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ME601 Slide 1

2-D Dynamic

Structural Analysis

Lecture 21

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ME601 Slide 2

Equation of Motion of a String

Assumptions:

(a) Small transverse displacements

(b) Tension does not change with

displacements

Equation of motion is derived usingNewtons Law, where:

l= total length

y(x,t)= transverse displacement

= mass per lengthT= tension in the string

2

2

2

2,,

t

txy

y

txyT

g

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ME601 Slide 3

2

2

22

2 ,1,t

txycy

txy

g

Tc

Equation of motion can be rewritten as:

where cis the velocity of wave propagation along the cable:

Free Response of the String

Using separation of variables, and solving the boundary value problem,

the free response of the string is found:

.. .3,2,1

sincossin,n

nnnn tDtCl

xntxy

where Cn and Dn are determined by initial conditions. We will not consider

the response of the string at this time. Instead, we will begin with an

investigation of the natural frequencies and mode shapes (i.e., a modal

analyses).

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ME601 Slide 4

Determining Natural Frequencies

g

T

l

n

l

cnn

g

T

l

nf nr

22

Natural frequencies are given by:

n = 1, 2, 3, .... n = 1, 2, 3, ....

(rad / sec) (Hz)

The string is a continuous system, so the number of natural

frequencies is infinite. In addition, each natural frequency will have acorresponding mode shape.

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ME601 Slide 5

. The sine term in the free response

equation is the spatial function

which represents the mode shapes

of the string.

Determining Natural Mode Shapes

...3,2,1sin

n

lxn

Mode 1

Mode 2

Mode 3

. Note that the mode number in this

case represents the number of half

sine waves across the length of the

string.

Node points are the locations of

zero displacement (excluding the

boundaries) as indicated in the

figure.

Node Point

Node Points

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ME601 Slide 6

Guitar String

Problem Description

A stainless-steel guitar string with length h and diameter d is

stretched a distance between two rigid supports.

Determine the stress and tension T in the string under theseconditions. Determine the first five natural frequencies

nof

lateral vibration of the stretched string.

Assume E= 190 GPa, = 7920 kg/m3, h = 710 mm, d= 0.254

mm, = 6.195 mm, l= h+

h

l

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Guitar String

The frequency at which a guitar string vibrates depends on how

tight it is stretched or pulled. Without tension, it is basically a wet, cheeseless (and therefore

tasteless) noodle.

In ANSYS, there are several ways to simulate the tension.

1) First do a static, prestress analysis.

- Apply a force at one end of the string.

- Apply a fixed displacement at one end of the string.

- Fix ends of string, specify an

, induce artificialtemperature drop.

2) If using Link 1 or Link 10 elements, use the initial strain

real constant. In this case, a static, prestress

analysis is not needed.

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Eigensolver ApplicationMemory

Required

Disk

Required

Block

Lanczos

Default. To find many modes (about 40+) of large models.

Recommended when the model consists of poorly shaped solid

and shell elements. This solver performs well when the model

consists of shells or a combination of shells and solids. Works

faster but requires about 50% more memory than subspace.

Medium Low

Subspace

To find few modes (up to about 40) of large models.

Recommended when the model consists of well -shaped solid and

shell elements. Works well if memory availability is limited.

Low High

Power

Dynamics

To find few modes (up to about 20) of large models.

Recommended for fast computation of eigenvalues of over 100K

DOF models. On coarse mesh models, the frequencies are

approximate. Missed modes are possible when repeated

frequencies are present.

High Low

Reduced

To find all modes of small to medium models (less than 10K

DOF). Can be used to find few modes (up to about 40) of largemodels with proper selection of master DOF, but accuracy of

frequencies depends on the master DOF selected.

Low Low

Modal Analysis Solvers

Well be using the subspace solver, but for this simple problem youll

probably get the same frequencies regardless of the solver chosen.

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Preprocessing

1. Launch ANSYS

2. Change Directory

Utility Menu: File Change Directory

3. Name file: string-modal

Utility Menu: File Change Title

4. Change jobname: string-modal

Utility Menu: File Change Jobname

Units: m, kg, s

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5. Set element type

Main Menu: Preprocessor Element type Add/Edit/Delete

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6. Set real constants

Main Menu: Preprocessor Real Constants

Add/Edit/Delete

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7. Set material properties

Main Menu: Preprocessor Material Props Material

Models

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8. Set material properties (cont.)

