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Lateral Natural Frequencyof a Cantilever Beam

By Dr. Chuan-Chiang Chen

Tuskegee University

Mechanical Engineering Dept.

cchen@tuskegee.edu

Copyright 2006

Expected completion time for this tutorial is 50 minutes

Companion Tutorial for Vibrations/Mechanical Design Course

Reference Text: Mechanical Vibrations, 2nd edition, by S. S. Rao

Table of Contents (TOC)

- Educational Objectives
- Problem Description
- Suggested Steps
- Step by Step Process
- Viewing the Results of the FE Analysis
- Analytical Analysis of a Cantilever Beam
- Comparison of Analytical to Finite Element Analysis
- Results of (Mode Shapes)
- To use the TOC hyperlinks left click the entire topic then right click your mouse and open a hyperlink to the PowerPoint slide of interest.

Table of Contents

- Summary and Discussion
- Cantilever Beam Exercise
- Appendix A:
- Background of Finite Elements
- Finite Element Theory
- Finite Element Textbooks and Sources
- Acknowledgement

Table of Contents

- Practice Problem
- Appendix A :
- Background Information
- Finite Element Theory
- Finite Element Theory sources

Educational Objectives

The educational goal is to provide undergraduate engineering students with understanding of a specific engineering topic and FE theory, along with an ability to apply commercial FE software to typical engineering problems. The educational goal will be accomplished through four educational objectives based upon Bloom’s Taxonomy and ABET Criteria 3 as follows:

1. Engineering Topics (Comprehension: 3a, 3k). Understand the fundamental basis of engineering topics through the use of finite element computer models.

Educational Objectives

2. FE Theory (Comprehension; 3a). Understand the fundamental basis of FE Theory.

3. FE Modeling Practice (Application; 3a, 3e, 3k). Be able to implement a suitable finite element model and construct a correct computer model using commercial FE software.

4. FE Solution Interpretation and Verification (Comprehension and Evaluation; 3a, 3e) Be able to interpret and evaluate finite element solution quality, including the importance of verification .

Problem Description

- Analysis Objectives
- Determine the natural frequencies and modes in a cantilever beam
- Use the finite element method to determine the modes shapes at resonant frequencies

50 “

Problem DescriptionModel Definition- Cantilever beam

(0.9”x0.9” cross section with a length of 50”)

- Determine the natural frequencies
- Plot the mode shapes

Overview of SolidWorks and COSMOSWorks

- SolidWorks
- SolidWorks is a parametric, solid, feature-based computer aided design (CAD) system. In SolidWorks, you sketch ideas and experiment with different designs to create 3D models. A SolidWorks model is made of : Parts, Assemblies or Drawings.
- A part is a single 3D object made up of features. Features are the shapes and operations that construct the part. A part can become a component in an assembly, and it can be represented in a 2D drawing.
- A assembly is a document in which parts, features and other assemblies ( sub-assemblies) are mated together.

Overview of SolidWorks and COSMOSWorks

- COSMOSWorks
- COSMOSWorks is finite element analysis software which is fully integrated with the solid modeling software SolidWorks. COSMOSWorks uses finite element analysis to simulate the working conditions of engineering designs and predict their behavior. Powered by fast solvers, COSMOSWorks makes it possible for designers to quickly check the integrity of their designs and search for optimum solutions. COSMOSWorks can perform static, thermal, buckling, frequency, compressible/incompressible fluid flow, drop test and optimization analysis of parts and assemblies.

Three-Dimensional Model Assumptions

- In constructing the 3-dimensionional model of the beam, there are some assumptions that can be made to assure that the model is accurate
- Assumptions
- The material is isotropic and elastic

The Steps to Analyze a Cantilever Beam

- A 3-dimensional model will be constructed using the SolidWorks CAD software and loads and restraints applied to this solid model.
- The 3-dimensional model is then meshed in prior to using COSMOSWorks to perform a Finite Element Analysis of the model
- Mode shapes with corresponding natural frequencies are produced

General Steps

- Create a geometric model of the cantilever beam
- Define the material properties
- Define the boundary conditions
- Create a finite element mesh of this model
- Submit the model to COSMOSWorks for analysis
- Post Process the results of analysis using COSMOSWorks
- Compare this finite element results with the analytical analysis

