Development of 2D to 3D FEA Conversion Factors for Natural
Frequencies of Beams Under Transverse Vibrations
by
James Austin Cosby
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Dr. Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
11, 2013
i
CONTENTS
LIST OF TABLES ............................................................................................................ iii
LIST OF FIGURES .......................................................................................................... iv
LIST OF SYMBOLS ......................................................................................................... v
KEYWORDS .................................................................................................................... vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. INTRODUCTION ....................................................................................................... 1
2. THEORY/METHODOLOGY ..................................................................................... 4
2.1
FEA Analysis and Modeling .............................................................................. 8
3. RESULTS and DISCUSSIONS ................................................................................ 16
3.1
Results Verification and Validation ................................................................. 16
3.2
Overview of Results ......................................................................................... 17
4. Conclusions................................................................................................................ 22
5. References.................................................................................................................. 23
6. Appendix A................................................................................................................ 24
6.1
Fixed – Fixed Solid element ANSYS INPUT file .......................................... 24
6.2
Fixed Fixed Beam Elemen ANSYS input file ................................................. 27
6.3
Fixed – Free Solid Element ANSYS Input File ............................................... 31
6.4
Fixed – Free Beam Element ANSYS Input File .............................................. 34
6.5
Simply Supported Solid Element ANSYS Input File ...................................... 38
6.6
Simply Supported Beam Element ANSYS Input File ..................................... 42
7. Appendix B ................................................................................................................ 46
8. Appendix C ................................................................................................................ 49
9. Appendix D................................................................................................................ 51
10. Appendix E ................................................................................................................ 52
ii
LIST OF TABLES
Table 1 Beam Boundary Conditions and Their Constraints .............................................. 7
Table 2 Titanium Material Properties Used for Beam..................................................... 12
Table 3 Geometric Input for Each FEA Case .................................................................. 13
Table 4 Number of Modes and Frequency Range for Each Case.................................... 14
Table 5 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Fixed Beam
......................................................................................................................................... 16
Table 6 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Free Beam
......................................................................................................................................... 16
Table 7 Frequency Comparison of Hand Calcs vs. FEA for a Simply Supported Beam 16
Table 8 Conversion Equations for Modes 1 and 2 .......................................................... 19
Table 9 Conversion Equations for Modes 3 and 4 .......................................................... 19
Table 10 Frequency Comparison of Mode 1 for a Fixed-Fixed Beam 12 Inches Long .. 20
Table 11 Test Case 2 Frequency Comparison for Mode 4 .............................................. 21
iii
LIST OF FIGURES
Figure 1 Typical Spring-Mass System .............................................................................. 1
Figure 2 Spring-Mass System with a Harmonic Force Input ............................................ 2
Figure 3 An Aircraft wing structure [1] and a Truss Bridge [2] Both Utilize Beams ....... 3
Figure 4 Beam under Transverse Loading......................................................................... 4
Figure 5 Beam Boundary Conditions and Closed-Form Natural Frequency Solutions .... 7
Figure 6 Beam Element ..................................................................................................... 8
Figure 7 Example of 3D FEA Element............................................................................ 11
Figure 8 Example of Beam Element Model .................................................................... 13
Figure 9 Example of Solid Element Model ..................................................................... 14
Figure 10 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Solid Elements
......................................................................................................................................... 15
Figure 11 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Beam Elements
......................................................................................................................................... 15
Figure 12 Frequency Comparisons W/ Fixed-Fixed Boundary Conditions .................... 17
Figure 13 Frequency Comparisons W/ Fixed-Free Boundary Conditions ...................... 18
Figure 14 Frequency Comparisons W/ Simply Supported Boundary Conditions ........... 18
Figure 15 Test Case 2 I-Beam Cross-section .................................................................. 20
iv
LIST OF SYMBOLS
h
Height of the Beam
𝐼𝑛𝑐ℎ𝑒𝑠
L
Length of the Beam
𝐼𝑛𝑐ℎ𝑒𝑠
M
Bending Moment
𝑖𝑛 ∙ 𝑙𝑏
F
Force Applied to beam
𝜌
Density
A
Cross-Sectional Area
𝑖𝑛2
u
Distance
𝑖𝑛3
E
Elastic Modulus
I
Area Moment of Inertia
V
Shear Force
𝑙𝑏/𝑖𝑛2
θ
Rotation
radians
m
Mass
𝑆𝑙𝑢𝑔𝑠
k
Spring Constant
𝐿𝑏/𝑖𝑛
𝜔𝑛
Natural Frequency
𝐻𝑒𝑟𝑡𝑧
𝜔
Forcing Frequency
𝐻𝑒𝑟𝑡𝑧
𝑟
Frequency Ratio
𝜗
Displacement
𝑙𝑏
𝑙𝑏/𝑖𝑛3
𝑙𝑏/𝑖𝑛2
𝑖𝑛4
𝑈𝑛𝑖𝑡𝑙𝑒𝑠𝑠
𝐼𝑛𝑐ℎ𝑒𝑠
v
KEYWORDS
Modal Analysis, Euler-Bernulli Beam Theory, ANSYS, Beam Elements, Solid Elements,
Conversion Factor, Finite Element Method
vi
ACKNOWLEDGMENT
Thank you to my wonderful wife and darling daughter for supporting me through this
process.
vii
ABSTRACT
In this project conversion factors for natural frequency predictions from 2D beam FEA
beam elements to 3D solid FEA element analysis was obtained. The conversion factors
allow an engineer to run a model using beam elements and convert the frequency solution
to a more accurate answer based on geometry while keeping solution time to a minimum.
