kilobytes
Product2.t51 = 5516.125 kilobytes
Product2.t52 = 145143.039 kilobytes
Product2.t54 = 445.652 kilobytes
31
period
(sec)
--------4.0608E-02
3.5015E-02
3.1618E-02
2.0411E-02
1.9027E-02
Product2.t55 = 13802.914 kilobytes
Product2.t56 = 27605.828 kilobytes
Product2.t58 = 891.258 kilobytes
total temporary disk storage (megabytes) = 200.565
Product2.ml = 9.012 kilobytes
Product2.mo = 8925.281 kilobytes
**** BEGIN DELETING TEMPORARY FILES
Processing completed for model:
[D:\ajay\beam -fillet\Product2]
**** TEMPORARY FILES DELETED
**** END OF SUCCESSFUL EXECUTION
**** Total actual hard disk space used = 209.290 megabytes
Sub -total elapsed time = 4.817 minutes
Frequency Analysis
Frequencies = 5
mode
number
circular
frequency
(rad/sec)
------ ----------1
1.54727980187160D+02
2
1.79444848658054D+02
3
1.98721912047240D+02
4
3.07834162947196D+02
5
3.30224605226713D+02
32
Other Analysis
The analysis out put file (D:\ajay \beam - fillet \Product2.MT X) w as not
found.
Weight and Center of Gravity Analysis
The weight and center of gravity analysis output file (D:\ajay \beam -fillet
\Product 2.WCG) w as not found.
Meshing Results
Par t 1
Status : the part successfully meshed.
Surface Mesh Statistics
Solid Mesh Statistics
33
Log File
_________________________________________________
SOLID MESH GENERATION BEFORE ANALYSIS
_________________________________________________
PROGRAM WILL USE THE FOLLOWING FILES:
Input: D:\ajay\beam -fillet\Product2.xgn
Output: D:\ajay\beam -fillet\Product2.esd
COMMAND LINE:
C:\Program Files\ALGOR\SolidX.exe -b=0 -zw=2 D:\ajay\beam fillet\Product2 -d=0 -u=13 -c=2 -t=1 -or=2033160 -op=1771134 za= -1
TYPE OF OPERATION:
Meshing only surface defined by part 1
Generating bricks, wedges, pyramids and tetrahedral elements
Automatically minimizing aspect ratio of solid elements
FINAL STATISTICS:
Elements built (4,5,6,8 noded): 8198 ( 5685, 2381, 92, 40 )
34
Volume (4,5,6,8 noded %): 30865.187066 ( 52.81, 39.49, 3.33,
4.39 )
Number of nodes: 5637
Length ratios (avg) 5.0, 2.9, 2.2, 1.7
Length ratios (max) 1007.6, 24.8, 6.1, 6.3
Aspect ratio: unconstrained ( 7.0, 2.7, 1.4, 1.2 )
Average aspect ratios: ( 1.3, 1.2, 1.1, 1.0 )
Used direct transfer of global surface data as input.
Total used memory: 11.76 MB
Number of restarts: 0
Elapsed time: 0 minutes 6 seconds
Par t 2
Status : the part successfully meshed.
Surface Mesh Statistics
35
Solid Mesh Statistics
Mesh type
Mix of bricks , wedges , pyramids and tetrahedral
Watertight
Ye s
Mesh h as micro holes
Total nodes
No
71 86
Log File
EXAMINING SURFACE MESH FOR ANOTHER PART
_________________________________________________
PROGRAM WILL USE THE FOLLOWING FILES:
Input: D:\ajay\beam -fillet\Product2.xgn
Output: D:\ajay\beam -fillet\Product2.esd
COMMAND LINE:
C:\Program Files\ALGOR\SolidX.exe -b=0 -zw=1 D:\ajay\beam fillet\Product2 -u=13 -c=2 -t=1 -or=788120 -op=1115508 -za=1 zg=1,2,3,4,5
TYPE OF OPERATION:
36
Meshing only surface defined by part 2
Generating bricks, wedges, pyramids and tetrahedral elements
Automatically minimizing aspect ratio of solid elements
FINAL STATISTICS:
Elements built (4,5,6,8 noded): 9013 ( 5695, 2797, 274, 247 )
Volume (4,5,6,8 noded %): 30705.841363 ( 37.92, 35.10, 8.35,
18.64 )
Number of nodes: 7186
Length ratios (avg) 5.0, 3.4, 2.0, 1.6
Length ratios (max) 1743.9, 33.6, 3.8, 6.3
Aspect ratio: unconstrained ( 7.5, 3.7, 1.2, 1.2 )
Average aspect ratios: ( 1.3, 1.3, 1.1, 1.0 )
Used direct transfer of global surface data as input.