Tutorial Step by Step Process

- Overview of SolidWorks and COSMOSWorks
- Left Side of SolidWorks/COSMOSWorks Window
- Use of the SolidWorks Interface
- Toolbars
- Tutorials and Getting Help
- Creating a SolidWorks Model of the beam
- Setting the Drawing Units to Inches in SolidWorks
- Creating a SolidWorks Model of the beam
- Verify that COSMOSWorks is loaded in your Computer

Tutorial Step by Step Process

- Opening your Model in COSMOSWorks
- The COSMOSWorks Study Folders
- Assigning Material Properties to the Model
- Assigning Material Properties to the beam
- Applying Restraints or Boundary Conditions to the beam
- Meshing the Model and Running the Study

Tutorial Step by Step Process

- Viewing the Results of the Finite Element Analysis
- Viewing Natural Frequencies
- Viewing Mode Shapes
- Animation of Lateral Vibration of a Cantilever Beam
- Analytical Analysis of a Cantilever Beam
- Comparison of Results between Analytical and Finite Element Analysis
- Summary and Discussion

Overview of the SolidWorks Window

- Menu Bar
- Sketch Toolbar/Sketch Relations Toolbar
- Confirmation Corner with Sketch Indicator
- Graphics area
- Sketch Origin
- Reference Triad

Left Side of SolidWorks/COSMOSWorks Window

- Feature Manager icon
- Feature Manager design tree
- COSMOSWorks icon
- COSMOSWorks Manager tree

Using the SolidWorks Interface

- Toolbar area
- Feature Manager window
- Graphical Interface window

Toolbars

- You can select which toolbars to display under View in the menu heading
- Toolbars are moveable to top, and sides of the window
- Icon Buttons for frequently used commands

Tutorials and Getting Help

- Click the Help menu and choose either
- COSMOSWorks Online Tutorials
- SolidWorks Online Tutorials
- Click the Help menu and choose Help Topics for either
- COSMOSWORKS
- SolidWorks

Creating SolidWorks Model of the Beam

- 1.Create a new part. Click on the Standard toolbar.
- The New SolidWorks Document dialog box appears
- 2. Select the Part icon
- 3.Click OK

Creating a SolidWorks Model of the Cantilever Beam

- Create a Base Feature (Rectangle) of the beam by Boss Extrusion on the Front Plane
- First left click on the Sketch Icon

Creating a SolidWorks Model of the Cantilever Beam

- Open the SolidWorks sketch on the Front Plane.
- Left click on the Front Plane Icon.

Setting the Drawing Units to Inches in SolidWorks

- Setting the Drawing Units
- 1. Left click on the Tools menu and select Options
- 2. SelectDocument Properties in the Systems Operations Menu.
- 3. Under Document Properties select Units.
- 4. Choose inches in both locations inside your Document Properties
- 5.Click Ok to close menu

Creating a SolidWorks Model of the Cantilever Beam

- 1. Left click on the Rectangle Icon and create a rectangle
- 2. Left click on theCenterline Icon and then left click the two points in diagonal direction.
- Click OK after diagonal centerline appears

Creating a SolidWorks Model of the Cantilever Beam

- 3.Left click on the Add Relation Icon and select the centerline and origin
- Left click Midpoint and then left Click OK to move the center of rectangle to the origin

Creating a SolidWorks Model of the Cantilever Beam

- 5. Left click on the Smart Dimension Icon and dimension the width of the rectangle to be 0.9”
- 6. Dimension the height of the rectangle to be 0.9”,
- Left click OK to close the dimension dialog box on the left

Creating a SolidWorks Model of the Cantilever Beam

- 7. Left click the Extruded Boss/Base Icon and the sketch will be extruded into a 3-D solid.
- 10. Select direction 1 to be Blind and d1 to be 50 in.
- 11. Finally click the green OK button to complete the rectangular beam.

Creating a SolidWorks Model of the Cantilever Beam

- Click zoom to fit . This is what your beam will look like as a 3D solidWorks model.
- Finally prior to opening your model inside, COSMOSWorks, Save it to your computer directory.
- Give the model a name that you can remember and SAVE it. The file has an extension name is .prt )
- (Vibration Tutorial 1.prt).

