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NAS122, Section 2, January 2004
Copyright 2004 MSC.Software Corporation
SECTION 2
NORMAL MODES ANALYSIS
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NAS122, Section 2, January 2004
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TABLE OF CONTENTS
Page
OVERVIEW 2-5
2 DOF EQUATION OF MOTION USING ENGINEERING APPROACH 2-6
SUMMARIZING SOME IMPORTANT IDEAS ABOUT NORMAL MODES THAT EMERGE 2-12
SETTING THE SAME PROBLEM USING A MATRIX APPROACH 2-13
CASE STUDY 1 – NORMAL MODES OF A 2 DOF STRUCTURE 2-20
WORKSHOP 2 – NORMAL MODES ANALYSIS OF A 2 DOF STRUCTURE 2-47
EXTENDING TO MULTI DOF PROBLEMS 2-48
EIGENVALUE EXTRACTION METHOD 2-56
STURM SEQUENCE THEORY 2-58
LANCZOS METHOD 2-59
CASE STUDY 2 – NORMAL MODES ANALYSIS OF A SATELLITE 2-62REASONS TO CALCULATE NORMAL MODES 2-76
WORKSHOP 13 – MODAL ANALYSIS OF A CAR CHASSIS 2-78
HOW ACCURATE IS THE NORMAL MODES ANALYSIS 2-79
MESH DENSITY 2-80
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NAS122, Section 2, January 2004
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TABLE OF CONTENTS (Cont.)Page
WORKSHOP 1a to 1c – NORMAL MODES ANALYSIS WITH VARIOUS MESH SIZE 2-81ELEMENT TYPE 2-84
WORKSHOP 15a to 15e – MODAL ANALYSIS OF A TUNING FORK 2-85
MASS DISTRIBUTION 2-88
WORKSHOP 14a – MODAL ANALYSIS OF A TOWER 2-89
DETAIL OF JOINTS 2-90
DETAIL OF CONSTRAINTS 2-91
WORKSHOP 14b – MODAL ANALYSIS OF A TOWER WITH SOFT GROUND CONNECTION 2-92
HAND CALCULATIONS 2-93
CHECK LIST FOR NORMAL MODES PRIOR TO DOING FURTHER ANALYSIS 2-94
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NAS122, Section 2, January 2004
Copyright 2004 MSC.Software Corporation
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NAS122, Section 2, January 2004
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OVERVIEW
The previous section looked at a SDOF problem of aspring mass system.
This section looks at Normal Modes analysis of Multi
Degree of Freedom problems and how to set these
problems up in MSC.Patran and MSC.Nastran. Steps to follow are:
Building a 2 DOF equation of motion using engineering approach.
Summarizing some important ideas about Normal Modes that
emerge.
Setting the same problem using a Matrix approach. Building the example in MSC.Patran/MSC.Nastran
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NAS122, Section 2, January 2004
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Consider the system with 2 masses and 3 spring stiffnesses asshown.
Use an engineering approach to solve the equations of motion.
First, set up free body diagrams for the masses.
Equating the Inertia and Elastic terms,
For 1st mass:
For 2nd mass:
2 DOF EQUATION OF MOTION USING ANENGINEERING APPROACH
k k k2MM
x1 x2
M 2Mkx1 k( x2- x1) k( x2- x1) k x2
1 xm
2 xm
1211x xk kx x M
2122
2kx x xk x M
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NAS122, Section 2, January 2004
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2 DOF EQUATION OF MOTION USING ANENGINEERING APPROACH (Cont.)
Assume the motion of x1 and x2 is harmonic so
The objective is to solve what the frequency is, and
the amplitudes. Now
Then putting the harmonic terms into the free body
equations.
For the 1st mass: so
For the 2nd mass:
so
t A x sin11
t A x sin22
This means they vibrate at the same frequency but have
different amplitudes A.
t A x sin1
2
1 t A x sin
2
2
2
t At Ak t kAt A M sinsinsinsin
1211
2
0221
2 kA A M k
t kAt At Ak t A M sinsinsinsin22122
2
0222
2
1 A M k kA
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NAS122, Section 2, January 2004
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2 DOF EQUATION OF MOTION USING ANENGINEERING APPROACH (Cont.)
Assemble these two equations in matrix form, theresult in:
3 unknowns exist; 2 and the pair of amplitudes
0
0
2
2
2
1
2
2
A
A
M k k
k M k
2
1
A
A
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NAS122, Section 2, January 2004
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2 DOF EQUATION OF MOTION USING ANENGINEERING APPROACH (Cont.)
Solve this by using the determinant of the aboveequation, letting 2=l .
Two roots of the equation are found as l 1 and l 2. These roots are called Eigenvalues.
mk 634.0
2
11 l
m
k 366.2
2
22 l
So the two frequencies where the inertia and elastic terms
balance are 1 and 2.
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2 DOF EQUATION OF MOTION USING ANENGINEERING APPROACH (Cont.)
The amplitudes are investigated by substituting backinto the equations of motion.
In turns out that only the ratio of the amplitudes can
be solved.
This is an important physical point in the analysis of
normal modes. The absolute amplitudes are notknown, only relative amplitudes.
731.0
1
2
1
A
A73.2
2
2
1
A
A
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NAS122, Section 2, January 2004
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2 DOF EQUATION OF MOTION USING ANENGINEERING APPROACH (Cont.)
Arbitrarily call A2 = 1.00, the relative amplitudes areexpressed as Mode Shapes or Eigenvectors.