Total used memory: 12.28 MB
Number of restarts: 0
Elapsed time: 0 minutes 7 seconds
Part 3
Status: the part successfully meshed.
Surface mesh statistics
37
Solid mesh statistics
38
Log File
EXAMINING SURFACE MESH FOR ANOTHER PART
PROGRAM WILL USE THE FOLLOWING FILES:
Input: D:\ajay\beam -fillet\Product2.xgn
Output: D:\ajay\beam -fillet\Product2.esd
COMMAND LINE:
C:\Program Files\ALGOR\SolidX.exe -b=0 -zw=1 D:\ajay\beam fillet\Product2 -u=13 -c=2 -t=1 -or=788120 -op=1115508 -za=1 zg=1,2,3,4,5
39
TYPE OF OPERATION:
Meshing only surface defined by part 2
Generating bricks, wedges, pyramids and tetrahedral elements
Automatically minimizing aspect ratio of solid elements
FINAL STATISTICS:
Elements built (4,5,6,8 noded): 9013 ( 5695, 2797, 274, 247 )
Volume (4,5,6,8 noded %): 30705.841363 ( 37.92, 35.10, 8.35,
18.64 )
Number of nodes: 7186
Length ratios (avg) 5.0, 3.4, 2.0, 1.6
Length ratios (max) 1743.9, 33.6, 3.8, 6.3
Aspect ratio: unconstrained ( 7.5, 3.7, 1.2, 1.2 )
Average aspect ratios: ( 1.3, 1.3, 1.1, 1.0 )
Used direct transfer of global surface data as input.
Total used memory: 12.28 MB
Number of restarts: 0
Elapsed time: 0 minutes 7 seconds
Part 4
Status : the part successfully meshed.
40
41
Solid Mesh Statistics
Log File
GENERATING SOLID MESH FOR ANOTHER PART
PROGRAM WILL USE THE FOLLOWING FILES:
Input: D:\ajay\beam -fillet\Product2.xgn
Output: D:\ajay\beam -fillet\Product2.esd
COMMAND LINE:
42
C:\Program Files\ALGOR\SolidX.exe -b=0 -zw=2 D:\ajay\beam fillet\Product2 -d=0 -u=13 -c=2 -t=1 -or=2098696 -op=1836670 za=1
TYPE OF OPERATION:
Meshing only surface defined by part 4
Generating bricks, wedges, pyramids and tetrahedral elements
Automatically minimizing aspect ratio of solid elements
FINAL STATISTICS:
Elements built (4,5,6,8 noded): 215 ( 163, 33, 2, 17 )
Volume (4,5,6,8 noded %): 153.864060 ( 32.04, 14.77, 2.25,
50.98)
Number of nodes: 165
Length ratios (avg) 23.4, 4.8, 2.2, 4.0
Length ratios (max) 1203.5, 18.9, 2.3, 4.7
Aspect ratio: unconstrained ( 7.4, 2.2, 1.1, 1.1 )
Average aspect ratios: ( 1.6, 1.4, 1.1, 1.1 )
Used direct transfer of global surface data as input.
Total used memory: 1.09 MB
Number of restarts: 0
Elapsed time: 0 minutes 0 seconds
Par t 5
Status : the part successfully meshed.
