Learning Module 6 Linear Dynamic · PDF file1 Learning Module 6 Linear Dynamic Analysis Title Page Guide What is a Learning Module? A Learning Module (LM) is a structured, concise,

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Learning Module 6

Linear Dynamic Analysis

Title Page Guide

What is a Learning Module?

A Learning Module (LM) is a structured, concise, and self-sufficient learning

resource. An LM provides the learner with the required content in a precise and

concise manner, enabling the learner to learn more efficiently and effectively. It has a

number of characteristics that distinguish it from a traditional textbook or textbook

chapter:

An LM is learning objective driven, and its scope is clearly defined and bounded.

The module is compact and precise in presentation, and its core material contains

only contents essential for achieving the learning objectives. Since an LM is

inherently concise, it can be learned relatively quickly and efficiently.

An LM is independent and free-standing. Module-based learning is therefore non-

sequential and flexible, and can be personalized with ease.

Presenting the material in a contained and precise fashion will allow the user to learn

effectively, reducing the time and effort spent and ultimately improving the learning

experience. This is the first module on structural analysis and covers a static structural

study in FEM. It goes through all of the steps necessary to successfully complete an

analysis, including geometry creation, material selection, boundary condition

specification, meshing, solution, and validation. These steps are first covered

conceptually and then worked through directly as they are applied to an example

problem.

Estimated Learning Time for This Module

Estimated learning time for this LM is equivalent to three 50-minute lectures, or one

week of study time for a 3 credit hour course.

How to Use This Module

The learning module is organized in sections. Each section contains a short

explanation and a link to where that section can be found. The explanation will give

you an idea of what content is in each section. The link will allow you to complete the

parts of the module you are interested in, while being able to skip any parts that you

might already be familiar with. The modularity of the LM allows for an efficient use

of your time.

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1. Learning Objectives

The objective of this module is to introduce the user to the process of Linear Dynamic

analysis using FEM. Upon completion of the module, the user should have a good

understanding of the necessary logical steps of an FEM analysis, and be able to perform

the following tasks:

Creating the solid geometry

Assigning material properties

Imposing displacement boundary conditions

Applying external forces

Meshing

Running the analysis

Verifying model correctness

Processing needed results

2. Prerequisites

In order to complete the learning module successfully, the following prerequisites are

required:

By subject area:

o Dynamics,

o Mechanics & vibrations of Materials.

By topic:

Knowledge of

o Displacement

o Velocity

o Acceleration

o Force balance

o Types of Vibrations

o Harmonic vibrations

o Modal analysis

o Damping

o Free vibrations

o Forced vibrations

o Degrees of Freedom

o Mode shapes

3. Pre-test

The pre-test should be taken before taking other sections of the module. The purpose of

the pre-test is to assess the user's prior knowledge in subject areas relevant to static

structural analysis such as Mechanics of Materials. Questions are focused towards

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fundamental concepts including stress, strain, displacement, kinematic relationship,

constitutive relationship, equilibrium, and material properties.

Pre-test

1. What is a free harmonic motion?

2. Define

a) critical damping

b) under damping

c) over damping

3. Differentiate free vibration and forced vibration. 4. What is steady state vibration?

5. Define Natural Frequency.

6. What is transient state vibration?

7. What is single Degree of Freedom?

8. What is a mode and its shape?

9. Differentiate Linear and Nonlinear vibrations.

10. What is torsional vibration?

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4. Tutorial Problem Statements

A good tutorial problem should focus on the logical steps in FEM modeling and

demonstrate as many aspects of the FEM software as possible. It should also be simple in

mechanics with an analytical solution available for validation. Three tutorial problems are

covered in this learning module.

Tutorial Problem 1

A Uniform Cantilever beam of 10 inches length, 2 inches breadth and 0.3 inches height is

fixed at one end as shown in the Figure 1. The other end of the beam is subjected to a

transverse end loading of 250 lbf in the direction as shown. The end loading varies with

time as a square wave for 0.1s. The beam is elastic and is made up of Alloy Steel.