000.1
731.01
2
1
A
A
00.1
73.22
2
1
A
A
0.731 1.000 -2.731 1.000
Mode 1 Mode 2
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SUMMARIZING SOME IMPORTANT IDEASABOUT NORMAL MODES THAT EMERGE
The motion of all displacements is assumedharmonic.
Resonance is found at a set of Natural Frequencies
where the Inertia terms balance the Elastic terms.
The Natural Frequencies are calculated by anEigenvalue Method
The relative amplitude, or mode shape, is found for
each natural frequency.
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Now Consider the system in terms of a MatrixSolution.
The individual „element‟ stiffness matrices [K1],[K2],
and [K3] are:
SETTING THE SAME PROBLEM USING AMATRIX APPROACH
11
11
321k K K K
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NAS122, Section 2, January 2004
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20
01m M
SETTING THE SAME PROBLEM USING AMATRIX APPROACH (Cont.)
To derive the model stiffness matrix [K], assemblethe individual „element‟ stiffness matrices [K1],[K2],
and [K3]:
Constrain out DOF‟s 1 and 4 as they are set to 0.0
11
1111
1111
11
k K
DOF: 1 2 3 4
and
21
12k K „Lump‟ masses at DOF
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NAS122, Section 2, January 2004
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SETTING THE SAME PROBLEM USING AMATRIX APPROACH (Cont.)
The equation of motion in matrix form is:
Substitute in
And
Then
So
0 x K x M
t ie x
This means there is a mode shape,
{} , which varies sinusoidal with a
frequency .
t ie x 2
02 K M
02 M K
This means we can find a mode
shape, {}, and frequency where
the inertia terms and elastic terms
balance
020
01
21
122
mk The Eigenvalue problem
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SETTING THE SAME PROBLEM USING AMATRIX APPROACH (Cont.)
If a set of n physical degrees of freedom exist (2 inour case) Then n sets of unique Eigenvalues i
2 and eigenvectors { i} exist.
where i = 1 to n
For each of these sets, the inertia terms balance the elastic terms
and this is the definition of resonance.
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SETTING THE SAME PROBLEM USING AMATRIX APPROACH (Cont.)
So at , the motion is defined by:
is in balance at this first resonant or natural frequency.
And at , the motion is defined by:
is also in balance at this second resonant or natural frequency
m
k
634.01
000.1
731.0
1
000.1
731.2
2 m
k 366.22
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NAS122, Section 2, January 2004
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SETTING THE SAME PROBLEM USING AMATRIX APPROACH (Cont.)
Let us add some values in and check out thenumbers: Let k = 1000 units of force / length
Let m = 20 units of mass
Then
Notice the conversion of Frequency from Radians/s to Cycles/s
(Hertz)
Hz s
radsm
k 896.0629.5634.0
1
Hz
s
rads
m
k 731.1875.10366.2
2
f 2
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SETTING THE SAME PROBLEM USING AMATRIX APPROACH (Cont.)
Load this model with a time dependant set of forcesat DOF 2 and 3. This results in a displacement
response which is a combination of the two mode
shapes calculated.
So
in our case
The scaling factors xi for each mode shape i are
called the Modal Displacements. There will be
further reference to this when loading is applied inlater sections.
i
n
i
it x x
1
2211
x x t x
0.731 1.000 -2.731 1.000
x1 x2
+ t x
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Model the system using MSC.Patran andMSC.Nastran
Allocate arbitrary dimensions to the model as above
and input the nodes and spring and mass elements
directly into Patran.
K K K
DOF: 1 2 3 4
MM
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Create the grids
using Create/Node/Edit
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Create the Bar2 generic Patran Elements
for the Springs using Create/Element/Edit
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Create the Point generic Patran Elements
for the Masses using Create/Element/Edit
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NAS122, Section 2, January 2004
Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Define the specific MSC.Nastran type of Spring
via the Physical Property definition:
Create/1D/Spring
Input the name
Input the properties (1000 force/length units)Select the bar2 elements
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The MSC.Nastran .bdf file will contain theconnectivity definition of the CELAS elements and
the PELAS property definition.
$ Elements and Element Properties for region : spring_stiffPELAS 1 1000.
$ Pset: "spring_stiff" will be imported as: "pelas.1"
CELAS1 1 1 1 1 2 1
CELAS1 2 1 2 1 3 1
CELAS1 3 1 3 1 4 1
ID
PID
End A
Grid DOF DOF
End B
Grid
Stiffness
PID
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Define the specific MSC.Nastran type of Mass
via the Physical Property definition:
Create/0D/Mass
Input the name
Input the properties 20 and 40 mass unitsChoose the Lumped option
Select the point elements
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The MSC.Nastran bdf file will contain the CONM2elements, there is one for each mass point.
$ Elements and Element Properties for region : m1
CONM2 4 2 20.
$ Elements and Element Properties for region : m2
CONM2 5 3 40.
ID Grid Mass
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
GRID: 1 2 3 4
„fixed‟ „make_1d‟
Two constraint
sets are created
„fixed‟:
constrains theends in all DOF
„make_1d‟:
constrains all
DOF except x at
GRIDS 2 and 3
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The Analysis Type is selected as
Normal Modes (SOL103)
Solution Parameters are set:
Mass Calculation is Lumped
Wt. Mass Conversion is 1.0
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The MSC.Nastran .bdf file will contain the solutionsequence definition and implied parameters for
weight mass conversion and choosing lumped mass.
The implications of the parameters will be discussedlater.
