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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
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