Perform a Modal Analysis for the cantilever beam for both no damping and with damping

cases. And plot time history response.

Figure 1. A Cantilever beam fixed on one end and loaded on the other end

5. Conceptual Analysis

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Conceptual analysis is the abstraction of the logical steps in performing a task or solving

a problem. Conceptual analysis for FEM simulation is problem type dependent but

software-independent, and is fundamental in understanding and solving the problem.

Conceptual analysis for a Linear Dynamics problem using finite element analysis reveals

that the following logical steps and sub-steps are needed:

1. Pre-processing (building the model)

1. Geometry creation

2. Material property assignment

3. Boundary condition specification

o Prescribed displacement boundary condition (holding the model)

o Applied force boundary condition (loading the model)

4. Mesh generation

2. Solution (running the simulation)

3. Post-processing (getting results)

4. Validation (checking)

The above steps are explained in some detail as follows.

1. Pre-processing

The pre-processing in FEM simulation is analogous to building the structure or making

the specimen in physical testing. Several sub-steps involved in pre-processing are

geometry creation, material property assignment, boundary condition specification, and

mesh generation.

The geometry of the structure to be analyzed is defined in the geometry creation step.

After the solid geometry is created, the material properties of the solid are specified in the

material property assignment step. The material required for the FEM analysis depends

on the type of analysis. For example, in the elastic deformation analysis of an isotropic

material under isothermal condition, only the modulus of elasticity and the Poisson’s

ratio are needed.

For most novice users of FEM, the boundary condition specification step is probably the

most challenging of all pre-processing steps. Two types of boundary conditions are

possible. The first is prescribed displacement boundary condition which is analogous to

holding or supporting the specimen in physical testing. The second is applied force

boundary condition which is analogous to loading the specimen. Several factors

contribute to the challenge of applying boundary conditions correctly:

1) Prescribed displacement boundary conditions expressed in terms such as

constuaboundary or const

x

u

bboundary

are mathematical simplifications, and

frequently only represent supports in real structures approximately. As a result,

choosing a good approximate mathematical representation can be a challenge.

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2) How a boundary is restrained depends also on the element type. For example, for

the "clamped" or "built-in" support, a boundary should be restrained as having

zero nodal displacement if solid element is used, while for the same support, the

boundary should be restrained as having zero nodal displacement and zero nodal

rotation if shell element is used.

3) Frequently, the structure to be analyzed is not fully restrained from rigid body

motion in the original problem statement. In order to obtain an FEM solution,

auxiliary restraints become necessary. Over-restraining the model, however, leads

to spurious stress results. The challenge is then adding auxiliary restraints to

eliminate the possibility of rigid body motion without over-restraining the

structure.

Because of the above challenges, one learning module will be devoted to boundary

condition specification.

Mesh generation is the process of discretizing the body into finite elements and

assembling the discrete elements into an integral structure that approximates the original

body. Most FEM packages have their own default meshing parameters to mesh the model

and run the analysis while providing ways for the user to refine the mesh.

2. Solution

The solution is the process of solving the governing equations resulting from the

discretized FEM model. Although the mathematics for the solution process can be quite

involved, this step is transparent to the user and is usually as simple as clicking a solution

button or issuing the solution command.

3. Post-processing

The purpose of an FEM analysis is to obtain wanted results, and this is what the post-

processing step is for. Typically, various components or measures of stress, strain, and

displacement at any given location in the structure are available for putout. Additional

quantities for output may include factory of safety, energy norm error, contact pressure,

reaction force, strain energy density, etc. The way a quantity is outputted depends on the

FEM software.

4. Validation

Although validation is not a formal part of the FEM analysis, it is important to be

included. Blindly trusting a simulation without checking its correctness can be dangerous.