To do so, FEA cases were run for beams with slenderness ratios between 20 and 4. The
beams studied had a square cross-section and had fixed-fixed, fixed-free, and simplysupported boundary conditions. To check the accuracy of the conversion factors 2 test
cases were ran. The first test had a square cross-section and simply supported boundary
conditions: the second test case had an I-Beam cross-section and fixed free boundary
conditions. In both cases the element error for beams with slenderness ratios below 10 was
decreased to below 10% from as high as 40%. Classical beam vibration theory assumes
that shear deformation and stress through the thickness of the beam is very small. This
assumption is good for very smaller slenderness ratios, but classical beam theory hand
calculations for natural frequencies diverge from the true solution for thicker beams.
viii
1. INTRODUCTION
When designing engineering structures, be it an airfoil in a gas turbine or a ventilation
stack of a factory, one must be mindful of vibratory failure modes. Vibration is the
oscillation of a mechanical structure. These vibratory oscillations can occur either freely
or forced. One of the simplest systems that can be used to describe free and forced
vibration is a spring-mass system shown in Figure 1.
Figure 1 Typical Spring-Mass System
The equation of motion for the system in Figure 1 is Equation 1, where k is the spring
constant and m is the mass of the system. .
𝑚𝑥̈ + 𝑘𝑥 = 0
[1]
The frequency at which the system will oscillate, or natural frequency, is described in
Equation 2.
𝜔𝑛 = √
𝑘
𝑚
[2]
The natural frequency of a system is a key parameter in the design process of a structure.
It becomes more important when an external force is applied to the system. Figure 2
shows a spring-mass system with an external, harmonic input also applied where 𝐹0 is the
amplitude, ω is the forcing frequency and t is time.
1
Figure 2 Spring-Mass System with a Harmonic Force Input
The equation of motion for this system is Equation 2.
𝑚𝑥̈ + 𝑘𝑥 = 𝐹0 sin 𝜔𝑡
[3]
The solution to this differential equation is Equation 3 assuming the initial boundary
conditions of 𝑥(0) = 𝑥̇ (0) = 0.
𝑥(𝑡) =
𝐹0 1
(sin 𝑤𝑡 − 𝑟 sin 𝑤𝑛 𝑡)
𝑘 1 − 𝑟2
[4]
Where r, also known as the frequency ratio, is:
𝑟=
𝜔
𝜔𝑛
It is noticed that if r=1, the forcing frequency and the natural frequency are equal, then the
displacement solution goes to infinity. This occurrence is known as a resonance and is a
major concern when designing engineering structures. Knowing the natural frequencies of
your structure is crucial to robust design that will not fail.
The simple spring-mass system that has been discussed up until now is also known
as a lumped-parameter system. In reality most structures, such as beams, are continuous
systems that have more complex equations of motion and have multiple natural
frequencies. Beams are a fundamental engineering structures used in multiple disciplines
of engineering such as civil and aerospace. The primary use of beam is to withstand
loading that induces bending or buckling. Figure 3 shows two such examples of how
beams are used in engineering structures.
2
Figure 3 An Aircraft wing structure [1] and a Truss Bridge [2] Both Utilize Beams
To solve these more complex problems finite element analysis software is used.
Finite element analysis software packages, such as ANSYS, have beam element classes.
These classes of elements are governed by Euler-Bernoulli beam theory. Beam elements
take much less time to solve than solid elements. However, as the beam becomes thicker
the accuracy of the solution decreases due to the assumptions of beam theory. To avoid
inaccuracy, it is recommended to use solid elements to model thick beams. This is very
computationally expensive compared to using beam elements.
It is the purpose of this project to find conversion factors between beam elements
and solid elements using ANSYS. The conversion factors will be for transverse vibration
of beams with simply supported, fixed-fixed and fixed-free boundary conditions. This
would allow an engineer to convert his beam-element solution based on the geometry of
the beam, thus reducing solution time without sacrificing solution accuracy.
3
2. THEORY/METHODOLOGY
Euler-Bernoulli beam theory for transverse vibration was developed as a simplification of
the theory of elasticity. The assumptions for classical beam theory are as follows [Ref 3]:
1.
2.
3.
4.
5.
6.
7.
The beam has uniform properties
The beam is slender (L/h is small)
The beam obeys Hooke’s Law
There is no axial load
Plane sections remain plane during motion
The plane of motion is the same as the beam symmetry plane
Shear Deformation is Negligible
Figure 4 shows a beam under transverse loading.
Figure 4 Beam under Transverse Loading
4
The bending moment is described by Equation 5.
𝑑2 𝑢
𝑀(𝑥, 𝑡) = 𝐸𝐼(𝑥) 2
𝑑𝑥
[5]
Where u is a distance from its equilibrium position at point x on the beam, M is the bending
moment, E is the elastic modulus and I is the area moment of inertia. Summing the forces
in the Y direction yields.
𝜌𝐴
𝜕 2𝑢
𝜕𝑉
= (𝑉 +
𝑑𝑥) − 𝑉 + 𝑓𝑑𝑥
2
𝜕𝑡
𝜕𝑥
[6]
Where V is the shear force, f is the applied force, 𝜌 is the density and A is the crosssectional area. Carrying out the subtraction, Equation 6 becomes Equation 7
𝜌𝐴
𝜕 2𝑢
𝜕𝑉
= (𝑓 + ) 𝑑𝑥
2
𝜕𝑡
𝜕𝑥
[7]
Summing the moments about point O yields 8
(𝑀 +
𝜕𝑀
𝜕𝑉
𝑑𝑥
) 𝑑𝑥 − 𝑀 + (𝑉 +
𝑑𝑥) 𝑑𝑥 + (𝑓𝑑𝑥)
=0
𝜕𝑥
𝜕𝑥
2
[8]
If (𝑑𝑥)2is much smaller than 𝑑𝑥, Equation 8 becomes Equation 9 assuming A and I are
both constant.
