previous

The beam below is cantilevered or "built in" on the left edge. This means that both the translations and the rotations are held to zero along this edge. The material properties for the beam are E= 70 x 10

Assume the element has the configuration shown below:

Our goal is to find the element stiffness matrix

ASSUME: 2 displacement degrees of freedom (dof) per node With :

[B] = the strain - displacement matrix such that [

where: {

[E] = the constitutive matrix such that [E]{

where {

This allows us to find the entries in [B]

where |J| is the determinant of the Jacobian matrix.

The elements as formed above must be assembled into a global stiffness matrix. In the same manner, element mass matrices are formed using the equation

a) Log on to the computer

b) Click START (lower left corner of the Windows Desktop), go to Programs, Select MSC (common), Select MSC Patran9.0.

From the TM choose

A SM called

Turn on (checked)

Click

A

Under

Set

Under

Choose

click

Set

Set the

Set

Turn off the

Enter the following into the

Click

A point will appear in the main viewport at coordinates [0,0,0]

Use this same procedure to create points at coordinates [1,0,0], [1,0.1,0] and [0,0.1,0]

Back at the top of the RM called

Set

Turn

Set

Set

Click

Back at the top of the RM called

Set

Set the

Turn

Set

Click

Back at the top of the RM called

Set

Set the

Set

Set

Set

Click

From the TM choose

A RM appears called

Set

Set

Set

Set

Set

Click in the

Click and drag to select the entire structure

The Words "Surface 1" should appear in the

From the TM choose

Set

Set

Click

Set

Set

Click

(back in the

Click

Turn on the

Click in box under

Use the cursor to highlight the set of nodes along the left vertical edge of the beam. There should be 5 nodes there.

Click

(The

Click

On the TM select

Set

Click

Input the name to be

SM called

Input

Input the

Click

Click

a RM will appear called

Set

Click

Enter

Click

a SM appears called

Click in the

Click on the word "aluminum" in the

( the words m:aluminum will appear in the

Click in the

Enter

Click

(Back in the

Click

a Patran

Click on the icon which contains the surface or face icon

Click

(The words Surface 1 appears in the

Click

(beam_prop will be added to the

On the TM select

a RM will appear called

Click Translation Parameters

In the SM that appears, set

Click

Click

On the TM select

Set

Click

A SM appears called

(You may need to look in your home or root directory to find the file. If this file does not exist, then you have made a mistake in constructing your model. Go to Explorer (right-click on Start and choose Explore) and find the file beam- vib.log and beam.f06. Open these files by double clicking on them and search for the word “error” to determine what your mistake is).

Click

Click
On the TM select

A RM will appear called

In the

In the Sel

In the

Turn on the

Click

On the TM select

From the pull down menu select

On the TM select

next

**MODELING A STABELIZATION FIXTURE WITH END PRESSURE USING SOLID ELEMENTS**

__MSC/PATRAN TUTORIAL # 6__

MODELING A CANTILEVERED BEAM’S VIBRATION USING 4 NODE SHELL ELEMENTSMODELING A CANTILEVERED BEAM’S VIBRATION USING 4 NODE SHELL ELEMENTS

**I. THE PHYSICAL PROBLEM**The beam below is cantilevered or "built in" on the left edge. This means that both the translations and the rotations are held to zero along this edge. The material properties for the beam are E= 70 x 10

^{9}Pascals (typical for Aluminum) and n = 0.3 . The beam has a solid rectangular cross section with thickness in the Z-direction t = 0.01 meters and height in the Y-direction h = 0.1 meters. We wish to find the mode shapes and associated vibration frequencies for this beam.
II. THINKING ABOUT THE MECHANICS

The analytic solution (modes shapes and natural frequencies) for this problem is readily available. Any vibrations text will provide equations for the mode shapes (eigenvectors) and the natural frequencies (eigenvalues). These equations are given below. For the cantilevered beam with bending moment of inertia “I”, Elastic (Young’s) modulus “E”, mass per unit length “m” and Length “L”, the first 3 natural frequencies W

The analytic solution (modes shapes and natural frequencies) for this problem is readily available. Any vibrations text will provide equations for the mode shapes (eigenvectors) and the natural frequencies (eigenvalues). These equations are given below. For the cantilevered beam with bending moment of inertia “I”, Elastic (Young’s) modulus “E”, mass per unit length “m” and Length “L”, the first 3 natural frequencies W

_{1-3}(rad/sec) are given by:
Note that these correspond to the following 3 mode shapes which are all bending modes in the plane of the smallest value of “I”.