The validation usually involves comparing FEM results at one or more selected positions

with exact or approximate solutions using classical approaches such as elasticity or

mechanics of materials. Going through validation strengthens conceptual understanding

and enhances learning.

Conceptual Analysis of the Given Problem

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The goal of the FEM simulation is to analyze the behavior of the solid with the given

forces acting on it. The problem shows a uniform cantilever beam which is fixed at one

end and has a traverse loading of 250 lbf acting at the other end. Conceptual analysis of

the current problem is described as follows.

1. Pre-processing (building the model)

The geometry of the structure is first created using the design feature of the FEM

package. Next, a material is assigned to the solid model. In the given problem, the

material of the beam is given as Alloy steel. Depending on the software, the material is

either directly selected as Alloy steel from the material library, or the properties of the

material given in the problem are inputted directly.

After assigning the material properties, the boundary conditions are specified. The end

that is attached to the wall will need a fixed restraint, which means zero displacement for

all boundary nodes due to the solid mesh. The load is applied on the other end. The

loading varies as a square wave with respect to time.

The next step is to mesh the solid to discretize it into finite elements. Generally,

commercial FEA software has automatic default meshing parameters such as average

element size of the mesh, quality of the mesh, etc. Here the default parameters provided

by the software is used.

2. Solution (running the simulation)

The next step is to run the simulation and obtain a solution. Usually the software provides

several solver options. The default solver usually works well. For some problems, a

particular solver may be faster or give more accurate results.

3. Post-processing (getting results)

After the analysis is complete, the post-processing steps are performed. Results such as

von Mises stress, various stresses, displacements, and strains can be viewed.

4. Validation (checking)

Validation is the final step in the analysis process. In this step, the stresses acting on the

beam are calculated by hand. These analytical solutions are compared with the software

generated results to check the validity of the analysis.

6. Abstract Modeling

Abstract modeling is a process pioneered by CometSolutions Inc. Abstract modeling

enables all attributes of an FEM model (such as material properties, constraints, loads,

mesh, etc.) to be defined independently in an abstract fashion, thus reducing model

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complexity without affecting model accuracy with respect to the simulation objective. It

detaches attributes from one another, and emphasizes conceptual understanding rather

than focusing on software specifics. Evidently, abstract modeling is independent of the

specific software being used. This is a fundamental departure from the way most FEM

packages operate.

Conceptual analysis focuses on the abstraction of steps necessary for an FEM simulation,

while abstract modeling focuses on the abstraction and modularization of attributes that

constitute an FEM model. They are powerful enabling instruments in FEM teaching and

learning.

7. Software-Specific FEM Tutorials

In software-specific FEM tutorial section, the tutorial problem is solved step by step in a

particular software package. This section fills in the details of the conceptual analysis as

outlined in previous section. It provides step by step details that correspond to the pre-

processing, solution, post-processing and validation phases using a particular software

package.

8. Post-test

The post-test will be taken upon completion of the module. The first part of the post-test

is from the pre-test to test knowledge gained by the user, and the second part is focused

on the FEM simulation process covered by the tutorial.

9. Assessment

The assessment is provided as a way to receive feedback about the module. The user

evaluates several categories of the learning experience, including interactive learning, the

module format, its effectiveness and efficiency, the appropriateness of the sections, and

the overall learning experience. There is also the opportunity to give suggestions or

comments about the module.

10. Practice Problems

The user should be able to solve practice problems after taking the module. The practice

problems provide a good reinforcement of the knowledge and skills learned in the

module, and can be assigned as homework problems in teaching or self study problems to

enhance learning. These problems are similar to the tutorial problem worked in the

module, but they involve different geometries and loading modes, stress concentration,

and statically indeterminate beams.

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Tutorial Problems

Overview: In this section, three tutorial problems will be solved using the commercial

FEM software SolidWorks. Although the underlying principles and logical steps of an

FEM simulation identified in the Conceptual Analysis section are independent of any

particular FEM software, the realization of conceptual analysis steps will be software

dependent. The SolidWorks-specific steps are described in this section.