𝜕 2𝑢
𝜕 4𝑢
2
+
𝑐
=0
𝜕𝑡 2
𝜕𝑥 4
Where
𝑐2 =
5
𝐸𝐼
𝜌𝐴
[9]
Equation 8 were simplified by assuming that shear deformation is negligible compared to
𝑢(𝑥, 𝑡). This assumption holds well with slender beams, however, when ℎ/𝐿 or 𝑤/𝐿 is
less than 10, this assumption becomes less valid.
Using the method of separation of variables to solve Equation 9 yields Equation 10 where
u(x, t) = F(x)G(t).
𝑐2
𝐹′′′′(𝑥)
𝐺̈ (𝑡)
=−
= 𝜔2
𝐹(𝑥)
𝐺(𝑡)
[10]
𝐹 ′′′′ (𝑥) − 𝛽 4 𝐹(𝑥) = 0
[11]
This yields Equation 11
Where,
𝛽4 =
𝜔2
𝑐2
[12]
Assuming the solution of the form 𝐹(𝑥) = 𝐶𝑒 𝜆𝑥 gives Equation 13.
𝐹(𝑥) = 𝐷1 𝑐𝑜𝑠ℎ𝛽𝑥 + 𝐷2 𝑠𝑖𝑛ℎ𝛽𝑥 + 𝐷3 𝑐𝑜𝑠𝛽𝑥 + 𝐷4 𝑠𝑖𝑛𝛽𝑥
[13]
To obtain the answer to Equation 9, the boundary conditions of the end of the beams must
be taken into account. The 4 different types of boundary condition and their constraints
are shown in Table 1.
6
Table 1 Beam Boundary Conditions and Their Constraints
End Condition
Free
Fixed
Simply
Supported
Sliding
Deflection
Slope
Bending Moment
Unconstrained Unconstrained
0
0
0
Unconstrained
Shear Force
0
Unconstrained
0
Unconstrained
0
Unconstrained
Unconstrained
0
Unconstrained
0
Where Deflection is 𝑢(𝑥, 𝑡), slope is 𝜕𝑢/𝜕𝑥, bending moment is 𝐸𝐼𝜕 2 𝑢/𝜕𝑥 2 and shear
force is 𝜕(𝐸𝐼𝜕 2 𝑢/𝑑𝑥 2 )/𝜕𝑥. For this project, 3 different beams will be considered using
fixed-fixed, fixed-free, and simply-supported boundary conditions. Using Equation 13 and
the boundary conditions in table 1, solutions for the 3 different beams can be found in
Figure 5.
Figure 5 Beam Boundary Conditions and Closed-Form Natural Frequency Solutions
7
2.1 FEA Analysis and Modeling
For this project ANSYS FEA models were created using both beam elements and solid
elements. Beam element equations are derived very similarly to that hand calculation
method seen in the in Methodology portion of this paper. A beam element is shown in
Figure 6.
Figure 6 Beam Element
The displacement vector for a beam element is defined in Equation 10.
𝑣1
𝜃1
𝑽 = [𝑣 ]
2
𝜃2
8
[14]
Boundary Conditions for the beam element are as follows:
𝑣(0, 𝑡) = 𝑣1 (𝑡)
𝜕𝑣(0, 𝑡)
= 𝜃1 (𝑡)
𝜕𝑥
𝑣(𝐿, 𝑡) = 𝑣2 (𝑡)
𝜕𝑣(0, 𝑡)
= 𝜃2 (𝑡)
𝜕𝑥
The solution form for the deflection of the beam element is assumed as follows:
𝑣(𝑥, 𝑡) = 𝑎(𝑡) + 𝑏(𝑡)𝑥 + 𝑐(𝑡)𝑥 2 + 𝑑(𝑡)𝑥 3
[15]
Applying the boundary conditions and solving, the following is obtained:
𝑎(𝑡) = 𝑣1(𝑡)
𝑏(𝑡) = 𝜃1 𝑡
1
[−3𝑣1 (𝑡) − 2𝜃1 (𝑡) + 3𝑣2 (𝑡) − 𝐿𝜃2 (𝑡)]
𝐿2
1
𝑑(𝑡) = 3 [2𝑣1 (𝑡) + 𝐿𝜃1 (𝑡) − 2𝑣2 (𝑡) + 𝐿𝜃2 (𝑡)]
𝐿
𝑐(𝑡) =
Equation 15 now becomes as follows:
𝑣(𝑥, 𝑡) = 𝑆1 (𝑥)𝑣1 (𝑡) + 𝑆2 (𝑥)𝜃2 (𝑡) + 𝑆3 (𝑥)𝑣2 (𝑡) + 𝑆4 (𝑥)𝜃2 (𝑡)
Where,
𝑥 2
𝑥 3
𝑆1 (𝑥) = 1 − 3 ( ) + 2 ( )
𝐿
𝐿
𝑥 2
𝑥 3
𝑆2 (𝑥) = 𝑥 − 2𝐿 ( ) + 𝐿 ( )
𝐿
𝐿
2
𝑥
𝑥 3
𝑆3 (𝑥) = 1 − 3 ( ) + 2 ( )
𝐿
𝐿
9
[16]
𝑥 2
𝑥 3
𝑆4 (𝑥) = −𝐿 ( ) + 𝐿 ( )
𝐿
𝐿
The kinetic energy of the element can expressed using the following equation:
1
𝜕𝑣 2
𝐾𝐸 = ∫ 𝜌𝐴 ( ) 𝑑𝑥
𝜕𝑡
0 2
𝐿
[17]
Substituting for 𝑑𝑣/𝑑𝑡 yields:
𝐿
1
2
𝐾𝐸 = 𝜌𝐴 ∫ [𝑆1 (𝑥)𝑣̇ 1 (𝑡)+𝑆2 (𝑥)𝜃̇1 (𝑡) + +𝑆3 (𝑥)𝑣̇ 2 (𝑡) + +𝑆4 (𝑥)𝜃̇2 (𝑡)] 𝑑𝑥
2
0
[18]
Integrating over the length of the beam yields:
𝐾𝐸 =
𝜌𝐴𝐿 𝑻
𝒗̇ 𝑴𝒗̇
420
[19]
Where,
156
22𝐿
𝑴=[
54
−13𝐿
22𝐿
4𝐿2
13𝐿
−3𝐿2
54
13𝐿
156
−22𝐿
−13𝐿
−3𝐿2
]
−22𝐿
4𝐿2
[20]
And
𝑣̇1
𝜃̇
𝑣̇ = 1
𝑣̇ 2
[𝜃̇2 ]
The potential energy of the beam can be expressed as the following:
𝐿
2
𝜕 2𝑣
𝑃𝐸 = 1/2𝐸𝐼 ∫ ( 2 ) 𝑑𝑥
𝜕𝑥
0
10
[21]
Substituting Equation 16 into Equation 21 in and integrating over the length of the beam
yields the stiffness matrix which is shown in Equation 22.