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
$ Normal Modes Analysis, Database
SOL 103
TIME 600
$ Direct Text Input for Executive Control
CEND
SEALL = ALL
SUPER = ALL
TITLE = MSC.Nastran job created on 10-Jan-02 at 12:52:18
ECHO = NONE
MAXLINES = 999999999
...
...
BEGIN BULK
PARAM,WTMASS,1.0 {do not appear as these are the defaults}
PARAM,COUPMASS,-1 {do not appear as these are the defaults}
...
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
A Subcase is created and Parameters
defined:
MSC.Nastran will select the upper and
lower bound on frequencies found using Lanczos.
No of roots is 2
Use the Maximum method in Normalization
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The MSC.Nastran .bdf file contains the SubcaseDefinition and the Lanczos data entry
...
SUBCASE 1
$ Subcase name : Default
SUBTITLE=Default
METHOD = 1
SPC = 2
VECTOR(SORT1,REAL)=ALL
...
...
EIGRL 1 2 0 MAX
...
a
b
ed
a. Eigenvalue Set definition
b. Eigenvector Output Request
c. EIGRL keyword indicates the
Lanczos method will be used
A description of the Lanczos method
occurs later
d. ID of Lanczos request
e. Number of Roots = 2f. Normalization Method (Maximum)
c f
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NAS122, Section 2, January 2004
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Once the analysis carried out, inspect theMSC.Nastran F06 File:R E A L E I G E N V A L U E S
MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED
NO. ORDER MASS STIFFNESS
1 1 3.169873E+01 5.630163E+00 8.960682E-01 5.071797E+01 1.607695E+03
2 2 1.183013E+02 1.087664E+01 1.731071E+00 2.535898E+01 3.000000E+03
a b
The MSC.Nastran .f06 file has an
Eigenvalue Summary. Important
pieces of information in this summary
are:
a. Values of the Eigenvalues in
(radians/sec)2.
b. The Natural Frequencies in hz and
Radians/Sec.
b
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S2-35NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The values agree with those calculated earlier. Themeaning of the Generalized Mass and Stiffness will
be discussed in the next few pages.
Hz srad
m
k 896.0/629.5796.0
1
Hz srad m
k 731.1/875.10538.1
2
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S2-36NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The Eigenvector results for the 2 modes are alsoshown in the F06 file.0 SUBCASE 1
EIGENVALUE = 3.169873E+01
CYCLES = 8.960682E-01 R E A L E I G E N V E C T O R N O . 1
POINT ID. TYPE T1 T2 T3 R1 R2 R3
1 G 0.0 0.0 0.0 0.0 0.0 0.0
2 G 7.320508E-01 0.0 0.0 0.0 0.0 0.0
3 G 1.000000E+00 0.0 0.0 0.0 0.0 0.0
4 G 0.0 0.0 0.0 0.0 0.0 0.0
1 MSC.NASTRAN JOB CREATED ON 10-JAN-02 AT 12:52:18 JANUARY 10, 2002 MSC.NASTRAN 6/11/01 PAGE8
DEFAULT
0 SUBCASE 1
EIGENVALUE = 1.183013E+02
CYCLES = 1.731071E+00 R E A L E I G E N V E C T O R N O . 2
POINT ID. TYPE T1 T2 T3 R1 R2 R3
1 G 0.0 0.0 0.0 0.0 0.0 0.0
2 G 1.000000E+00 0.0 0.0 0.0 0.0 0.0
3 G -3.660254E-01 0.0 0.0 0.0 0.0 0.0
4 G 0.0 0.0 0.0 0.0 0.0 0.0
000.1
731.2
2
000.1
731.0
1
Eigenvector 1 clearly matches
the hand-calculation from
before, but Eigenvector 2 does
not. The reason is because of
a difference in the way
MSC.Nastran and the hand
calculation normalize the
results.
CASE STUDY 1 NORMA MODES OF A
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S2-37NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The difference in the normalization methods betweenthe hand calculation and MSC.Nastran is interesting: In the hand calc. displacement was normalized at Grid 3 to 1.000
quite arbitrarily
The MSC.Nastran method selected in this run was „Maximum‟. Thismeans the maximum value in the e-vector list is set to 1.000 for each e-vector, so
This emphasizes all that is known about the e-vectors in a NormalModes analysis is their relative values – the shape is known, butnot the amplitude.
There is another commonly used normalization method called Massnormalization that will be discussed.
000.1
731.2
2
366.0
000.1
2
CASE STUDY 1 NORMAL MODES OF A
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S2-38NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Mass normalization is a useful way of normalizing anEigenvector because it can be thought of as a
universal standard.
Scale { } so that for each mode:
If there is comparison between modes of different
analyses, or even to test data, then it becomes
meaningful to compare the mass normalizedeigenvectors as:
I M T 0.1i
T i M
21 MODEL
T
MODEL M TEST
T
MODEL M
CASE STUDY 1 NORMAL MODES OF A
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S2-39NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The term { }T
[ M ]{ } is called the Generalized Mass. It is clearly orthogonal if you can equate it to an Identity matrix in
Mass Normalization.
In general, orthogonality is defined as:
Generalized Stiffness, { }T [ K ]{ } can be defined in a
similar way:
0.0
j
T
i
ii
T
i
M
m M
0.0
j
T
i
ii
T
i
K
k K
CASE STUDY 1 NORMAL MODES OF A
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Reviewing the Generalized Stiffness and Mass termsfrom the current example, it is obvious that the
Eigenvector is not mass normalized in this analysis:
R E A L E I G E N V A L U E S MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED
NO. ORDER MASS STIFFNESS
1 1 3.169873E+01 5.630163E+00 8.960682E-01 5.071797E+01 1.607695E+03
2 2 1.183013E+02 1.087664E+01 1.731071E+00 2.535898E+01 3.000000E+03
CASE STUDY 1 NORMAL MODES OF A
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S2-41NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
It is difficult to visualize the mode shapes by lookingat printed e-vectors, so Patran and its deformation
plots and animation methods will be used to help
understand the behavior.