This is a step-by-step tutorial. However, it is designed such that those who are familiar

with the details in a particular step can skip it and go directly into the next step.

Tutorial Problem 1. A uniform cantilever beam fixed at one end and under a

transverse end square loading.

0. Launching SolidWorks

SolidWorks Simulation is an integral part of the SolidWorks computer aided design

software suite. The general user interface of SolidWorks is shown in Figure 4.

Figure 4: General user interface of SolidWorks.

Main menu Frequently used command icons Help icon

Roll over to

display

“File”,

“Tools” and

other menus

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In order to perform FEM analysis, it is necessary to enable the FEM component,

called SolidWorks Simulation, in the software.

Step 1: Enabling SolidWorks Simulation

o Click "Tools" in the main menu. Select "Add-ins...". The Add-ins dialog

window appears, as shown in Figure 2.

o Check the boxes in both the “Active Add-ins” and “Start Up” columns

corresponding to SolidWorks Simulation.

o Checking the “Active Add-ins” box enables the SolidWorks for the

current session. Checking the “Start Up” box enables the SolidWorks for

all future sessions whenever SolidWorks starts up.

Figure 2: Location of the SolidWorks icon and

the boxes to be checked for adding it to the panel.

1. Pre-Processing

Purpose: The purpose of pre-processing is to create an FEM model for use in the next

step of the simulation, Solution. It consists of the following sub-steps:

Geometry creation

Material property assignment

Boundary condition specification

Mesh generation.

1.1 Geometry Creation

The purpose of Geometry Creation is to create a geometrical representation of the solid

object or structure to be analyzed in FEM. In SolidWorks such a geometric model is

called a part. In this tutorial, the necessary part has already been created in SolidWorks.

The following steps will open up the part for use in the FEM analysis.

Step 1: Opening the part for simulation. One of the following two options can be

used.

Check

“SolidWorks

Simulation” boxes

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o Option1: Double click the following icon to open the embedded part file,

part1.SLDPRT, in SolidWorks.

Click SolidWorks part file icon to open it ==>

o Option 2: Download the part file “part1.SLDPRT” from the web site

http://www.femlearning.org/. Use the “File” menu in SolidWorks to open the

downloaded part.

The SolidWorks model tree will appear with the given part name at the top. Above the

model tree, there should be various tabs labeled “Features”, “Sketch”, etc. If the

“Simulation” tab is not visible, go back to steps 1 and 2 to enable the SolidWorks

Simulation package.

Step 2: Creating a Study

o Click the “Simulation” tab above the model tree

o Click on the drop down arrow under “Study” and select “New Study” as in

Figure 3

o In the “Name” panel, give the study the name “Frequency”

o Select “Frequency” in the “Type” panel to study the static equilibrium of the

part under the load

o Click “OK” to accept and close the menu

Figure 3: The SolidWorks “Study” menu.

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1.2 Material Property Assignment

The Material Property Assignment sub-step assigns materials to different components of

the part to be analyzed. All components must be assigned with appropriate material

properties.

Step 3: Opening the material property manager

o In the upper left hand corner, click “Apply Material”.

o The “Material” window appears as shown in Figure 4.

Figure 4: The “Material” window.

This will apply one material to all components. If the part is made of several components

with different materials, open the model tree and apply this process to individual

components.

SolidWorks has a built-in material library that can be directly selected for the part.

Alloy Steel is selected in the Material List.

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1.3 Boundary Condition Specification

In the Boundary Condition Specification sub-step, the restraints and loads on the part are

defined. Here, the face of the beam attached to the wall needs to be restrained, and the

force in the proper direction needs to be applied on the other end of the beam.

Step 5: Opening the fixtures property manager

o Right click on “Fixtures” in the model tree and select “Fixed Geometry”

o Move the cursor into the graphic window.