12
6𝐿
𝐸𝐼 6𝐿
4𝐿2
𝑲= 3[
𝐿 −12 −6𝐿
6𝐿
2𝐿2
−12 6𝐿
−6𝐿 2𝐿2
]
12 −6𝐿
−6𝐿 4𝐿2
[22]
Finally, the equation of motion can be expressed as the following:
𝑴𝒗̈ + 𝑲𝒗 = 0
[23]
3D finite element takes a structure, such as a beam, and discretizes it into a finite number
of elements. An example 3D element is shown in Figure 7.
Figure 7 Example of 3D FEA Element
11
Each node is allowed to move in all three directions (x,y, and z). For free vibration analysis
of a 3D mesh, the equation of motion can be described as Equation 24.
𝑴𝒔 𝑿̈ + 𝑲𝒔 𝑿 = 0
[24]
Where 𝑴𝒔 and 𝑲𝒔 are the global mass and stiffness matrix of the entire system. These are
developed from the individual elements and nodes.
The models considered for this project utilize 4 types of boundary conditions commonly
used in engineering. The types of boundary conditions and their effects can be seen in
Figure 5. For each set of boundary conditions both a beam and solid element model will
be run for 13 cases. Cases were selected such that geometries had slenderness ratios
between 5 and 20. The geometric inputs for each case are shown in Table 3. For this
analysis Ti 6-4 material, a very common aerospace alloy, will be used. Ti 6-4 material
properties are summarized in Table 2 Titanium Material Properties Used for Beam. An
ANSYS input file was created to for each boundary condition and element type to
automate the model generation process. All the input files can be found in Appendix A.
Table 2 Titanium Material Properties Used for Beam
12
Table 3 Geometric Input for Each FEA Case
BEAM3 elements will be used for the beam cases and SOLID45 elements for the solid
cases. Examples of each type model can be seen in Figure 8 and Figure 9.
Figure 8 Example of Beam Element Model
13
Figure 9 Example of Solid Element Model
In addition to the boundary conditions above, the solid element cases will have roller
boundary conditions on the vertical faces of the beams. As transverse vibrations are the
only mode shapes being considered for this project, the roller boundary conditions
eliminates extraneous, non-transverse modes shapes. This assists with modal
identification later in the process. The number of mode shapes and frequency range for
each type of boundary conditions are shown in Table 4.
Table 4 Number of Modes and Frequency Range for Each Case
After the solutions are obtained, further modal identification was required to find
the first 4 transverse vibration mode shapes. This was done by visual observation of the
mode shapes themselves. The first four mode shapes of the fixed-fixed boundary condition
using solid and beam elements can be seen in Figure 10 and Figure 11.
14
Figure 10 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Solid Elements
Figure 11 First Four Mode Shapes For Fixed-Fixed Beam Modeled with Beam Elements
15
3. RESULTS and DISCUSSIONS
3.1 Results Verification and Validation
To ensure that our FEA modeling techniques are correct the closed form solutions for each
case was obtained using the results listed in Figure 5. A frequency comparison of mode 1
for each type of boundary conditions is shown in Table 5 and Table 6. Mode 2 through 4
frequency comparisons can be seen in Appendix B.
Table 5 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Fixed Beam
Table 6 Mode 1 Frequency Comparison of Hand Calcs vs. FEA for a Fixed-Free Beam
Table 7 Frequency Comparison of Hand Calcs vs. FEA for a Simply Supported Beam
The hand calculations follow the beam element and solid element solutions closely
between slenderness ratios of 20 and 10. For example, the percent difference for the solid
16
element and beam element solution compared to the hand calculations for a fixed-fixed
beam with a slenderness ratio equal to 20 are 2.5% and 0.5% respectively. For slenderness
ratios below 10, the hand calculations and beam elements are still close, but both diverge
from the solid element solution. The percent difference for the solid element and beam
element compared to the hand calculations for a fixed-fixed beam with a slenderness ratio,
L/W, equal to 5.7 are -12.5% and -1.9% respectively for mode 1.
3.2 Overview of Results
All cases were run for each set of boundary conditions and the first 4 modes for each case
were identified. Comparisons of beam element and solid element frequencies for each
mode and boundary condition can be seen in Figure 12 through Figure 14 .