The analysis just carried out will create an XDB file
containing binary results data. Attach that to the
Patran database, so the results can be viewed.
CASE STUDY 1 NORMAL MODES OF A
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
The results file mass2_k3_ex.xdb is attached
to the database
CASE STUDY 1 NORMAL MODES OF A
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Select Results: Create/Deformation and select
the first Modal result.
The deformation we want to plot isEigenvector: Translational
Do not hit Apply yet because we want to set up
some plotting options.
Select the Plot Options Icon
See next slide for Plot Options
CASE STUDY 1 NORMAL MODES OF A
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
Using the Plot Options form we can:
Increase the line width of the spring elements
Switch off the Undeformed plot option, to make
things clearer
Switch off Maximum Label
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CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
1. The size of nodes can be
increased to help visualization.
Select the Node Size Increase
Icon from the Main Toolbar.
This icon is magnified for clarification.
2. Notice the PCL call in the
History Window
3. Copy this to the Command line
by clicking on it
4. Modify it to read node_size(30)
and hit Enter
5. Now select the Animation
Options
2
34
5
1
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1. In Animation Options increase the Number
of Frames to 20 to get a smooth animation2. Check the Animate box and hit Apply
The first Mode will be animated
Repeat for the second Mode
The Tutor will play the mpeg files if available
CASE STUDY 1 – NORMAL MODES OF A2 DOF STRUCTURE
1
2
WORKSHOP 2
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Please now carry out Workshop 2 in the WorkshopSection to allow you to set up this model and
visualize the results.
The workshop will take you through step by step if
you are unfamiliar with MSC.Nastran or MSC.Patran.
If you have some experience, then try to set up the
analysis without referring to the step by step guide.
Please feel free to ask your tutor for help.
WORKSHOP 2NORMAL MODES ANALYSIS OF A 2 DOF
STRUCTURE
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EXTENDING TO MULTI DOF PROBLEMS
Consider Normal Modes of Multi Degree of Freedom(MDOF) problems like the beam shown below.
The beam model here contains 50 DOF.
When a Normal Modes analysis is ran, results are
obtained as shown in the table on the next page.
Built in at both ends
EXTENDING TO MULTI DOF PROBLEMS
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EXTENDING TO MULTI DOF PROBLEMS(Cont.)
MSC.Nastran .F06 excerpt showing Eigenvalue tableR E A L E I G E N V A L U E S
MODE EXTRACTION EIGENVALUE RADIANS CYCLES GENERALIZED GENERALIZED
NO. ORDER MASS STIFFNESS
1 1 8.260010E+04 2.874023E+02 4.574150E+01 1.000000E+00 8.260010E+04
2 2 6.253773E+05 7.908080E+02 1.258610E+02 1.000000E+00 6.253773E+05
3 3 6.940343E+05 8.330872E+02 1.325899E+02 1.000000E+00 6.940343E+054 4 2.387144E+06 1.545039E+03 2.459005E+02 1.000000E+00 2.387144E+06
5 5 5.195627E+06 2.279392E+03 3.627764E+02 1.000000E+00 5.195627E+06
6 6 6.435358E+06 2.536801E+03 4.037444E+02 1.000000E+00 6.435358E+06
7 7 1.397762E+07 3.738666E+03 5.950272E+02 1.000000E+00 1.397762E+07
8 8 1.952659E+07 4.418890E+03 7.032882E+02 1.000000E+00 1.952659E+07
9 9 2.586864E+07 5.086123E+03 8.094815E+02 1.000000E+00 2.586864E+07
10 10 4.159941E+07 6.449761E+03 1.026511E+03 1.000000E+00 4.159941E+07
a. Note the
generalized mass is
1.0, this is a result
of Mass
Normalization
Method being
chosen, as
explained earlier.
a
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EXTENDING TO MULTI DOF PROBLEMS(Cont.)
It is important to be able to describe all the modes.Patran is used to identify and characterize the modes
as described in the table. The modes increase in complexity for a given type of mode. This
leads us to the use of a subset of modes to describe the physical
behavior. Mode F (Hz) Description
1 45.74 1st bend xz plane
2 125.86 2nd bend xz plane
3 132.59 1st bend xy plane
4 245.9 3rd bend xz plane
5 362.8 2nd bend xy plane
6 409.74 4th bend xz plane
7 595.08 5th bend xz plane
8 703.29 3rd bend xy plane
9 809.48 6th bend xz plane
10 1026.5 7th bend xz plane
EXTENDING TO MULTI DOF PROBLEMS
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EXTENDING TO MULTI DOF PROBLEMS(Cont.)
Modes in XZ plane
Mode 1 = 45.74 Hz Mode 2 = 125.86 Hz
Mode 4 = 245.9 Hz Mode 6 = 409.74
EXTENDING TO MULTI DOF PROBLEMS
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Modes in XZ plane (cont.)
EXTENDING TO MULTI DOF PROBLEMS(Cont.)
Mode 7 = 595 Hz Mode 9 = 809 Hz
Mode 10 = 1027 Hz
EXTENDING TO MULTI DOF PROBLEMS
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Modes in XY plane
EXTENDING TO MULTI DOF PROBLEMS(Cont.)