As the cursor traverses the image of the model, notice a small icon accompany the cursor.

This icon will change shapes when the cursor is at different locations on the part. This

indicates that SolidWorks is in graphical selection mode, and different shapes indicate

different identities would be selected: a square (icon) indicates the surface to be selected

if the mouse is clicked, a line (icon) for an edge or a line, and a dot (icon) for a point. In

this tutorial problem, the entire end surface is restrained.

Figure 5: Applying an immovable restraint to the beam.

At the initial orientation, the end to be restrained is not visible, and cannot be selected.

The model should be rotated to make the fixed end visible. To rotate the model either

hold down the scroll bar and rotate with the mouse or change the orientation by clicking

on the “View Orientation” icon in the top middle area of the workspace.

Once the desired face is visible, select the face on which to apply the restraint. Note that

in the display panel, within the second box in the “Type” panel, “Face<1>” appears,

indicating that one surface is being selected. Clicking on this face in the graphics panel

would deselect the face.

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Step 6: Restraining the member

o Select the face as in Figure 5

o Once the face has been selected, click “OK” to close the “Fixture” menu

1.4 Creation of Point:

o Click on Insert icon from the pull down menu and select Reference Geometry

and select Point.

o Then select the face as shown in the below figures.

o Then the midpoint of the face is created.

o This point is used for defining the sensor to obtain the response graph.

Figure . Creation of point in 3D-coordinates using center of face

Figure . Creation of point in 3D-coordinates using center of face

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1.5 Defining the Part properties:

o Right click on the frequency study on the feature tree and then select

properties, next click on the options window as shown in the below figure.

o Then enter 3 for number of frequencies.

o Select Direct Sparse for the solver.

o Then click on run icon.

o Three frequency plots can be identified in the results folder.

o Now we need to incorporate these frequency results to our actual problem.

1.6 Creation of a New Dynamic Study:

Step 1: A new dynamic study is created by using the results from the frequency study.

o Right click on the frequency study icon in the study tree.

o Then select the “copy to new dynamic study”.

o Then name the new study as “Modal Analysis-Without Damping” and select

study type as Modal time History analysis.

Then a new dynamic study is created with the existing frequency study results.

The figures shown below depict the procedure to be followed.

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o Right click on the Modal Analysis-Without Damping study on the feature tree

and then select properties, next click on the options window as shown in the

below figure.



o Then click on dynamic options, there we need to define the start time, end

time and frequency as shown in the figure.

The next step is to apply a load of 250 lb force vertically downwards on the face of the

beam as shown in the below figure.

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After selecting the face and the direction, we need to define the curve parameter as the

loading here we are going to define is a square wave loading with time interval of 0.1

seconds. The load curve is defined similar to the figure shown below.

After the load is defined, the result options are defined by right clicking on the results

options and are updated as shown in the below figure.

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1.8 Mesh Generation

Purpose: The purpose of the Mesh Generation sub-step is to discretize the part into

elements. The mesh consists of a network of these elements.

Step 9: Creating the mesh

o Right click “Mesh” in the model tree and select “Create mesh”

o Leave the mesh bar on its default value

o Expand the “Advanced” menu and ensure the mesh is high quality, not draft

quality, by making sure the “Draft Quality Mesh” checkbox is unchecked

o Figure shows the completed mesh

Click “OK” to close the menu and generate the mesh

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“Mesh Control” in SolidWorks may be used to refine the mesh locally. The guiding

principle is to refine mesh at locations of high stress gradient, such as regions around

stress concentrators and locations of geometric changes. For the current problem, local

mesh refinement is not pursued.

2. Solution

Purpose: The Solution step is where the computer solves the simulation problem and

generates results for use in the Post-Processing step.