Figure 12 Frequency Comparisons W/ Fixed-Fixed Boundary Conditions
It is observed from the frequency comparisons for the fixed-fixed beam that as the
slenderness ratio decreases below 10, the 2 types of elements solutions diverge. The higher
order modes have a larger difference between solutions at slenderness ratios greater than
10 than the lower order modes. It is also noticed that the difference between mode 1
17
frequencies is less than the higher order modes throughout the slenderness ratio regime
considered. This is due to the mode shapes being more complicated in the higher order
modes. The more complicated the mode shapes the less the assumption that plane sections
remain plane holds. The same observations hold for the fixed-free and simply supported
beams frequency comparisons which can be seen in Figure 13 and Figure 14
Figure 13 Frequency Comparisons W/ Fixed-Free Boundary Conditions
Figure 14 Frequency Comparisons W/ Simply Supported Boundary Conditions
18
To obtain a conversion factor between the solid and beam element models, the
solid element frequencies are divided by the beam element frequencies for each mode.
This non-dimensional parameter is then plotted against the slenderness ratio. Using excel,
a curve fit was obtained and a convsersion factor equation based on slenderness ratio is
generated. The polynomial orders of the equations correspond to the lowest order that had
a 𝑅 2 value of greater than .99 and the best visual fit. The conversion factor equations can
be seen in Table 8 and Table 9 where y is the conversion factor and x is the slenderness
ratio.
Table 8 Conversion Equations for Modes 1 and 2
Table 9 Conversion Equations for Modes 3 and 4
To ensure that the conversion factors are accurate two test cases were ran. The first
test case was a fixed-fixed beam with the same geometric inputs seen in Table 3 except
the length was reduced by half to 12 inches. This simple test case was used as a first pass
attempt at proving the assumption that the conversion factors obtained would be valid for
different geometries. Frequency comparisons between solid element frequencies, beam
element frequencies, and converted beam element frequencies for mode 1 of test case 1
can be seen in Table 10. A frequency comparison for modes 2 through 4 are documented
in Appendix D.
19
Table 10 Frequency Comparison of Mode 1 for a Fixed-Fixed Beam 12 Inches Long
Comparing the unconverted beam element error with the converted error shows an
improvement throughout the entire slenderness ratio band considered. For the smallest
slenderness ratio considered, which has the large amount of error associated with it, the
percent error decreased from 37.82% to 6.65% using the conversion factors obtained.
The second test case was selected to test the bounds with which the conversion
factors would hold. This case utilizes fixed-free boundary conditions, an I-beam crosssection and a length increase from 24 inches to 32 inches. The cross-sectional geometry
is shown in Figure 15.
Figure 15 Test Case 2 I-Beam Cross-section
A frequency comparison of mode 4 for test case 4 is shown in Table 11. A
frequency comparison for modes 1 through 3 are documented in Appendix E.
20
Table 11 Test Case 2 Frequency Comparison for Mode 4
Once again a significant improvement in frequency predictions is noticed. Converting the
frequencies using the conversion factors decreases the error from a max of 47% to 7%. It
is also interesting that the Beam element solution is, once again, higher than the solid
element solution throughout the range of slenderness ratios considered. It is also of note
that for a slenderness ratio above 10 the beam element solution for test case 2 has an error
associated with it above 10%. This increase in error is most likely due to the more complex
cross-sectional geometry. This just further expands the regime of slenderness ratios where
the conversion factors obtained in this project are useful.
21
4. Conclusions
The ability to accurately predict natural frequencies of engineering structures is
vital in any design. Having a resonance within the operating parameters of a structure
during its life will lead to failure. Beams are a fundamental engineering structure with
multiple applications, be it in civil, aerospace, or mechanical engineering.
Euler-Bernoulli beam theory was developed to aid in analysis of beams, but due
to its basic assumptions it becomes less valid as the beam thickness increases. FEA
software packages have elements that are based on beam theory and have very fast solution
time. However, these elements have the same limitations for thicker beams so solid
elements are advised. Using solid elements greatly increase the solution times for solving
a model. Large models can be extremely computationally expensive.
Three different sets of boundary conditions were used in this project which was
discussed previously which all had a square cross-section. The beam element solutions
were very close to hand calculation predictions throughout the slenderness ratio regime
considered. In each case, a large increase in solution accuracy using solid elements was
observed for beams with slenderness ratios less than 10. It was also noticed that for all
boundary conditions the higher order mode shapes had a greater amount of error than the
lower order modes shapes. The conversion factors obtained with this project can improve
the beam element natural frequency predictions to within 10% error from as high as 38%.
The validity of the conversion factors were checked using 2 test cases. The first
test case considered was very similar to the original fixed-fixed case but with a decrease
in length. The second test had an I-Beam cross-section and an increase in length compared
to the original fixed-free boundary condition case. Both of the test cases beam element
solution error decreased to below 10% for the mode when converted using the conversion
factors.
While this project only covered free transverse vibration of beams using a select
group of boundary conditions, further work could be completed for the remaining
conditions. Also, conversion factors for more complex structures based on beams such as
trusses could be done.
