Module 10: Free Vibration of an Undampened 1D Cantilever Beam

UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam Page 1

Module 10: Free Vibration of an Undampened 1D Cantilever Beam

Table of Contents Page Number

Problem Description 2

Theory 2

Geometry 4

Preprocessor 6

Element Type 6

Real Constants and Material Properties 7

Meshing 9

Displacement 10

Solution 11

General Postprocessor 12

Results 14

Validation 15


Problem Description:

Nomenclature:

L =5m Length of beam

b =0.5m Cross Section Base

h =0.5m 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)


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:

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

At x=0:

y=0

𝑑𝑦

𝑑𝑥

At x=L:

𝑑 𝑦

𝑑𝑥

𝑑 𝑦

𝑑𝑥


Geometry

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

3

1

2

1

2

3


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

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

The resulting graphic should be as shown:

6

2

3

2

3

4

7

5


Preprocessor

Element Type

1. Go to Main Menu -> Preprocessor -> Element Type -> Add/Edit/Delete

2. Click Add

3. Click Beam -> 2D Elastic 3

4. Click OK

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.

3

4


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

2

3

4

56

9

7

8


Now we must specify Young’s Modulus, Poisson’s Ratio and Density

1. Go to Main Menu -> Preprocessor -> Material Props -> Material Models

2. Go to Material Model Number 1 -> Structural -> Linear -> Elastic -> Isotropic

3. Input 7E10 for the Young’s Modulus in EX.

4. Input 0.35 for Poisson’s Ratio in PRXY

5. Click OK

6. Go to Material Model Number 1 -> Structural -> Density

7. Input 2700 for the Density in DENS

8. Click OK

9. Of Define Material Model Behavior window

3 4

5

2

6

7

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

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

2

3

4

5

6

9

10


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

The resulting graphic should look as shown below:

2

3

4

5

6


Solution

Analysis Type

1. Go to Main Menu -> Solution -> Analysis Type -> New Analysis

2. Choose Modal

3. Click OK

4. Go to Main Menu -> Solution -> Analysis Type ->Analysis Options

5. Under No. of modes to extract enter 8

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

2

3

5

6


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

3. Click Close

3

2

*

*


4. Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape

5. Under KUND Items to be plotted select Def + undeformed

6. Click OK

The graphics area should look as below:

You can repeat these steps to view the other mode shapes produced by different frequencies.

6

5


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.


0

10

20

30

40

50

60

70

80

90

100

ω1 ω2 ω3 ω4 ω5 ω6

Erro

r (%

)

Natural frequency

Deviation from Theoretical Solution

2 Elements

10 Elements

500 Elements

Validation

Documents

Module 10: Free Vibration of an Undampened 1D Cantilever Beam