Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Table of Contents
Page Number
Problem Description
2
Theory
2
Geometry
4
Preprocessor
Element Type
Real Constants and Material Properties
Meshing
Displacement
6
6
7
9
10
Solution
11
General Postprocessor
12
Results
14
Validation
15
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 1
Problem Description:
Nomenclature:
L =5m
b =0.5m
h =0.5m
Length of beam
Cross Section Base
Cross Section Height
E=7*
Young’s Modulus of Aluminum
=0.35
Poisson’s Ratio of Aluminum
=2700
Density of Aluminum
Moment of Inertia
In this module, we will introduce the ANSYS Mechanical APDL Vibration Analysis Type. This
uses the Modal solution method. This tutorial will explore the free vibration of a cantilever
beam modeled with 1D BEAM elements and we will extract the natural frequencies and mode
shapes at these frequencies.
Theory
Natural Frequency
Using Euler-Bernoulli Beam Theory we find:
(10.1)
Normal Mode solution to the above equation is:
(10.2)
This makes equation 10.1:
(10.3)
Solution for displacement is:
(10.4)
Where:
(10.5)
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 2
For a cantilever beam, the displacement and slope are zero at the fixed end, while at the free end,
the moment and shear are zero. Thus the boundary conditions are:
At x=0:
y=0
At x=L:
𝑑 𝑦
𝑑𝑥
𝑑𝑦
𝑑 𝑦
𝑑𝑥
𝑑𝑥
This proves that:
Solving for
and
we find:
Cos(βL)Cosh(βL) = -1
(10.6)
The roots of this equation are
(10.7)
The equation for time breaks down into:
√
√
(10.8)
So the frequency in rad/s is:
√
(10.9)
Converting to Hz, we get the natural frequency as shown below:
√
(10.10)
Extracting the first few natural frequencies, we get:
n
1
1.8751
103.3607
16.45
2
4.69409
647.753
103.0931
3
7.8539
1813.332
288.6007
4
10.99557
3554.2012
565.6687
5
14.1372
5875.344
935.09
6
17.279
8776.95
1396.895
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 3
Geometry
3
Opening ANSYS Mechanical APDL
1. On your Windows 7 Desktop click the Start button
2. Under Search Programs and Files type “ANSYS”
3. Click on
Mechanical APDL (ANSYS) to start
ANSYS. This step may take time.
Preferences
1. Go to Main Menu -> Preferences
2. Check the box that says Structural
3. Click OK
1
2
1
2
3
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 4
Key points
Since we will be using 1D Elements, our goal is to model the length of the beam.
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Keypoints -> On Working Plane
2. Click Global Cartesian
3. In the box underneath, write: 0,0,0. This will create a key point at the
origin.
4. Click Apply
5. Repeat Steps 3 and 4 for 5,0,0
6. Click Ok
7. The Triad in the top left corner is blocking keypoint 1.
To get rid of the triad, type
/triad,off in Utility Menu -> Command Prompt
2
3
6
7
2
8. Go to Utility Menu -> Plot -> Replot
Line
1. Go to Main Menu -> Preprocessor -> Modeling -> Create ->
Lines -> Lines -> Straight Line
2. Select Pick
3. Select List of Items
4. Type 1,2 for points previously generated.
5. Click Ok
3
The resulting graphic should be as shown:
4
5
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 5
Preprocessor
Element Type
1.
2.
3.
4.
Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete
Click Add
Click Beam -> 2D Elastic 3
Click OK
3
4
Beam3 is a uniaxial element with tension, compression, and bending capabilities. The element
has three degrees of freedom at each node: translations in the nodal x and y directions and
rotation about the nodal z-axis. For more information consult the ANSYS HELP.
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 6
Real Constants and Material Properties
Now we will dimension our beam.
1. Go to Main Menu -> Preprocessor ->
Real Constants -> Add/Edit/Delete
2. Click Add
3. Choose Type 1 Beam3
4. Click OK
5. Under Cross-sectional area AREA
enter 1/4
6. Under Area moment of inertia IZZ
Enter 1/192
7. Under Total beam height HEIGHT
enter 0.5
8. Click OK
9. Click Close
3
2
9
4
5
6
7
8
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 7
Now we must specify Young’s Modulus, Poisson’s Ratio and Density
1.