22
5. References
1. http://thexodirectory.com/wp-content/uploads/2011/05/Air-to-air-overhead-frontview-of-an-SR-71A-460×361.jpg; 11/05/2013
2. http://www.texasescapes.com/TexasTowns/Burkett-Texas.htm; 11/05/2013
3. Palm, W. [2007], Mechanical Vibrations, Hoboken: Wiley & Sons
23
6. Appendix A
6.1 Fixed – Fixed Solid element ANSYS INPUT file
/prep7
et,1,45
! sets element type 1 to beam3, the 2d beam element in ANSYS
mp,ex,1,16e6
! sets modulus of mtl 1
mp,prxy,1,0.3
! sets poissons ratio of mtl 1
mp,dens,1,0.41443E-03 ! defines density
mat,1
! sets material to 1, not needed as it defaults to 1
real,1
! sets the property set to 1 (this is the default)
ESIZE,0.1,0,
mode_num=10
!mode numbers to extract
/INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized
*DIM, input_parms, TABLE, rw-1, 3 !dimension table
*DIM, inputs, ARRAY, rw-1, 4 !dimension array
*TREAD, input_parms,input_parms, csv,,1 !reads input file into table
*MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array
fini
rw=rw-2
*DIM, frequencies, ARRAY, mode_num + 1, rw,
*do,j,1,rw
fini
run_name='modal_analysis_%j%'
24
/FILNAME,run_name,0
/syp,mkdir, run_name
/CWD,W:\m310186\RPI_project\fx-fx-solid\%run_name%
/prep7
BLOCK,0,24,0,inputs(j,1),.25,-.25,
vmesh,1
! creates a mesh of elements on volume 1
asel,s,area,,5
asel,a,area,,6
nsla,s,1
cm,nn,node
d,all,all,0
! constrains displacements on forward face to zero
d,all,all,0
!constrains displacements on rear face to zero
asel,s,area,,1
asel,a,area,,2
nsla,s,1
cmsel,u,nn
d,all,uz,0
!constrains the side faces in teh z direction to zero
allsel,all
/solu
antype,2
LUMPM,0
PSTRES,0
MODOPT,LANB,mode_num,0,20000,,OFF
25
ematwrite,yes
solve
psolve,eiglanb
ematwrite,yes
MXPAND,mode_num, , ,0
PSOLVE,EIGEXP
fit
/post1
/output,run_name,freq
set,list
/output
set,first
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_1.jpg'
r_r='%run_name%000.jpg'
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(1,j)=j
frequencies(2,j)=reqf
/syp,ren ,r_r,i_i
looper=mode_num-1
*do,k,1,looper
l=k+1
z=k+2
set,next
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_%l%.jpg'
26
r_r='%run_name%000.jpg'
/syp,ren ,r_r,i_i
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(z,j)=reqf
*enddo
/CWD,W:\m310186\RPI_project\fx-fx-solid
/prep7
vclear,1
vdele,all,all,,1
*enddo
*cfopen,freq_tot,out
*vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies(
1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10
),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1,
15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19)
(f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f
10.1,f10.1,f10.1,f10.1)
*cfclose
6.2 Fixed Fixed Beam Elemen ANSYS input file
/prep7
k,2,24,0
! sets a keypoint at (24,0)
l,1,2,36
! creates a line with 5 divisions from keypoints 1 to 2
et,1,3
! sets element type 1 to beam3, the 2d beam element in ANSYS
27
mp,ex,1,16e6
! sets modulus of mtl 1
mp,prxy,1,0.3
! sets poissons ratio of mtl 1
r,1,.75,9,
! defines property set 1 for the beam element (A,I,height)
mp,dens,1,0.41443E-03 ! defines density
mat,1
! sets material to 1, not needed as it defaults to 1
real,1
! sets the property set to 1 (this is the default)
lmesh,1
d,1,all,0
! creates a mesh of elements on line1 (undo with lclear)
! constrains displacements at node 1 to zero
d,2,all,0
/INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized
*DIM, input_parms, TABLE, rw-1, 3 !dimension table
*DIM, inputs, ARRAY, rw-1, 4 !dimension array
*TREAD, input_parms,input_parms, csv,,1 !reads input file into table
*MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array
fini
rw=rw-2
mode_num=10
*DIM, frequencies, ARRAY, mode_num + 1, rw,
*do,j,1,rw
fini
run_name='modal_analysis_%j%'
/FILNAME,run_name,0
/syp,mkdir, run_name
28
/CWD,S:\csmc_struct_64\RPI_project\fx-fx-beam\%run_name%
/solu
r,1,inputs(j,3),inputs(j,2),
! defines property set
1 for the beam element (A,I,height)
antype,2
LUMPM,0
PSTRES,0
MODOPT,LANB,mode_num,0,25000,,OFF
ematwrite,yes
solve
psolve,eiglanb
ematwrite,yes
MXPAND,mode_num, , ,0
PSOLVE,EIGEXP
fit
/post1
/output,run_name,freq
set,list
/output
set,first
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_1.jpg'
r_r='%run_name%000.jpg'
*GET,reqf, ACTIVE, 0, set, frequency,
29
frequencies(1,j)=j
frequencies(2,j)=reqf
/syp,ren ,r_r,i_i
looper=mode_num-1
*do,k,1,looper
l=k+1
z=k+2
set,next
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_%l%.jpg'
r_r='%run_name%000.jpg'
/syp,ren ,r_r,i_i
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(z,j)=reqf
*enddo
/CWD,S:\csmc_struct_64\RPI_project\fx-fx-beam
*enddo
*cfopen,freq_tot,out
*vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies(
1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10
),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1,
15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19)
(f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f
10.1,f10.1,f10.1,f10.1)
*cfclose
30