Step 1: Running the simulation

o At the top of the screen, click “Run”

o When the analysis is complete, the “Results” icon will appear on the model

tree

3. Post-Processing

Purpose: The purpose of the Post-Processing step is to process the results of interest. For

this problem, the Displacement plots in Y-direction for various modes of vibration and

the time response graph are of interest.

The following plots are obtained:

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Figure. Displacement in Y direction for the first mode of vibration.

Figure. Displacement in Y direction for the Second mode of vibration.

21

Figure. Displacement in Y direction for the Third mode of vibration.

1. We need to plot the Time History response graph. For that we need to right click on

the results folder and then select Define Response Graph. And then follow the

procedure shown in the below figure.

The following plot represents the time history plot for the model which we have done

analysis.

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2. Then the Absolute maximum Y-displacement is plotted as shown below. And this

value is being compared with the analytic results.

Next our task is to do analysis for a Damped case. Where, a Damping coefficient of 0.02

is defined in the analysis.

23

Creation of a new study for the Damped case:

o We can create a new study and follow the same procedure we have done in the

previous study or we can create a duplicate of the existing study and we can

modify it by defining the damping ratio.

o Either wise we can do the following study.

So the procedure followed for the above analysis holds good for this case also. Except we

need to define the damping coefficient.

We can define the damping like as shown in the Figure.

o Right click on the damping icon in the study tree.

o Then select modal damping and enter 0.02 for damping ratios.

o Then hit ok.

24

Then similar to the previous study, we will run the solution and observe the results as

shown below.

Figure. Displacement in Y direction for the first mode of vibration.

Figure. Displacement in Y direction for the Second mode of vibration.

25

Figure. Displacement in Y direction for the Third mode of vibration.

The above plot represents the time history plot for the model which we have done

analysis.

26

Then the Absolute maximum Y-displacement is plotted as shown below. And this value

is being compared with the analytic results.

4. Validation

Purpose: The purpose of the Validation step is to compare FEM solutions with analytical

solutions, or known published results, to validate the correctness of the FEM model.

For the current problem, closed form solutions based on beam analysis from vibrations of

Materials by using the Mat Lab software are computed and compared with the FEM

results. This will reveal whether or not the results of the finite element analysis are

reasonable.

From Lumped Parameter Modeling,

Kequivalent =

= 411.182

From the above equation, the Kequivalent can be calculated.

For a cantilever beam, the mass equivalent is given by:

mequivalent =

= 0.2357m = 0.3934

then natural frequency of the cantilever beam is calculated by using the following

equation:

27

ωn = √

= 32.33

by using the above equations we calculate the natural frequency of the cantilever beam.

And by considering the periodic loading for mass-spring dashpot system, we will find out

the fourier expansion and corresponding Mat Lab codes are written and executed as

follows:

The given cantilever beam can be assumed as a spring system then,

For Without-Damping Case:

o The following figure shows the program code in Mat Lab to determine the

frequency response.

o The Frequency response plot is shown below.

28

o The plot matches with the input square wave frequency hence our results achieved

so far are considered to be acceptable.

Now we look into the With-Damping case.

For With-Damping Case:


frequency response.


29

Tutorial Problem 2. Harmonic analysis of a uniform cantilever beam having a mass

and loaded at one end and the other end is fixed.




General user interface of SolidWorks.












Roll over to

display

“File”,

“Tools” and

other menus

30

Location of the SolidWorks icon and


1. Pre-Processing



Geometry creation



Mesh generation.







used.

Download the part file “part2.sldassm” from the web site


downloaded part.




Simulation package.



Check

“SolidWorks

Simulation” boxes

31


Figure 3



part under the load


The SolidWorks “Study” menu.




properties.



o The “Material” window appears as shown in the below figure.

32

The “Material” window.



components.










33







Applying an immovable restraint to the beam.

















34








o Then name the new study as “Harmonic Analysis” and select study type as

Harmonic analysis.