Verifying that COSMOSWorks is loaded on your computer

- 1. Click the Tools item on the menu and select Add-Ins.
- 2. The Add-Ins box should appear and verify that the COSMOSWorks 2007 box is checked.

Opening your Model in COSMOSWorks

- 1. Click the COSMOSWorks Analysis Manager tab
- 2. In the COSMOSWorks Manager tree right click the Vibration Tutorial 1 and select Study
- 3. The Study dialog box appears

Opening your Model in COSMOSWorks

- 4. In the Study dialog box type in a name for study:Cantilever Beam and select Solid Mesh.
- 5. Under Type of study select Frequency.
- 6. Click theOK check when done
- 7. To specify the range of frequency, right click Cantilever Beam, and select Properties
- 8.A Frequency window will pop up, choose 5 for the Number of frequencies

The COSMOSWorks Study Folders

- COSMOSWorks automatically creates a folder called Cantilever Beam with the following sub folders: Solids, Load/Restraint, Design Scenario, Contac/Gaps, Mesh and Report folders
- We are now ready to define the mathematical model. This process generally consists of the following steps:
- Geometry preparation
- Material properties assignment
- Restraints application
- Contract/Gap assignment
- Load application
- The Contact/Gaps Icon will not be used in this Tutorial

Assigning Material Properties to the Model

- We now have a 3D model of the beam and need to assign material properties to it.
- We will select AISI 1020 Steel as the material for this model.
- COSMOSWorks will then assign the material properties of AISI 1020 Steel to the beam.

Assigning Material Properties

- 1. Right click the Solids Icon and select Apply Materials to All Bodies
- 2. Once the Material dialog box appears for Sourceselect fromLibrary Files.
- 3.Select the AISI 1020 Steel from the list of materials.
- 4.Click OK

Applying Restraints or Boundary Conditions to the Beam

- 1. In the COSMOS Tree Right click the Loads/Restraints Icon and select Restraints.
- 2.In the Restraints dialog under Type select Fixed
- 3. Left click inside the dialog box then select the front face of the beam model.

Applying Restraints or Boundary Conditions

- 4. The Restraints dialog box will contain Face<1> and the Top Surface of the Beam Model will have restraints symbols on it.
- 5. Now left click to close.

Applying Restraints or Boundary Conditions

- To keep the beam in y-z plane, let’s keep the displacement in the x-direction on the right face to be zero
- 1. In the COSMOS Tree Right click the Loads/Restraints Icon and select Restraints.
- 2.In the Restraints dialog under Type selectOn flat face.
- 3. Left click inside the dialog box, then select the right face of the beam.
- 4. Under Translation, left click the third icon (Normal to face) and make the translation 0
- 5. The Restraints dialog box will contain Face<1> and the Right Surface of the beam will have restraints symbols on it.
- 6. Now left click the check symbol

to close.

Meshing the Model and Running the Study

- 1. In the COSMOSWorks Manager tree, right click the Mesh Icon and select Create Mesh.
- 2.The Mesh Parameters Manager will appear, check the Run box and left click the Options (Run analysis after meshing) box.
- 3. Expand Options and a larger window appears:
- Mesh Quality-High
- Mesh Control- Smooth Surface
- Jacobian Check-4 point
- Mesher Type-Standard
- Finally Click
- 4. Click the green in the Mesh dialog box. Now, COSMOSWorks is performing analysis.

Viewing the Results of Finite Element Analysis

- After the analysis, all the results will be shown in Results folder, expand the folder in tree, 5 Displacements and 5 Deformations will appear.
- Double click on this Displacement 1Icon to view
- Natural Frequency and deformation scale are shown on the left top corner in the graphical window

Viewing the Results of Finite Element Analysis

- Folder plots can be edited by right clicking on the Plot and making a selection from :
- Edit Definition
- Animate
- Section Clipping
- Iso Clipping
- Chart Option
- Setting
- Axes
- Probe
- List Selected
- Save as
- Add to New Folder
- Copy
- Delete

Natural Frequenciesby Finite Element Analysis

- Right click the Results folder and select List Resonant Frequency to view the first five natural frequencies of the transverse motion,
- A ‘List Modes’ window listing the frequencies will appear