!*get,name,active,,jobnam
6.3
Fixed – Free Solid Element ANSYS Input File
/prep7
et,1,45
! sets element type 1 to beam3, the 2d beam element in ANSYS
mp,ex,1,16e6
! sets modulus of mtl 1
mp,prxy,1,0.3
! sets poissons ratio of mtl 1
mp,dens,1,0.41443E-03 ! defines density
mat,1
! sets material to 1, not needed as it defaults to 1
real,1
! sets the property set to 1 (this is the default)
ESIZE,0.1,0,
mode_num=10
!mode numbers to extract
/INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized
*DIM, input_parms, TABLE, rw-1, 3 !dimension table
*DIM, inputs, ARRAY, rw-1, 4 !dimension array
*TREAD, input_parms,input_parms, csv,,1 !reads input file into table
*MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array
fini
rw=rw-2
*DIM, frequencies, ARRAY, mode_num + 1, rw,
31
*do,j,1,rw
fini
run_name='modal_analysis_%j%'
/FILNAME,run_name,0
/syp,mkdir, run_name
/CWD,W:\m310186\RPI_project\fx-free-solid\%run_name%
/prep7
BLOCK,0,24,0,inputs(j,1),.25,-.25,
vmesh,1
! creates a mesh of elements on volume 1
asel,s,area,,5
nsla,s,1
cm,nn,node
d,all,all,0
! constrains displacements on forward face to zero
asel,s,area,,1
asel,s,area,,2
nsla,s,1
cmsel,u,nn
d,all,uz,0
!constrains the side faces in teh z direction to zero
allsel,all
/solu
antype,2
32
LUMPM,0
PSTRES,0
MODOPT,LANB,mode_num,0,20000,,OFF
ematwrite,yes
solve
psolve,eiglanb
ematwrite,yes
MXPAND,mode_num, , ,0
PSOLVE,EIGEXP
fit
/post1
/output,run_name,freq
set,list
/output
set,first
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_1.jpg'
r_r='%run_name%000.jpg'
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(1,j)=j
frequencies(2,j)=reqf
/syp,ren ,r_r,i_i
looper=mode_num-1
*do,k,1,looper
l=k+1
z=k+2
set,next
plnsol,u,sum
33
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_%l%.jpg'
r_r='%run_name%000.jpg'
/syp,ren ,r_r,i_i
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(z,j)=reqf
*enddo
/CWD,W:\m310186\RPI_project\fx-free-solid
/prep7
vclear,1
vdele,all,all,,1
*enddo
*cfopen,freq_tot,out
*vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies(
1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10
),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1,
15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19)
(f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f
10.1,f10.1,f10.1,f10.1)
*cfclose
6.4 Fixed – Free Beam Element ANSYS Input File
/prep7
k,1,0,0
! sets a keypoint at (0,0)
k,2,24,0
! sets a keypoint at (24,0)
l,1,2,36
! creates a line with 5 divisions from keypoints 1 to 2
34
et,1,3
! sets element type 1 to beam3, the 2d beam element in ANSYS
mp,ex,1,16e6
! sets modulus of mtl 1
mp,prxy,1,0.3
! sets poissons ratio of mtl 1
r,1,.75,9,
! defines property set 1 for the beam element (A,I,height)
mp,dens,1,0.41443E-03 ! defines density
mat,1
! sets material to 1, not needed as it defaults to 1
real,1
! sets the property set to 1 (this is the default)
lmesh,1
d,1,all,0
! creates a mesh of elements on line1 (undo with lclear)
! constrains displacements at node 1 to zero
/INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized
*DIM, input_parms, TABLE, rw-1, 3 !dimension table
*DIM, inputs, ARRAY, rw-1, 4 !dimension array
*TREAD, input_parms,input_parms, csv,,1 !reads input file into table
*MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array
fini
rw=rw-2
mode_num=10
*DIM, frequencies, ARRAY, mode_num + 1, rw,
*do,j,1,rw
fini
run_name='modal_analysis_%j%'
35
/FILNAME,run_name,0
/syp,mkdir, run_name
/CWD,S:\csmc_struct_64\RPI_project\fx-free-beam\%run_name%
/solu
r,1,inputs(j,3),inputs(j,2),
! defines property set
1 for the beam element (A,I,height)
antype,2
LUMPM,0
PSTRES,0
MODOPT,LANB,mode_num,0,20000,,OFF
ematwrite,yes
solve
psolve,eiglanb
ematwrite,yes
MXPAND,mode_num, , ,0
PSOLVE,EIGEXP
fit
/post1
/output,run_name,freq
set,list
/output
set,first
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_1.jpg'
36
r_r='%run_name%000.jpg'
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(1,j)=j
frequencies(2,j)=reqf
/syp,ren ,r_r,i_i
looper=mode_num-1
*do,k,1,looper
l=k+1
z=k+2
set,next
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_%l%.jpg'
r_r='%run_name%000.jpg'
/syp,ren ,r_r,i_i
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(z,j)=reqf
*enddo
/CWD,S:\csmc_struct_64\RPI_project\fx-free-beam
*enddo
*cfopen,freq_tot,out
*vwrite,frequencies(1,1),frequencies(1,2),
frequencies(1,3),frequencies(1,4),frequencies(1,5),frequencies(1,6),frequencies(1,7),freq
uencies(1,8),frequencies(1,9),frequencies(1,10),frequencies(1,11),frequencies(1,12),freq
uencies(1,13),frequencies(1,14),frequencies(1,15),frequencies(1,16),frequencies(1,17),fr
equencies(1,18),frequencies(1,19)
(f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f
10.1,f10.1,f10.1,f10.1)
37
*cfclose
6.5
Simply Supported Solid Element ANSYS Input File
/prep7
et,1,45
! sets element type 1 to beam3, the 2d beam element in ANSYS
mp,ex,1,16e6
! sets modulus of mtl 1
mp,prxy,1,0.3
! sets poissons ratio of mtl 1