35


o Right click on the Harmonic analysis study on the feature tree and then select








Similar to the previous tutorial, the loading is defined.



seconds.

36



37

Defining Damping coefficient:

Damping coefficient is being defined as shown in the following figure.

38

1.8 Mesh Generation














2. Solution





39


tree

3. Post-Processing





Displacement in Y direction for the first mode of vibration.

40

Displacement in Y-direction for the second mode of vibration.

Displacement in Y-direction for the third mode of vibration.

4. The Response plot for displacement is being plotted similar to the method we

followed in the previous tutorial.

41

1. The Response plot for acceleration is being plotted similar to the method we

followed in the previous tutorial.

42

A continuous solution can be obtained for the cantilever beam or an analytical model can

be developed using the finite element technique in a software package such as MATLAB

[3]. (Reference 4 contains some brief notes concerning the finite element modeling

process as well as some rudimentary MATLAB script files for the generation of a simple

cantilever beam model). Using some basic strength of materials approximations along

with the continuous beam vibration equation, an equivalent model can be developed for

analysis purposes.

43

44

45

Tutorial Problem 3. A uniform cantilever beam fixed at one end and having a mass

at other end with a transverse end sinusoidal loading.




General user interface of SolidWorks.












Roll over to

display

“File”,

“Tools” and

other menus

46

Location of the SolidWorks icon and


1. Pre-Processing



Geometry creation



Mesh generation.







used.

Download the part file “part3.SLDASSM” from the web site


downloaded part.




Simulation package.



Check

“SolidWorks

Simulation” boxes

47


Figure 3



part under the load


The SolidWorks “Study” menu.




properties.



o The “Material” window appears as shown in Figure 4.

48

The “Material” window.



components.










49







Applying an immovable restraint to the beam.

















50








o Then name the new study as “Modal Analysis-Without Damping” and select

study type as Modal time History analysis.



51


o Right click on the Modal Analysis-Without Damping study on the feature tree

and then select properties, next click on the options window as shown in the

below figure.







52



seconds. The load curve is defined similar to the figure shown below.



53

1.8 Mesh Generation














2. Solution






tree

54

3. Post-Processing





Displacement in Y direction for the first mode of vibration.

55

Displacement in Y-direction for the second mode of vibration.


56

6. We need to plot the Time History response graph. For that we need to right click on

the results folder and then select Define Response Graph. And then follow the

procedure shown in the below figure.


analysis.

57

7. Then the Absolute maximum Y-displacement is plotted as shown below. And this


58

Next our task is to do analysis for a Damped case. Where, a Damping coefficient of 0.02

is defined in the analysis.

Creation of a new study for the Damped case:

o We can create a new study and follow the same procedure we have done in the

previous study or we can create a duplicate of the existing study and we can

modify it by defining the damping ratio.

o Either wise we can do the following study.

So the procedure followed for the above analysis holds good for this case also. Except we

need to define the damping coefficient.

We can define the damping like as shown in the Figure.

o Right click on the damping icon in the study tree.

o Then select modal damping and enter 0.1 for damping ratios.

o Then hit ok.

59

Then similar to the previous study, we will run the solution and observe the results as

shown below.

Displacement in Y-direction for the First mode of vibration.

60

Displacement in Y-direction for the Second mode of vibration.



analysis.

61

Then the Absolute maximum Y-displacement is plotted as shown below. And this


62

4. Validation

Purpose: The purpose of the Validation step is to compare FEM solutions with analytical

solutions, or known published results, to validate the correctness of the FEM model.

For the current problem, closed form solutions based on beam analysis from vibrations of

Materials by using the Mat Lab software are computed and compared with the FEM

results. This will reveal whether or not the results of the finite element analysis are

reasonable.

From Lumped Parameter Modeling,

Kequivalent =

= 1903.62

From the above equation, the Kequivalent can be calculated.