Mode 3 = 132.6 Hz Mode 5 = 362.8 Hz
Mode 8 = 703.3 Hz
EXTENDING TO MULTI DOF PROBLEMS
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EXTENDING TO MULTI DOF PROBLEMS(Cont.)
Remember, the contribution of each mode is definedas the modal displacement: So
In the 2 DOF case
For the beam example, it may be possible to represent the
response to loading in the XZ plane only using the first two modes.The assumption is that the higher modes do not contributesignificantly to the solution.
This is a significant advantage of modal methods, the response of thebeam, { x(t)}, has 50 physical degrees of freedom.
But the response can be represented by 2 Modal DOF, x i
EXTREME CARE must be taken when assumingwhich modes contribute and this will be discussedmore in later sections. As a taster consider the following over page
ii i
t x x
nto1
2211
x x t x
x
EXTENDING TO MULTI DOF PROBLEMS
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EXTENDING TO MULTI DOF PROBLEMS(Cont.)
Possible Problems with 2 modes only: If loading in the XY plane exists, it simply will not be able to be
represented and will therefore be ignored
If the XZ loading excites at multiple inputs as shown, then the first
two modes may not represent the response, and the mode number
4 (the third bending mode in the x-z plane) may be needed
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EIGENVALUE EXTRACTION METHOD
Now consider the method of solving the Eigenvalue problem in
MSC.NASTRAN.
3 types of methods for eigenvalue extraction are available:
Tracking Methods
Eigenvalues (or natural frequencies) are determined one at a time using an
iterative technique. Two variations of the inverse power method are provided, INV
and SINV. In general, SINV is more reliable than INV.
Transformation Methods
The original eigenvalue problem ([K] - l [M]) { } = 0
is transformed to the form [A] { } = l { }
where [A] = [M]-1 [K]
Then the matrix A is transformed into a tridiagonal matrix using either the Givens
technique or the Householder technique. Finally, all the eigenvalues are
extracted at once using the QR algorithm. Two variations of the Givens techniqueand two variations of the Householder technique are provided: GIV, MGIV, HOU,
and MHOU. These methods are more efficient when a large proportion of
eigenvalues are needed.
EIGENVALUE EXTRACTION METHOD
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EIGENVALUE EXTRACTION METHOD(Cont.)
Lanczos Method (recommended method)
This method is a combined tracking-transformation method
Features are:
A trial eigenvalue (called a shift point) is assumed, and an attempt is made
to extract all of the eigenvalues close to this value.
It is called a block method because it extracts several eigenvectors within a
frequency block close to the trial eigenvalue.
A Sturm sequence check is made at each of the shift points to determinethe number of eigenvalues below that shift point.
This information is used to determine when all of the eigenvalues of interest
have been found.
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STURM SEQUENCE THEORY
Tracking method example: Sturm Sequence Theory Choose l .
Factor ([K - li M]) into [L][D][LT]
The number of negative terms on the factor diagonal is the number
of eigenvalues below l .
0.0
No. Neg
Terms=7
No. Neg
Terms=8
(must be in the range)l8
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LANCZOS METHOD
Real Eigenvalue Extraction Data, Lanczos Method Defines data needed to perform real eigenvalue (vibration of
buckling) analysis with the Lanczos method.
Example
1 2 3 4 5 6 7 8 9 10
EIGRL SID V1 V2 ND MSGLVL MAXSET SHFSCL NORM
EIGRL 1 0.1 3.2 10
This example requests 10
Eigenvalues calculated between
0.1 and 3.2 Hz using the Lanczos
method.
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LANCZOS METHOD (Cont.)
Field Contents: SID: Set identification number. (Unique Integer > 0) V1, V2: The V1 field defines the lower frequency bound; the V2 field
defines the upper frequency field. (Real or blank, )
For vibration analysis: frequency range of interest.
For buckling analysis: eigenvalue range of interest. ND Number of eigenvaluesand eigenvectors desired. (Integer > 0 or blank)
MSGLVL: Diagnostic level. (0 < Integer < 4; Default = 0)
MAXSET: Number of vectors in block or set. (1 < Integer < 15; Default = 7)
SHFSCL: Estimate of the first flexible mode natural frequency. (Real or blank)
NORM: Method for normalizing eigenvectors (Character: "MASS" or "MAX")
MASS: Normalize to unit value of the generalized mass.
Not available for buckling analysis. (Default for normal modes analysis.) MAX: Normalize to unit value of the largest displacement in the analysis
set.
Displacements not in the analysis set may be larger than unity. (Default for buckling analysis.)
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LANCZOS METHOD (Cont.)
Tips when using Lanczos method: If User Fatal Message 5299 is reported, note that this is often
caused by a massless mechanism, such as a grid point floating in
space or a BAR not connected to the rest of the structure in torsion.
The DOF should be eliminated or mass added as appropriate.
Another potential error, which may or may not indicate a modeling
problem, may occur when a shift point is chosen too close to anactual eigenvalue.
So, if UFM 5299 occurs and massless mechanisms are not present in
the model, try adjusting the frequency range on the EIGRL entry. This
adjustment forces the Lanczos method to use different shift points that
may result in a better numerically conditioned solution.
If rigid body modes are not present or are not needed, it issometimes helpful to increase V1 to a small positive number.