Modeshape 1:

Modeshape 2:

Modeshape 3:

Some basic questions to consider before creating the computational model are:- Are there any other types of mode shapes that might occur (torsional, axial or bending in a different plane)?
- What would be a reasonable frequency for the first mode shape?
- Are there any constraint force checks that will help me validate the accuracy of my model?

Answering these questions qualitatively, along with the quantitative analytical solutions for the mode shapes and their associated natural frequencies will provide reinforcement that your computational model is correctly constructed.

**III. GEOMETRIC AND FINITE ELEMENT MODEL**As is the standard procedure for building MSC/Patran models, we will build the geometry first and then construct a finite element mesh on that geometry. The geometry will proceed from creation of points to curves to surfaces for this simple model. Next, we will use 4 node shell elements to model the beam. Next, the material and element properties will be entered. We will constrain the 3 displacement and 3 rotational degrees of freedom on the left edge (for all nodes). This creates the cantilevered or built-in, end condition. Finally, the nodes must be equivalenced before the analysis is ready to run.

**IV. FINITE ELEMENT THEORY**The exact details of the formulation of the 4 node shell elements in MSC/Nastran is rather complicated. However, the basic formulation of an isoparametric 4 node membrane element is not extremely difficult and will provide us with sufficient background information to begin to understand the vibration model studies. This basic form is constructed as follows:

__Isoparametric Formulation of a 2-D Membrane Element [K] Matrix__Assume the element has the configuration shown below:

The physical and natural coordinate locations of the 4 nodes are:

Our goal is to find the element stiffness matrix

ASSUME: 2 displacement degrees of freedom (dof) per node With :

[B] = the strain - displacement matrix such that [

*B*]{*u*} = {*ε*}where: {

*u*} is the dof vector and {*ε*} is the strain vector[E] = the constitutive matrix such that [E]{

*ε*} = {*σ*}where {

*σ*} is the stress vector and V = volume.**Step 2**: Find the [B] matrix:**Step 3**: Use the Jacobian to find derivatives:
i.e. the isoparametric assumption is that geometry can be interpolated using the same interpolation functions as the displacements.

This allows us to find the entries in [B]

**Step 4**: Perform the numerical integration:where |J| is the determinant of the Jacobian matrix.

Gaussian numerical integration is then used to find the final numbers for the element stiffness.

Where ng

_{j}and ngi are the number of gaussian integration points in the “j” and “i” directions respectively and w_{j}and w_{i}are the associated gaussian weighting factors.__Understanding the Computational Vibration Analysis :__The elements as formed above must be assembled into a global stiffness matrix. In the same manner, element mass matrices are formed using the equation

[M] = ρʃ[N]T [N] J dx dh . A similar form exists for the Rayleigh damping matrix . A similar form exists for the Rayleigh damping matrix [C]. The stiffness, mass and damping matrices are then used in the dynamics equilibrium relationship where the over-dots indicated derivatives with respect to time and {f} is the forcing function. This set of equations can be solved for the time history of the motion (transient dynamics) or for the eigenvalues and eigenvectors. For the vibration analysis, the damping and the forcing function are assumed to be zero. The resulting eigenvalue problem of the second kind is : [M] {w} + [K] {d} = {0} where eigenvalues are the natural frequencies w and the eigenvectors {d} give the node shapes.

V. STEP BY STEP INSTRUCTIONS FOR MODELING THE VIBRATION OF THE CANTILEVERED BEAM USING MSC/PATRAN

Preliminaries for using PATRAN include: a) Log on to the computer

b) Click START (lower left corner of the Windows Desktop), go to Programs, Select MSC (common), Select MSC Patran9.0.

In the instructions below, the following abbreviations and terms will be used:

**TM = Top Menu**. This refers to the horizontal menu options residing at the top of the screen after PATRAN has been initiated.**RM = Right Menu**. This refers to the menus that pop up after an option has been chosen from the top menu. These menus reside on the far right side of the PATRAN desktop.