For a cantilever beam, the mass equivalent is given by:

mequivalent =

= 0.2357m = 0.4065

then natural frequency of the cantilever beam is calculated by using the following

equation:

ωn = √

= 68.432

by using the above equations we calculate the natural frequency of the cantilever beam.

And by considering the periodic loading for mass-spring dashpot system, we will find out

the fourier expansion and corresponding Mat Lab codes are written and executed as

follows:

The given cantilever beam can be assumed as a spring system then,

For Frequency F=0.1*(natural frequency):


frequency response.

63


o The plot matches with the input square wave frequency hence our results achieved

so far are considered to be acceptable.

For Frequency F=1.0*(natural frequency):


frequency response.

64


65

For Frequency F=10*(natural frequency):


frequency response.


66

Post-test

2. What is a harmonic analysis?

2. Define

d) critical damping

e) under damping

f) over damping

3. Differentiate free vibration and forced vibration. 4. Define Damping Ratio.

5. Define Natural Frequency.

6. What are the effects of resonance?

7. What is single Degree of Freedom?

8. List the types of damping techniques available.

9. Differentiate Linear and Nonlinear vibrations.

10. what is torsional vibration?

67

Attachment F. Assessment

1. Do you feel it was bad to not have a teacher there to answer any questions you

might have?

O It didn’t matter

O It would have been nice

O I really wanted to ask a question

2. How did the interactivity of the program affect your learning?

O Improved it a lot

O Improved it some

O No difference

O Hurt it some

O Hurt it a lot

3. The six levels of Bloom’s Taxonomy are listed below. Rank how well this

learning module covers each level. 5 meaning exceptionally well and 1 meaning

very poor.

1. Knowledge (remembering previously learned material)

O 5

O 4

O 3

O 2

O 1

2. Comprehension (the ability to grasp the meaning of the material and give

examples)

O 5

O 4

O 3

O 2

O 1

3. Application (the ability to use the material in new situations)

O 5

O 4

O 3

O 2

O 1

68

4. Analysis (the ability to break down material into its component parts so that

its organizational structure may be understood)

O 5

O 4

O 3

O 2

O 1

5. Synthesis (the ability to put parts together to form a new whole)

O 5

O 4

O 3

O 2

O 1

6. Evaluation (the ability to judge the value of the material for a given purpose)

O 5

O 4

O 3

O 2

O 1

4. Do you think the mixed text and video format works well?

O Yes

O Indifferent

O No

5. Do you think the module presents an affective method of learning FEA?

O Yes

O Indifferent

O No

6. Did you prefer this module over the traditional classroom learning experience?

Why or why not.

69

7. How accurate would it be to call this module self-contained and stand-alone?

O Very accurate

O Accurate

O Indifferent

O Inaccurate

O Very inaccurate

8. What specifically did you like and/or dislike about the module.

9. How useful were the practice problems?

O Very helpful

O Helpful

O Indifferent

O Unhelpful

O Very unhelpful

10. Was there any part of the module that you felt was unnecessary or redundant?

Was there a need for any additional parts?

11. Please list any suggestions for improving this module.

12. Overall, how would you rate your experience taking this module?

O Excellent

O Fair

O Average

O Poor

O Awful

70

Practice Problems:

1. Follow the first tutorial and redo the tutorial problem for 5 frequency nodes and

also for a damping ratio of 0.2.

2. For the same first tutorial problem change the time period to 0.5 seconds and

observe the results.

3. Follow the second tutorial and redo the tutorial problem for 5 frequency nodes

and do it without damping and with 0.2 damping.

4. Redo the second tutorial problem with sine wave loading of 0.1 seconds time

period.

5. Follow the third tutorial and do the problem with a damping ratio of 0.5 and do it

for 5 frequency nodes.

Documents

Learning Module 6 Linear Dynamic · PDF file1 Learning Module 6 Linear Dynamic Analysis Title Page Guide What is a Learning Module? A Learning Module (LM) is a structured, concise,