If the Lanczos method cannot find all of the roots after modifying V1
and V2 several times, this generally indicates a modeling problem.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 – NORMAL MODESANALYSIS OF A SATELLITE
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 – NORMAL MODESANALYSIS OF A SATELLITE
Import a MSC.Nastran .bdf
file
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
Select the Solution Type
and enter the correct
Weight-Mass conversion
value for the Normal
Modes Analysis.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
This model uses a value of 0.00259 as the weight mass
conversion parameter - what does this mean? MSC.Nastran requires consistent units.
Some systems of Units (including the US system) define density as being a
Weight Per Unit Volume (eg. lbs/in3).
This is not a consistent unit if used with loads of lbf and dimensions of inches.
The weight mass conversion parameter converts „weight mass‟ units to
mass units by scaling by the appropriate units of acceleration due to gravity.
So for our model defined in lbs and inches, g = 386.4 in/s2
PARAM,WTMASS,0.00259 converts the mass of the structure to the correct
units of (lbf/in/s^2).
Some industries also mix SI units for convenience, so density may be given
in N/m3
instead of the correct term Kg/m3
In this case PARAM,WTMASS,0.102 will scale by g = 9.81m/s^2
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
Another common method used in the industry is tohave length measured in mm for convenience, butstill want to apply forces in N.
When converting a non-standard system of units thegolden rule is to apply Newton‟s Law of Motion and
then dimensional equivalence Force = Mass * Acceleration
N = (mass units) * mm/s2
Kg *m/s2 = (mass units) *mm/s2
Kg *m/s2 = (Kg*103) * mm/s2
So mass is in Kg*103 or Metric Tonnes and density is then Tonnes/mm3 and these units should be used in the model
In this case if units of Kg mass and Kg/mm3 density are used in themodel PARAM,WTMASS,0.001 will scale the mass and densityunits correctly.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
Notice that in the case of the satellite, the base of the
structure is constrained. The assumption is that it is
rigidly built in to the launcher.
A „free-free‟ analysis could have been carried out
where the structure is not connected to the launcher.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
For every DOF in which a structure is not totally
constrained, it admits a Rigid Body Mode (stress-free
mode) or a mechanism. There should be six Rigid Body Modes.
The natural frequency of each Rigid Body Mode
would be zero. You will try this in the next workshop. A later section is devoted to
discussion of Rigid Body Modes
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
Select the imported
subcase that contained the
correct boundary
conditions for this Normal
Modes Analysis.
Submit the model to
MSC.Nastran for analysis.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
The F06 file contains the Natural Frequencies or Eigenvalues.
The results in terms of Cycles/sec or Hz are highlighted
The Generalized Mass is calculated by:
{ }T [ M ] { }
In this case the Mass normalization method for Eigenvectors
was used, so the term reverted to:
{ }T [ M ] { } = 1
The Generalized Stiffness is:
{ }T [ K ] { }
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
Use the Quick Plot option
to see the Mode Shape
and Frequency of each
mode graphically.
It can be useful to use
fringes to see the
displaced shape,
particularly in high order
panel modes.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
The Mode Shape and
Frequencies for the 10
Normal Modes requested
are shown.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
It is very important to be able to characterize the
Modes that are found, particularly when comparing
one analysis with other, or against test. To state “I get 32 Hz for mode 4, against test of 36 Hz” is
meaningless unless it is confirmed that we are comparing the same
physical modes
To help with this it is recommended that the modes are labeled with
some simple description to help identification.
CASE STUDY 2 – NORMAL MODES
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CASE STUDY 2 NORMAL MODESANALYSIS OF A SATELLITE
So in the case of the satellite:
Mode F (Hz) Description
1 22.55 1st bend main body
2 22.59 1st bend main body Orthogonal to mode 1
3 38.41 1st Torsion main body and all panels pant 1st order, same sense +ve
4 41.85 1st order Panel pant + + + + + +
5 42.94 1st order Panel pant + + + + + +
6 42.94 1st order Panel pant, orthogonal to mode 5
7 44.46 1st order Panel pant + + + + + +
8 44.46 1st order Panel pant, orthogonal to mode 7
9 55.13 2nd order Panel pant
10 55.13 2nd order Panel pant, orthogonal to mode 9
CASE STUDY 2 – NORMAL MODES
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C S S U O O SANALYSIS OF A SATELLITE
In this case the first three modes include motion of
the main body, the remainder are local panel modes.
It is very likely that the main response will be
dominated by the first three modes.
Notice that many of the modes are described as
orthogonal, and have identical or nearly identical
frequencies. This means that repeated roots have been found in the eigenvalue
analysis, due to symmetry of the structural response. The
eigenvectors are orthogonal to each other, meaning each is unique
mathematically.
REASONS TO CALCULATE NORMAL
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MODES Now consider some reasons to compute natural frequencies
and normal modes of structures To assess the dynamic characteristics of a structure.
For example, if a structure is going to be subject to rotational or cyclic loading
input, to avoid excessive vibrations, it might be necessary to see if the frequency
of the input is close to one of the natural frequencies of the structure.
Passing blade frequencies of a helicopter
Rotational speed of an automobile wheel Rotational speed of a lathe
Vortex shedding or flutter of bridge and deck structures
Assess the possible dynamic amplification of the loads.
If a structure is loaded near a natural frequency with an input that matches that
frequency then the dynamic amplification can be significant for a lightly damped
structure, perhaps being an order of magnitude higher than an equivalent static
loading Dynamic response of aircraft structure due to landing loads can exceed static loading
Dynamic response of Tacoma Narrows bridge, runaway loading
REASONS TO CALCULATE NORMAL
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MODES (Cont.) Use the Modal Data (natural frequencies and mode shapes) in a
subsequent dynamic analysis Later you will see that there is a class of transient and frequency response
analysis methods that use modal techniques, using Modal data.