Viewing Mode Shapes at Natural Frequencies

- Double click the Deformation1, the mode shape at the first natural frequency appears.
- Right click the mouse in the graphic window, and select View Orientation and *Right to get a better view the mode shape in y-z plane

Animation of Lateral Motion at

- Right click the Displacement1, and select animate, an Animation dialog Box will appear (The Displacement1 needs to be in Bold font. If not, right click and select show)
- Left click the play Icon, a motion animation will be played.
- The playing speed can be also changed by adjusted the bar

W(x, t): Deflection : Mass Density A: Cross Sectional Area

L: length E: Young Modulus I: Second Moment of Inertia

x

Analytical Analysis of a cantilever beam- Expectations The analytical solution for the cantilever beam is readily available in the text “Mechanical Vibration” by Rao, page 394, 2nd Edition, 1990. The neutral frequencies are found from eqn (8.93) and Figure 8.15.

Comparison of Results between Analytical to Finite Element Analysis

The natural Frequency can be found by plugging the numerical values into the equation.

=0.2875 lb/in3 A=0.81 in2

L=50” E=30x106 psi I=0.05468 in4

Summary and Discussion

The FEA can provide insight in determining the natural frequency and mode shapes of a uniform cross sectional cantilever beam. The FEA method provides another analysis tool for the engineer to develop reliable designs that can avoid the severe vibrations due to the excitation close to the resonant frequency.

Appendix A:

- Background Information on Finite Elements
- Finite Element Theory

Background Information

- The purpose of this tutorial is to provide an introduction into using the finite element method* of engineering analysis and to gain experience using a commercial (FEA) software to analyze a structural problem with static loads. Finite element analysis, commonly called FEA is a numerical analysis method. FEA is a numerical analysis method used for solving field problems described by a set of partial differential equations. Other numerical methods include the Finite Difference Method, the Boundary Element Method, or the Finite Volume Method to name a few. COSMOSWorks is a commercial FEA software code capable of solving problems commonly found in design engineering. COSMOSWorks was developed by Structural Research & Analysis Corporation and has since merged with SolidWorks Corporation and both are currently owned by Dassault Systems.
- * The use of the finite element method of analysis is rarely used for basic design work, because the factor of safety doesn’t justify the higher precision.

Background Information

- We will be creating a rectangular beam in SolidWorks 3D for this finite element analysis. The solid model of the beam will then be submitted to the COSMOSWorks software to perform the finite element analysis. The finite element method of analysis includes three major steps : pre-processing, Computing the solution, and post-processing. Pre-processing involves preparing the 3-D model , discretizing the model, defining the model material properties, defining boundary conditions. Computing solutions involve using the software algorithms to solve the partial differential equations of the problem. Post-processing involves analysis and displaying the results in a useful format.

Finite Element Theory

- The discretization process, better known as meshing, splits the continuous 3D mathematical models into finite elements. The type of elements created in this process depends on the type of geometry meshed, the type of analysis that needs to be executed. More advance FEA software codes have three types of finite elements: one-dimensional elements or line elements, two-dimensional elements or shell elements and three-dimensional elements or solid elements to solve a varied number of problems. COSMOSWorks offers only two types of elements: three-dimensional tetrahedral solid elements, for meshing solid geometry, and two-dimensional triangular shell elements, for meshing surface geometry. These two types of elements will solve most typical engineering problems.

FEA Analogy: Area

What do we do to improve the accuracy of the area measurement?

FEA Mesh: Elements

- Each element is a simple solid.
- Elements are connected together at locations called NODES.

Background Information

- The beginning point for COSMOSWorks is a 3D geometric model of the problem, a part or assembly, representing the object that needs to be analyzed. We then assign material properties and defined structural boundary conditions for the model. The model must be constrained to generate stresses in the model, if it is not properly constrained we could have free body motion in space whereby no loads or stresses are developed. We next split the geometry into relatively small and simple shaped entities called finite elements. Creating finite elements is commonly called meshing.
- The COSMOSWorks mathematical solver approximates a solution to the constitutive partial differential (PD) equations of the meshed model. COSMOS has three high speed math solvers one using a direct method of solution to the PD equations and two using a iterative method of solution to the PD equations.

