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Module 10: Free Vibration of an Undampened 1D Cantilever Beam
- Problem Description Table of Contents Page Number
- Theory
- Geometry
- Preprocessor
- Element Type
- Real Constants and Material Properties
- Meshing
- Displacement
- Solution
- General Postprocessor
- Results
- Validation
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)
Geometry
Opening ANSYS Mechanical APDL
- On your Windows 7 Desktop click the Start button
- Under Search Programs and Files type “ ANSYS ”
- Click on Mechanical APDL (ANSYS) to start ANSYS. This step may take time.
Preferences
- Go to Main Menu -> Preferences
- Check the box that says Structural
- Click OK
Key points
Since we will be using 1D Elements, our goal is to model the length of the beam.
- Go to Main Menu -> Preprocessor -> Modeling -> Create -> Keypoints -> On Working Plane
- Click Global Cartesian
- In the box underneath, write: 0,0,0. This will create a key point at the origin.
- Click Apply
- Repeat Steps 3 and 4 for 5,0,
- Click Ok
- 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
- Go to Utility Menu -> Plot -> Replot
Line
- Go to Main Menu -> Preprocessor -> Modeling -> Create -> Lines -> Lines -> Straight Line
- Select Pick
- Select List of Items
- Type 1,2 for points previously generated.
- Click Ok
The resulting graphic should be as shown:
Real Constants and Material Properties
Now we will dimension our beam.
- Go to Main Menu -> Preprocessor -> Real Constants -> Add/Edit/Delete
- Click Add
- Choose Type 1 Beam
- Click OK
- Under Cross-sectional area AREA enter 1/
- Under Area moment of inertia IZZ Enter 1/
- Under Total beam height HEIGHT enter 0.
- Click OK
- Click Close
Now we must specify Young’s Modulus, Poisson’s Ratio and Density
- 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
- Go to Material Model Number 1 -> Structural -> Density
- Input 2700 for the Density in DENS
- Click OK
- Of Define Material Model Behavior window
2
6
7
8
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
- Go to Main Menu -> Preprocessor -> Loads -> Define Loads ->Apply ->Structural -> Displacement -> On Nodes
- Select Pick -> Single -> and click node 1
- Click OK
- Under Lab2 DOFs to be constrained select All DOF
- Under Value Displacement value enter 0
- Click OK
The resulting graphic should look as shown below:
Solution
Analysis Type
- Go to Main Menu -> Solution -> Analysis Type -> New Analysis
- Choose Modal
- Click OK
- Go to Main Menu -> Solution -> Analysis Type ->Analysis Options
- Under No. of modes to extract enter 8
- Click OK
- Since there is no added Frequency, click OK in Block Lanczos Window
- Go to Main Menu -> Solution -> Solve -> Current LS
- Go to Main Menu -> General Postproc -> Plot Results -> Deformed Shape
- Under KUND Items to be plotted select Def + undeformed
- 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.
103.09 102.56 101.73 101.
288.6 338.91 279.86 279.
565.67 907.38 535.21 534.
935.09 N/A 859.22 857.
1396.895 N/A 1242.2 1236.
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.