Assess requirements of subsequent dynamic analysis
For Transient response, calculate time steps based on the highest
frequency of interest For Frequency response, calculate the range of frequencies of interest
Guide the experimental analysis of structures
Identify optimum location of accelerometers, etc.
Avoid overstressing of components
Evaluate the effect of design changes A normal modes analysis will give a clear indication of frequency shifts,
changes in mode shapes to allow an early judgment on effect of design
changes to be made
WORKSHOP 13
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MODEL ANALYSIS OF A CAR CHASIS
Objectives: Importing the car model from a MSC.Nastran
.bdf file into MSC.Patran and carry out anormal modes analysis.
Identify rigid body modes and describe theelastic modes.
Then carry out Workshop 13b, which sets the
lowest frequency in the eigenvalue extractionto 0.1 Hz.
Note: The workshop will take you through step by
step if you are unfamiliar with MSC.Nastran or MSC.Patran.
If you have some experience, then try to setup the analysis without referring to the step bystep guide.
Please feel free to ask your tutor for help.
HOW ACCURATE IS THE NORMAL
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MODES ANALYSIS?
The following section and workshops explore the key
ingredients: mesh density
element type
mass distribution
detail of constraints detail of joints
MESH DENSITY
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MESH DENSITY
Mesh Density The mesh must be fine enough to permit a representation of the of
the highest mode considered
In the case of the beam, it was assumed that the second order mode
was sufficient. The mesh is adequate for this.
However, if the higher order mode shown is required, then the mesh is
inadequate.
WORKSHOP 1a to 1cNORMAL MODES ANALYSIS WITH
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S2-81NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
NORMAL MODES ANALYSIS WITHVARIOUS MESH SIZE
Objectives:
Building a simple plate model inMSC.Patran
Perform a normal modes analysis of the structure.
Workshop 1b and Workshop 1cvary mesh densities to investigatetheir effect on the results.
Identify and describe the elasticmodes.
Note: The workshop will take you through
step by step if you are unfamiliar with MSC.Nastran or MSC.Patran.
If you have some experience, thentry to set up the analysis withoutreferring to the step by step guide.
Please feel free to ask your tutor for help.
WORKSHOP 1a to 1cNORMAL MODES ANALYSIS WITH
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S2-82NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
NORMAL MODES ANALYSIS WITHVARIOUS MESH SIZE
Comparison of results from Workshops 1a to 1c: The mesh density of the plate structure was varied to check out the
influence on the Normal Modes results.
These results show that the low frequency, first order modes are
captured reasonable well in the coarser models, but as the higher
modes are met then the frequency mode order drifts.
This illustrates how essential it is to describe each mode to be able
to compare it.
WORKSHOP 1a to 1cNORMAL MODES ANALYSIS WITH
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S2-83NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
NORMAL MODES ANALYSIS WITHVARIOUS MESH SIZE
Workshop 1a – 1c results.Mode f Mesh 1a (10 x 4)
Description
Mode f Mesh 1b (2 x 1)
Description
Mode f Mesh 1c (50 x 20)
Description
1 133.1 1st order Bending 1 120.1 1st Order Bending 1 133.6 1st order Bending
2 348.7 1st Order Torsion 2 395.7 1st Order Torsion 2 689.6 1st Order Torsion
3 821.4 2
nd
Order Bending 3 624.5 2
nd
Order Bending 3 832.8 2
nd
Order Bending
4 2043 2nd Order Torsion 4 1003. 2nd Order Torsion 4 2133. 2nd Order Torsion
5 2278. 3rd Order Bending 5 2144. 1st Order Shear 5 2332. 3rd Order Bending
6 2358. 1st Order Shear 6 8722. 2nd Order Shear 6 2358. 1st Order Shear
7 3705. 3rd Order Torsion 7 9988. 1st Order Extension 7 4051. 3rd Order Torsion
8 4344. 4th Order Bending 8 16667 2nd Order Shear II 8 4552. 4th Order Bending
9 4763. 1st Order Axial
Bending
9 20793 2nd Order Extension 9 5633. 1st Order Axial
Bending
10 5569. 2nd Order Axial
Bending
10 22799 2nd Order Shear III 10 6433. 4th Order Torsion
ELEMENT TYPE
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S2-84NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
ELEMENT TYPE
Element Type The type of element chosen is very important in dynamic analysis,
in that it can control the stiffness representation and to a lesser
extent the mass distribution of the structure.
Examples of poor choices are:
Using TET4 elements to model solid structures. If they are used in
relatively thin regions that have plate or shell the results can be verypoor. TET10 or preferably HEXA are a better choice.
If RBE2 is used instead of an RBE3 on a flexible structure such as a
satellite platform then it may over stiffen the structure and influence the
frequencies badly.
In the next workshop a classic structure is analyzed using different
elements.
WORKSHOP 15a – 15e
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S2-85NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
TUNING FORK
Objectives: Building a Tuning Fork in MSC.Patran
Perform a normal modes analysis of thestructure.
Vary element types (tet10, tet4, beam) andmesh densities to investigate their effect
on the results. Identify and describe the elastic modes.
Note: The workshop will take you through step
by step if you are unfamiliar with Nastranor Patran.
If you have some experience, then try toset up the analysis without referring to thestep by step guide.
Please feel free to ask your tutor for help.