Finite Element Theory

- The tetrahedral solid elements can be either first order (draft quality) or second order elements (high quality). The user decides whether to use draft quality or high quality elements for meshing the 3D geometric model. However only high quality elements are used in analysis of importance. First order tetrahedral elements have four nodes, straight edges and flat faces. Second order tetrahedral elements have ten nodes and are more accurate in modeling problems. The second order elements are the elements of choice for accurate results.
- The use of the elements with the higher number of nodes has improved accuracy with but with additional computational time over the elements with less nodes. Each tetrahedral element with either 4 or 10 nodes per element has three degrees of freedom (DOF) for each node. The degrees of freedom of a node in a finite element mesh define the ability of the node to perform translation or rotation. The number of DOF that a node posses depends on the type element that the element belongs to.

Finite Element Theory

- Node of solid elements have three degrees of freedom while nodes of shell elements have six degrees of freedom. This means that in order to describe transformation of a solid element from the original to the deformed shape, we need to know three translational components of nodal displacement usually x, y and z. In the case of a shell element we need to know six DOF or three translations and three rotations for each node.
- Each degree of freedom of each node in a finite element mesh constitutes an unknown. A partial differential equation defining the physics of the problem is solved for displacements at specific locations on each finite element and extrapolated to each node. Once the displacements are calculated the strains and stresses can be calculated for the model.

Finite Element Theory

- In thermal analysis, which determines temperatures and heat flow, the primary unknowns are nodal temperatures. Since temperature is a scalar displacement, and not a vector-like displacement, then regardless of what type of elements used, there is only one unknown temperature to be found for each node. The fact that there is only one unknown to be found for each node, rather than three or six, makes thermal analysis less computationally intensive than structural analysis.
- Errors in FEA. The process of creating a mathematical model and discretizing it into a finite element model introduces unavoidable errors. FEA errors can be categorized into three areas: 1. mathematical modeling errors, 2. discretization errors during meshing, and 3. solution errors which are round-off errors accumulated by the solver. In most instances these errors are usually very low (3% or less) when compared with classical analysis.

Mesh Accuracy : Mesh Size or Mesh Nodes

Coarse Mesh:

1773 nodes

Moderate Mesh:

7009 nodes

Fine Mesh:

16,107 nodes

Sol. Time: 5 sec.

Max. Stress: 27.8 ksi

Sol. Time: 10 sec.

Max. Stress: 27.6 ksi

Sol. Time: 2 sec.

Max. Stress: 25.8 ksi

Finite Element Theory

- Limitations of COSMOSWorks and other FEA software. We need to appreciate some important limitations of this FEA software: material is assumed as linear, deformations are small, and loads are static. Material we assign to be analyzed will be assumed to be linear or that the stress is proportional to strain in linear manner.
- In “real-life” there is a yield or ultimate stress that the material cannot exceed without rupturing. A linear model omits these “real-life” end conditions. We therefore must review the level of stresses very carefully in our FEA results. The fact that we assume small deformations requires that those deformations be “small” in relation to the size (3% or less) of the structure and that the “structural-stiffness” matrix remains relatively the same during the deformation process. All loads, as well as restraints, are assumed not to change with time, meaning that dynamic loading conditions can not be analyzed with COSMOSWorks. This time limitation implies that loads are applied slowly enough to ignore inertial effects.

Reference List of Texts

- 1. Sprakos, C. , “Finite Element Modeling”, Algor, Inc.,1996.
- 2.Moaveni, S., “Finite Element Analysis Theory and Application with ANSYS, Prentice Hall,1999
- 3. Chandrupatla, T., Belegundu,A., “Introduction to Finite Elements”, Prentice Hall, 2001.
- 4. Hutton,D., “Fundamentals of Finite Element Analysis”, McGraw Hill,2004.
- 5.Kurowski, P.M., Engineering Analysis with COSMOSWorks”. Schroff Development Corporation, 2006
- 6.”COSMOSWorks Designer 2006 Training Manual, Structural Research Corporation, 2006

Acknowledgement

- This tutorial was developed under the NSF Grant Number 0536197

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