03 Section2 Normal Modes 012904

7/28/2019 03 Section2 Normal Modes 012904

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S2-1

NAS122, Section 2, January 2004

Copyright 2004 MSC.Software Corporation

SECTION 2

NORMAL MODES ANALYSIS



S2-2



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



S2-3



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



S2-4





S2-5



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



S2-6



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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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






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






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.






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






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





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.





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





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






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






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.






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






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






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





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






Create the grids

using Create/Node/Edit






Create the Bar2 generic Patran Elements

for the Springs using Create/Element/Edit



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Create the Point generic Patran Elements

for the Masses using Create/Element/Edit



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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



S2-25




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



S2-26




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



S2-27




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



S2-28




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



S2-29




The Analysis Type is selected as

Normal Modes (SOL103)

Solution Parameters are set:

Mass Calculation is Lumped

Wt. Mass Conversion is 1.0



S2-30




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.



S2-31




$ 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}

...



S2-32




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



S2-33




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



S2-34




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



S2-35NAS122, Section 2, January 2004Copyright 2004 MSC.Software Corporation


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





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





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





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






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






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


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






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.






The results file mass2_k3_ex.xdb is attached

to the database






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






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






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





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


1

2

WORKSHOP 2




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




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(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


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






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


8 703.29 3rd bend xy plane








Modes in XZ plane

Mode 1 = 45.74 Hz Mode 2 = 125.86 Hz

Mode 4 = 245.9 Hz Mode 6 = 409.74





Modes in XZ plane (cont.)


Mode 7 = 595 Hz Mode 9 = 809 Hz

Mode 10 = 1027 Hz





Modes in XY plane


Mode 3 = 132.6 Hz Mode 5 = 362.8 Hz

Mode 8 = 703.3 Hz






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






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




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




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.




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




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.




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.)




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




CASE STUDY 2 – NORMAL MODESANALYSIS OF A SATELLITE





CASE STUDY 2 – NORMAL MODESANALYSIS OF A SATELLITE

Import a MSC.Nastran .bdf

file





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.






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






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.






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.






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






Select the imported

subcase that contained the

correct boundary

conditions for this Normal

Modes Analysis.

Submit the model to

MSC.Nastran for analysis.






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 ] { }






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.






The Mode Shape and

Frequencies for the 10

Normal Modes requested

are shown.






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.






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 + + + + + +


6 42.94 1st order Panel pant, orthogonal to mode 5


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





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




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




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




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.


HOW ACCURATE IS THE NORMAL




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




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




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.







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 – 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




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




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.


WORKSHOP 15a – 15e




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 )




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




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




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




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




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




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




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



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

03 Section2 Normal Modes 012904

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