43
Surface Mesh Statistics
Solid Mesh Statistics
Log File
GENERATING SOLID MESH FOR ANOTHER PART
_________________________________________________
PROGRAM WILL USE THE FOLLOWING FILES:
Input: D:\ajay\beam -fillet\Product2.xgn
Output: D:\ajay\beam -fillet\Product2.esd
COMMAND LINE:
44
C:\Program Files\ALGOR\SolidX.exe -b=0 -zw=2 D:\ajay\beam fillet\Product2 -d=0 -u=13 -c=2 -t=1 -or=2164232 -op=1902206 za=1
TYPE OF OPERATION:
Meshing only surface defined by part 5
Generating bricks, wedges, pyramids and tetrahedral elements
Automatically minimizing aspect ratio of solid elements
FINAL STATISTICS:
Elements built (4,5,6,8 noded): 220 ( 175, 29, 1, 15 )
Volume (4,5,6,8 noded %): 150.390332 ( 35.31, 13.37, 1.66,49.72)
Number of nodes: 168
Length ratios (avg) 9.8, 4.2, 2.2, 3.9
Length ratios (max) 91.7, 11.3, 2.2, 4.6
Aspect ratio: unconstrained ( 2.8, 2.1, 1.2, 1.2 )
Average aspect ratios: ( 1.4, 1.4, 1.2, 1.1 )
Used direct transfer of global surface data as input.
Total used memory: 1.13 MB
Number of restarts: 0
Elapsed time: 0 minutes 1 seconds
45
Chapter 5
RESULTS
46
FIG:1
The above figure shows the model that is being used for analysis.
47
FIG:2
This figure indicates the boundary condition.
48
FIG:3
This figure shows the 1st mode of the analysis.
49
FIG:4
This figure shows the second mode of the analysis.
50
FIG:5
This figure shows the third mode of the analysis.
51
FIG:6
This figure shows the fourth mode of the analysis.
52
FIG:7
This figure shows the fifth mode of the analysis.
53
Chapter 6
CONCLUSION
54
The analysis of the given problem by using the software ALGOR is available for the
determination of natural frequencies and the associated mode shapes of a elastically restrained
beam.Moreover.it is found that it is better that the conventional finite element method(FEM) in
saving much computer time and also better than the existing analytical method for tackling
problems.
55
Chapter 7
REFERENCE
56
References
1. Mechanical vibration by Benaroya
2. Vibration problems in engineering by S. P. TIMOSHENKO, D. H. YOUNG and W.
WEAVER
3. K. R. CHUN 1972 Journal of Applied Mechanics 39, 1154-l 155. Free vibration of
a beam with one end spring-hinged and the other free.
4. T. W. LEE 1973 Journal of AppZied Mechanics 40, 813-815. Vibration
frequencies for a uniform beam with one end spring-hinged and carrying a mass at the
other free end.
5. J. C. MACBAIN and J. GENIN 1973 Journal of the Franklin Institute 296, 259273. Effect of a support flexibility on the fundamental frequency of vibrating beams.
6. D. A. GRANT 1975 Journal of Applied Mechanics 42, 878-880. Vibration
frequencies for a uniform beam with one end elastically supported and carrying a
mass at the other end.
7. R. P. GOEL 1976 Journal of Sound and Vibration 47, 9-14. Free vibrations of a
beam-mass system with elastically restrained ends.
8. User manual and guide ALGOR Version 4.0008.12.11
9. E. M. NASSAR and W. H. HORTON 1976 American Institute of Aeronautics
and Astronautics Journal 14, 122-123. Static deflection of beams subjected to
elastic rotational restraints.
10. M. J. MAURIZI, R. E. Rossr and J. A. REYES 1976 Journal of Sound and
Vibration 48,565-568.Vibration frequencies for a uniform beam with one end
spring-hinged and subjected to a translational restraint at the other end.
11. B. A. H. ABBAS 1973 M.Sc. Thesis, University of Surrey. Vibration
characteristics of Timoshenko beam.
12. J. THOMAS and B. A. H. ABBAS 1975 Journal of Sound and Vibration 41, 291299. Finite element model for dynamic analysis of Timoshenko beam.
13. B. A. H. ABBAS 1979 Aeronautical Journal 450-453. Simple finite elements for
dynamic analysis of thick pre-twisted blades.
14. D. H. YOUNG and R. P. FELGAR 1949 The University of Texas, Austin,
Publication No. 4913, (Engineering Research Series, No. 44). Tables of
characteristic functions representing normal modes of vibration of a beam.
15. S. P. TIMOSHENKO, D. H. YOUNG and W. WEAVER, JR 1974 Vibration
Problems in Engineering. New York: Wiley.
16. P. W. TRAILL-NASH and A. R. COLLAR 1953 Quarterly Journal of
Mechanics and Applied Mathematics 6, 186-222. The effects of shear flexibility
and rotary inertia on the bending vibration of beams.
57