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

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   =2700    

Poisson’s Ratio of Aluminum 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:

     UCONN ANSYS –Module 10: Free Vibration of an Undampened 1D Cantilever Beam

(10.5)

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:



  

  

 





  

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

2

1. Go to Main Menu -> Preferences 2. Check the box that says Structural 3. Click  OK

1

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

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

Page 11

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 70

    )    %     (    r    o    r    r    E

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