2.
3.
4.
5.
Go to Main Menu -> Preprocessor -> Material Props -> Material Models
Go to Material Model Number 1 -> Structural -> Linear -> Elastic -> Isotropic
Input 7E10 for the Young’s Modulus in EX.
Input 0.35 for Poisson’s Ratio in PRXY
Click OK
3
4
2
6
5
6. Go to Material Model Number 1 -> Structural -> Density
7. Input 2700 for the Density in DENS
8. Click OK
7
8
9.
Of Define Material Model Behavior window
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 8
Meshing
1. Go to Main Menu -> Preprocessor ->
Meshing -> Mesh Tool
2. Go to Size Controls: -> Global -> Set
3. Under NDIV No. of element divisions put 10.
This will create a mesh of a total 10 elements
4. Click OK
5. Click Mesh
6. Click Pick All
2
3
4
5
6
7. Go to Utility Menu -> Plot -> Nodes
8. Go to Utility Menu -> Plot Controls -> Numbering…
9. Check NODE Node Numbers to ON
10. Click OK
9
10
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 9
The resulting graphic should be as shown:
ANSYS numbers nodes from the left extreme to the right extreme and then numbers from left to
right.
Displacement
1. Go to Main Menu -> Preprocessor -> Loads ->
Define Loads ->Apply ->Structural -> Displacement -> On Nodes
2. Select Pick -> Single -> and click node 1
3. Click OK
4. Under Lab2 DOFs to be constrained select All DOF
5. Under Value Displacement value enter 0
6. Click OK
2
4
3
5
6
The resulting graphic should look as shown below:
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 10
Solution
Analysis Type
1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis
2. Choose Modal
3. Click OK
2
3
4. Go to Main Menu -> Solution -> Analysis Type ->Analysis Options
5. Under No. of modes to extract enter 8
5
6
6. Click OK
7. Since there is no added Frequency, click OK in Block Lanczos Window
8. Go to Main Menu -> Solution -> Solve -> Current LS
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
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General Postprocessor
Natural Frequencies
Go to Main Menu -> General Postproc -> List Results -> Detailed Summary
*
*
* Ignore frequencies 3 and 6 in comparison with the theoretical values, these are the torsional
frequencies and were not calculated.
Mode Shape
To view the mode shapes that correspond to these frequencies:
1. Go to Main Menu -> General Postproc -> Read Results -> By Pick
2. Select the lowest Eigenvalue: Set 1 -> Click Read
2
3
3. Click Close
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
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4. Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape
5. Under KUND Items to be plotted select Def + undeformed
6. Click OK
5
6
The graphics area should look as below:
You can repeat these steps to view the other mode shapes produced by different frequencies.
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 13
Results
The percent error (%E) in our model can be defined as:
(
Frequency
)
Theoretical
2 Elements
10 Elements
500 Elements
16.45
16.427
16.419
16.419
103.09
102.56
101.73
101.73
288.6
338.91
279.86
279.79
565.67
907.38
535.21
534.73
935.09
N/A
859.22
857.22
1396.895
N/A
1242.2
1236.1
With an unreasonably coarse mesh, ANSYS only allows you to extract a certain amount of
modes. This is shown with the 2 elements where ANSYS only calculated the first four
translational frequencies and first 2 torsional frequencies. Since the theory section uses
approximations to calculate the natural frequencies, the theoretical answer is not precise. As
mesh is refined, our analysis shows that the answers converge to the solution. This again can be
seen as ANSYS does not provide natural frequencies if the mesh is too coarse.
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 14
Validation
Deviation from Theoretical Solution
100
90
80
Error (%)
70
60
2 Elements
50
10 Elements
40
500 Elements
30
20
10
0
ω1
ω2
ω3
ω4
ω5
ω6
Natural frequency
UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam
Page 15