WORKSHOP 15a – 15e
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S2-86NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
TUNING FORK (Cont.)
Comparison of results from Workshops 15a to 15e: The primary interest is the first elastic mode of the tuning fork. The
theoretical value is 440 Hz, and is known as the “A above middle C”
musical note.
As an aside, it is interesting to note that the vertical translation of
the stem is what excites an instrument or another object that the
tuning fork is placed against.
Workshop Frequency
1st Elastic (Hz)
% error
(fr. theo. value)
Element
Type
Mesh
Density
# of Elements
15a 440.92 +0.21 TET10 0.226 2222
15b 404.69 -8.02 TET10 0.500 351
15c 539.05 +22.5 TET4 0.113 3090
15d 560.29 +27.3 TET4 0.250 355
15e 436.85 -0.72 BAR 0.100 65
WORKSHOP 15a – 15eG O (C )
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S2-87NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
TUNING FORK (Cont.)
Comparison of results from Workshops 15a to 15e
(cont.): The results clearly show the superiority of the TET10 Solid
Elements over the TET4 Elements. In general, TET4 elements areNOT recommended in dynamic analysis.
Increasing the mesh density of the TET4 elements does not yield asignificant change and it is likely that the element type isconverging to an incorrect solution. This again underscores theneed to avoid TET4‟s.
Increasing the mesh density improves the results significantly in theTET10 models, and this type of sensitivity is typical of what shouldbe carried out in an analysis.
It is interesting to see that the BAR elements perform very well,
because the quality of the idealization is high. Improvement maybe possible by increasing the number of BAR‟s around the curvedregions.
MASS DISTRIBUTION
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S2-88NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
MASS DISTRIBUTION
Mass Distribution A poor stiffness representation can influence a structure badly and
a poor mass representation can also have the same effect.
The mass values may be wrong due to user error. The values can
be checked in MSC.Patran and also the MSC.Nastran .f06 file (the
next workshop shows how to do this)
There are two forms of mass representation in MSC.Nastran – lumped and coupled. Differences may occur in the analysis
depending on which is selected. This is discussed in Section 3.
WORKSHOP 14aMODEL ANALYSIS OF A TOWER
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S2-89NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
MODEL ANALYSIS OF A TOWER Objectives:
Build a Tower in MSC.Patran.
Perform a normal modes analysis
of the structure.
Check the mass of the model.
Identify and describe the elastic
modes.
Note:
The workshop will take you
through step by step if you are
unfamiliar with MSC.Nastran or
MSC.Patran.
If you have some experience,
then try to set up the analysis
without referring to the step by
step guide.
Please feel free to ask your tutor
for help.
DETAIL OF JOINTS
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S2-90NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
DETAIL OF JOINTS
Detail of Joints: Is the joint flexibility correct?
For example a corner of a formed sheet structure will have an internal
radius which increases its torsional stiffness. It may be important in this
case to include the torsional stiffness via ROD element.
If bolts are used to connect components together then the bolt
stiffness may play an important role in dynamic analysis.
CQUAD4
CQUAD4
CROD
DETAIL OF CONSTRAINTS
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S2-91NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
DETAIL OF CONSTRAINTS
Detail of Constraints When idealizing a structure assumptions are always made about
the connection to an adjacent structure or to ground.
Hence if a panel is surrounded on all sides by reinforcing structure, is it
represented as fully built in, simply supported, or is it necessary to
model an equivalent edge stiffness using CELAS or CBUSH elements?
A particularly difficult case is where the connectivity is ill-defined,such as the push-fit and snap connectors of a typical car dashboard
assembly.
Remember there is no such thing in nature as an infinitely stiff
connection or structure.
In the workshop that follows, the mode shapes of the tower are
significantly changed by the fact that the connection to ground isnot rigid. A major redesign of this structure is needed.
WORKSHOP 14bMODEL ANALYSIS OF A TOWER WITH
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S2-92NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
MODEL ANALYSIS OF A TOWER WITHSOFT GROUND CONNECTION
Objectives: Using the model from Workshop
14a:
Account for the soil-base interaction
using CBUSH elements.
Note: The workshop will take you through
step by step if you are unfamiliar
with MSC.Nastran or MSC.Patran.
If you have some experience, then
try to set up the analysis without
referring to the step by step guide. Please feel free to ask your tutor for
help.
Soil Stiffness
modeled with CBUSH
Elements
Tower Leg
RBE2
HAND CALCULATIONS
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S2-93NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation
HAND CALCULATIONS
Hand Calculations Manual checking of the frequencies in an analysis to make sure
answers are in the right ballpark can involve:
Using simple analogies of the structure to match standard solutions in
Roark or Blevins
Applying a 1g load in relevant directions and using the resultant
displacement at the cg. to calculate an equivalent SDOF frequency.
Using idealization techniques to create simple FE models to verify
important modes of a complex model.
Remember the frequency is proportional to (k/m)1/2, therefore
consider whether stiffness or mass dominates errors and that
frequency can be relatively insensitive to errors in both.
CHECK LIST FOR NORMAL MODESPRIOR TO DOING FURTHER ANALYSIS
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PRIOR TO DOING FURTHER ANALYSIS RBM‟s - are they as expected
Is the frequency range adequate (this will be discussed thismore in the section on modal effective mass)
Are the modes clearly identified
Is Mesh Density adequate
Is the Element Type appropriate Is the Mass distribution correct
Is coupled vs. lumped mass important
Are the internal joints modeled correctly
Are the constraints modeled correctly
Do the results compare with hand calculations, previousexperience or test
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