**SM = Subordinate Menu**. This referees to the menus that pop up from options selected in the right menu.**Click**= Unless otherwise stated, this indicates a click with the left mouse button.**Boldface**will indicate text that occurs in the PATRAN menus.*Italics*text will indicate text that you must enter into text boxes in the PATRAN menus or text that you choose in a menu scroll box.__1. Our first step is to create a new database:__From the TM choose

**File**In the resulting pull down menu choose**New**A SM called

**New Database**pops upTurn on (checked)

**Modify Preferences**Under**File Name**enter*beam-vib.db*Click

**OK**__2. Next set the analysis preference:__A

**New Model Preferences**window will appear as a RMUnder

**Tolerance**choose*Based on Model*Set

**Model Dimension**to*10.0*Under

**Analysis Code**choose*MSC/NASTRAN*Choose

**Analysis Type**=*Structural*click

**OK**__3. The geometry of the beam will be determined next:__From the TM choose**Geometry**A RM called**Geometry**will resultSet

**Action**=*Create***Object**=*Point***Method**=*XYZ*Set the

**Point ID**list to*1*Set

**Reference Coordinate Frame**to*Coord 0*Turn off the

**Auto Execute**buttonEnter the following into the

**Point Coordinates**list:*[0,0,0]*(note that PATRAN will accept either commas or blanks as separators between coordinates)Click

**Apply**A point will appear in the main viewport at coordinates [0,0,0]

Use this same procedure to create points at coordinates [1,0,0], [1,0.1,0] and [0,0.1,0]

Back at the top of the RM called

**Geometry**Set

**Action**=*Create***Object**=*Curve***Method**=*Point*Set the**Curve ID**list to*1*Turn

**Autoexecute**offSet

**Starting Point List**=*Point 1*Set

**Ending Point List**=*Point 2*Click

**Apply**Back at the top of the RM called

**Geometry**Set

**Action**=*Create***Object**=*Curve***Method**=*Point*Set the

**Curve ID**list to*2*Turn

**Autoexecute**offSet

**Starting Point List**=*Point 3*Set**Ending Point List**=*Point 4*Click

**Apply**Back at the top of the RM called

**Geometry**Set

**Action**=*Create***Object**=*Surface***Method**=*Curve*Set the

**Surface ID**list to*1*Set

**Patran 2 Convention***off***Option**=*2 Curve*Set

**Manifold***off*(not checked)Set

**Starting Curve List**=*Curve 1*Set**Ending Curve List**=*Curve 2*Click

**Apply**__2. The finite element mesh is specified next:__From the TM choose

**Elements**A RM appears called

**Elements**Set

**Action**=*Create***Object**=*Mesh***Type**=*Surface*Set

**Node Id**=*1*Set

**Element Id****List**=*1*Set**Global Edge Length**=*0.025*Set

**Element Topology**=*Quad4*Set

**Mesher**=*Isomesh*Click in the

**Surface List**boxClick and drag to select the entire structure

The Words "Surface 1" should appear in the

**Surface List**Click**Apply**
Set

(The purpose here is to tie the nodes together that lie on top of one another)

Set the

Click

**Action**=*Equivalence***Object**=*All***Type**=*Tolerance Cube*(The purpose here is to tie the nodes together that lie on top of one another)

Set the

**Equivalencing Tolerance**to*.003*Click

**Apply**(The command window at the bottom of the PATRAN desktop will tell you that 0 nodes were deleted. This step will become critical if, in more complicated models, you are attempting to join portions of a model which have been meshed separately.)__3. The boundary conditions are specified next:__From the TM choose

**Load/BC's**A RM called**Load/Boundary Conditions**will appearSet

**Action**=*Create***Object**=*Displacement***Type**=*Nodal*Set

**Current Load Case**=*Default*Enter**New Set Name**as*l_cant*( The name can be whatever name you wish. The name*l_cant*is chosen as this is for the cantilever of the left most nodes)Click

**Input Data...**a SM called Input Data appearsSet

**Load/BC Scale factor**=*1*Set

**Translations**to*<0,0,0>*Set**Rotations**to*<0,0,0>*Be sure**Analysis Coordinate Frame**is*Coord0*Click

**OK**(back in the

**Load/Boundary Conditions**RM)Click

**Select Application Region**A SM called**Select Application Region**appearsTurn on the

**FEM**(button down)Click in box under

**Select Nodes**Use the cursor to highlight the set of nodes along the left vertical edge of the beam. There should be 5 nodes there.