Main Menu: Preprocessor Material Props Material

Models

ANSYS Toolbar: SAVE_DB

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9. Create nodes

Main Menu: Preprocessor Modeling Create Nodes In

Active CS

- Create nodes according to table. We will create the first and

last node (node 31) and use the fill between nodes function to

create the rest of the nodes.

Node

No.X Y

1 0 0

31 0.71 0

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ME601 Slide 15

10. Fill in nodes.

Main Menu: Preprocessor Modeling Create Nodes

Fill Between Nds

Click on

node 1 andnode 31,

confirm that

Count = 2,

then click

OK.

Note: the nodes should now appear in your output window.

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ME601 Slide 16

11. Create elements

Main Menu: Preprocessor Modeling Create Elements

Auto Numbered Thru Nodes

ANSYS Toolbar: SAVE_DB

Click on nodes 1 and

2, confirm that

Count = 2, then

click OK.

One element should

now appear in the

output window.

Well start out by creating just 1 element manually, and on the

next slide well copy the first element to create the others.

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ME601 Slide 17

12. Create elements (cont.)

Main Menu: Preprocessor Modeling Copy Elements

Auto Numbered

Click on theelement

between

nodes 1 and

2, confirm

that Count =1, then click

OK.

All 30 elements

should now appear in

the output window.

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ME601 Slide 18

13. Constrain left end of string.

Main Menu: Preprocessor Loads Define Loads Apply

Structural Displacement On Nodes

Click on node

1 (at x = 0)

and click OK.

Select All DOF

and click Apply.

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ME601 Slide 19

14. Apply constraints at right end of string.

Main Menu: Preprocessor Loads Define Loads Apply

Structural Displacement On Nodes

Click on node31 (at x = 0.71)

and click OK.

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ME601 Slide 20

15. Specify analysis type.

Main Menu: Solution

Analysis Type Soln Controls

ANSYS Toolbar:

SAVE_DB

First we have to perform a static analysis to calculate

prestress effects (determine stress in string).

Solution

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ME601 Slide 21

16. Solve.

Main Menu: Solution Solve Current LS

17. Exit the solution processor.

Main Menu: Finish

a. Click OK on the Solve Current Load Step window

b. Close the /STATUS Command window

c. Close the Solution is done! window

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ME601 Slide 22

18. Specify the analysis type.

Main Menu: Solution Analysis Type New Analysis

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ME601 Slide 23

19. Specify analysis options.

Main Menu: Solution Analysis Type Analysis Options

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ME601 Slide 24

20. Specify subspace solver options.

Well use the default options.

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ME601 Slide 25

21. Solve.

Main Menu: Solution Solve Current LS

22. Check any warnings or error messages

Utility Menu: File List Error File

a. Click OK on the Solve Current Load Step windowb. Close the /STATUS Command window

c. Close the Solution is done! window

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ME601 Slide 26

Postprocessing

23. View results summary.

Main Menu: General Postproc Results Summary

What frequencies did you get? Does the first frequency

match the analytical solution?

g

T

l1

AE

Th

h

AET

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ME601 Slide 27

24. Plot fundamental mode.

Main Menu: General Postproc Read Results First Set

Main Menu: General Postproc Plot Results DeformedShape

25. Plot second mode.

Main Menu: General Postproc Read Results Next Set

Main Menu: General Postproc Plot Results Deformed

Shape

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ME601 Slide 28

25. Animate fundamental mode.

Main Menu: General Postproc Read Results First Set

Utility Menu: PlotCtrls Animate Mode Shape

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ME601 Slide 29

25. Animate fundamental mode (cont).

Use the Animation Controller to decrease the delay between

frames and to increase and decrease the speed of theanimation.

26. Animate other modes.

Main Menu: General Postproc Read Results Next Set

Utility Menu: PlotCtrls Animate Mode Shape

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1. Why are all of the modes in the XY plane? Why doesnt thestring vibrate in the XZ plane? What would you have to change

in the model to see modes in the XZ plane?

2. Should the slope of the guitar string be zero at the fixed ends?

3. The mode shapes are, by default, plotted normalized. Use thedscale command to adjust the scaling. Utility Menu: Plotctrls

Style Displacement Scaling

4. What should be the value of to obtain a tone of Middle C?

5. Apply a temperature drop of -20C. What is the correspondingtension Tand fundamental frequency 1? (Note: for stainless

steel, = 1.0e-5/ C)

Practice