mp,dens,1,0.41443E-03 ! defines density
mat,1
! sets material to 1, not needed as it defaults to 1
real,1
! sets the property set to 1 (this is the default)
ESIZE,0.1,0,
mode_num=10
!mode numbers to extract
/INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized
*DIM, input_parms, TABLE, rw-1, 3 !dimension table
*DIM, inputs, ARRAY, rw-1, 4 !dimension array
*TREAD, input_parms,input_parms, csv,,1 !reads input file into table
*MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array
fini
rw=rw-2
*DIM, frequencies, ARRAY, mode_num + 1, rw,
*do,j,1,rw
fini
run_name='modal_analysis_%j%'
38
/FILNAME,run_name,0
/syp,mkdir, run_name
/CWD,/pw/data/csmc_struct_64/RPI_project/simply-supportedsolid/%run_name%
/prep7
CSYS,0
WPAVE,0,0,0
wpcsys,,,
BLOCK,0,24,0,inputs(j,1),.25,-.25,
CSYS,0
WPAVE,0,0,0
CSYS,0
ofst=inputs(j,1)/2
wpoff,,ofst,
wpro,,90.000000,
VSBW,
1
vmesh,2
vmesh,3
! creates a mesh of elements on volume 1
lsel,s,line,,19
lsel,a,line,,20
nsll,s,0
cm,nn,node
d,all,ux,0,,,,uy,
! constrains displacements at node 1 to zero
d,all,ux,0,,,,uy,
39
asel,s,area,,7
asel,a,area,,8
asel,a,area,,13
asel,a,area,,12
nsla,s,1
cmsel,u,nn
d,all,uz,0
!constrains the side faces in teh z direction to zero
allsel,all
/solu
antype,2
LUMPM,0
PSTRES,0
MODOPT,LANB,mode_num,0,20000,,OFF
ematwrite,yes
solve
psolve,eiglanb
ematwrite,yes
MXPAND,mode_num, , ,0
PSOLVE,EIGEXP
fit
/post1
/output,run_name,freq
set,list
/output
set,first
plnsol,u,sum
fit
40
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_1.jpg'
r_r='%run_name%000.jpg'
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(1,j)=j
frequencies(2,j)=reqf
/syp,ren ,r_r,i_i
looper=mode_num-1
*do,k,1,looper
l=k+1
z=k+2
set,next
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_%l%.jpg'
r_r='%run_name%000.jpg'
/syp,ren ,r_r,i_i
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(z,j)=reqf
*enddo
/CWD,/pw/data/csmc_struct_64/RPI_project/simply-supported-solid
/prep7
vclear,all
vdele,all,all,,1
*enddo
*cfopen,freq_tot,out
41
*vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies(
1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10
),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1,
15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19)
(f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f
10.1,f10.1,f10.1,f10.1)
*cfclose
6.6 Simply Supported Beam Element ANSYS Input File
/prep7
k,1,0,0
! sets a keypoint at (0,0)
k,2,24,0
! sets a keypoint at (24,0)
l,1,2,36
! creates a line with 5 divisions from keypoints 1 to 2
et,1,3
! sets element type 1 to beam3, the 2d beam element in ANSYS
mp,ex,1,16e6
! sets modulus of mtl 1
mp,prxy,1,0.3
! sets poissons ratio of mtl 1
r,1,.75,9,
! defines property set 1 for the beam element (A,I,height)
mp,dens,1,0.41443E-03 ! defines density
mat,1
! sets material to 1, not needed as it defaults to 1
real,1
! sets the property set to 1 (this is the default)
lmesh,1
d,1,ux,0,,,,uy
! creates a mesh of elements on line1 (undo with lclear)
! constrains displacements at node 1 to zero
d,2,ux,0,,,,uy
/INQUIRE,rw,LINES,input_parms,csv ! number of lines in input file parameterized
*DIM, input_parms, TABLE, rw-1, 3 !dimension table
*DIM, inputs, ARRAY, rw-1, 4 !dimension array
42
*TREAD, input_parms,input_parms, csv,,1 !reads input file into table
*MFUN, inputs(1,1), COPY, input_parms(1,0) !converts table into an array
fini
rw=rw-2
mode_num=10
*DIM, frequencies, ARRAY, mode_num + 1, rw,
*do,j,1,rw
fini
run_name='modal_analysis_%j%'
/FILNAME,run_name,0
/syp,mkdir, run_name
/CWD,S:\csmc_struct_64\RPI_project\simply_supported_beam\%run_name%
/solu
r,1,inputs(j,3),inputs(j,2),
! defines property set
1 for the beam element (A,I,height)
antype,2
LUMPM,0
PSTRES,0
mode_num=10
43
MODOPT,LANB,mode_num,0,20000,,OFF
ematwrite,yes
solve
psolve,eiglanb
ematwrite,yes
MXPAND,mode_num, , ,0
PSOLVE,EIGEXP
fit
/post1
/output,run_name,freq
set,list
/output
set,first
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_1.jpg'
r_r='%run_name%000.jpg'
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(1,j)=j
frequencies(2,j)=reqf
/syp,ren ,r_r,i_i
looper=mode_num-1
*do,k,1,looper
l=k+1
z=k+2
set,next
plnsol,u,sum
fit
/ui,copy,save,jpeg,grap,colo,vid,port,,100
i_i='mode_%l%.jpg'
44
r_r='%run_name%000.jpg'
/syp,ren ,r_r,i_i
*GET,reqf, ACTIVE, 0, set, frequency,
frequencies(z,j)=reqf
*enddo
/CWD,S:\csmc_struct_64\RPI_project\simply_supported_beam
*enddo
*cfopen,freq_tot,out
*vwrite,frequencies(1,1),frequencies(1,2),frequencies(1,3),frequencies(1,4),frequencies(
1,5),frequencies(1,6),frequencies(1,7),frequencies(1,8),frequencies(1,9),frequencies(1,10
),frequencies(1,11),frequencies(1,12),frequencies(1,13),frequencies(1,14),frequencies(1,
15),frequencies(1,16),frequencies(1,17),frequencies(1,18),frequencies(1,19)
(f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f10.1,f
10.1,f10.1,f10.1,f10.1)
*cfclose
45
7. Appendix B
46
47
48
8. Appendix C
49
50
9. Appendix D
51
10.Appendix E
52