Click

**OK**(The

**Load / Boundary Condition**RM appears again)Click

**Apply**
(3 displacement constraint arrows and 3 rotation constraint arrows should now appear on each node in the main viewport window on the extreme left edge of the beam. Numbers 1,2,3,4,5,6 will appear with the arrows to show that all 6 of the dof are constrained there)

__4. The materials are specified next:__On the TM select

**Materials**a RM will appear called**Materials**Set

**Action**=*Create***Object**=*Isotropic***Method**=*Manual Input*Click

**Material Name**boxInput the name to be

*aluminum*Click**Input Properties**boxSM called

**Input Options**appearsInput

**Elastic Modulus**=*70.0E9*Input**Poisson**=*0.3*Input the

**Density**to be*2700*Click

**OK**Back in the**Materials**RMClick

**Apply**__5. The properties for each element are assigned next:__On the TM select**Properties**a RM will appear called

**Element Properties**Set

**Action**=*Create***Dimension**=*2d***Type**=*Shell*Click

**Property Set Name**boxEnter

*beam_prop*Click

**Input Properties**a SM appears called

**Input Properties**Click in the

**Material Name**boxClick on the word "aluminum" in the

**Material Property Sets**box at the bottom of the SM( the words m:aluminum will appear in the

**Material Name**box at the top of the SM)Click in the

**Thickness**boxEnter

*0.01*Click

**OK**(Back in the

**Element Properties**RM)Click

**Select Members**boxa Patran

**Select menu**will appear on the left edge of the RMClick on the icon which contains the surface or face icon

Move the cursor arrow to a point to the left and above the highest, leftmost point on the beam. Click and hold down the left mouse button. Drag the cursor (while holding down the mouse button) to a point to the right of and below the right-most bottom node. A "selection box" is formed while you drag. Release the button.

(The words Surface 1 will appear in the **Select Members**box)Click

**Add**(The words Surface 1 appears in the

**Application Region**box)Click

**Apply**in the**Element Properties**menu(beam_prop will be added to the

**Existing Property Sets**box)__6. The analysis is to be done is specified next:__On the TM select

**Analysis**a RM will appear called

**Analysis**Set**Action**=*Analyze***Object**=*Entire Model***Method**=*Full Run*Click Translation Parameters

In the SM that appears, set

**Data Output**=*Op2 and Print*Click**OK**Back in the RM**Analysis**Set**Solution Type**=*Normal Modes*(button down)Click

**OK**Click

**Apply**(The analysis will take a few seconds to run. A SM indicating that MSC/Nastran is working may appear)__7. A graphical representation of the mode shapes can be produced.__A graphical representation of the mode shapes provides an easy way to begin to determine if you have constructed your model correctly.

On the TM select

**Analysis**

Set

**Action**=

*Read Output2*

**Object**=

*Results Entities*

**Method**=

*Translate*

Click

**Select Results File**

A SM appears called

**Select File**Click the file

**beam-vib.op2**

(You may need to look in your home or root directory to find the file. If this file does not exist, then you have made a mistake in constructing your model. Go to Explorer (right-click on Start and choose Explore) and find the file beam- vib.log and beam.f06. Open these files by double clicking on them and search for the word “error” to determine what your mistake is).

*Beam-vib.op2*then appears in the

**File Name**box

Click

**OK**(back in the

**Analysis**menu)

Click

**Apply**

**Results**A RM will appear called

**Results**Set**Action**=*Create***Object**=*Quick Plot*In the

**Select Result Case**box click**Default**,*Mode 1…*In the Sel

**ect Fringe Result**box click*Eigenvectors, translational*In the

**Apply Fringe Result**box click*Eigenvectors, translational*Set**Quantity**=*Magnitude*Turn on the

**animation**button (so it displays a check)Click

**Apply**(This will create the animation of the first mode)
Investigate other, higher order mode shapes. Be sure to record data and screen captures needed to answer the questions below.

__8. Next you will end your MSC/PATRAN session by saving your database and exiting.__On the TM select

**File**From the pull down menu select

**Save**On the TM select

**File**From the pull down menu select**Quit**next

**THERMAL ANALYSIS OF A COOLING FIN USING SHELL ELEMENTS**
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