nas101, page 6 - 1 section 6 real eigenvalue analysis

NAS101, Page 6 - NAS101, Page 6 - 11

Section 6

Real Eigenvalue Analysis

NAS101, Page 6 - NAS101, Page 6 - 22

Real Eigenvalue Analysis

PAGE

Governing Equations 4

Mass Matrix 6

Theoretical Results 10

Reasons to Compute Natural Frequencies 16

and Normal Modes

Important Facts and Results Regarding Normal Modes 18

and Natural Frequencies

Methods of Computation 22

Normal Modes Analysis Entries 25

Mass Properties 26

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Real Eigenvalue Analysis (cont.)

PAGE

Output from Grid Point Weight Generator 29

SUPORT Entry 30

Normal Modes Analysis Entries 32

Workshop 8 36

Normal Mode Analysis of Stiffened Plate 37

Partial Input File for Workshop # 8 39

F06 Output for Workshop # 8 41

Mode # 1 for Workshop 8 42

Mode # 2 for Workshop 8 43

Solution for Workshop # 8 44

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GOVERNING EQUATIONS

Consider the undamped single-degree-of-freedom system shown below

where m = mass

k = stiffness

The equation of motion for free vibrations (i.e., without external load or damping) is:

m

x

k

mx·· kx–=

or

mx·· kx 0=+

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GOVERNING EQUATIONS (Cont.)For a multi-degree-of-freedom system, this equation becomes

where [K] = the stiffness matrix of the structure (the same asin static analysis)

[M] the mass matrix of the structure. (It represents the inertia properties of the structure.)

[K] and [M] must be real and symmetric.

Remember: The number of degrees of freedom is equal to the number of coordinates necessary to describe the deformed shape of the structure at any given time.

M x·· K x 0=+

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MASS MATRIX

The mass matrix represents the inertia properties of the structure. MSC/NASTRAN provides the user with two choices:

1. Lumped mass matrix (default)

Contains only diagonal terms associated with translational degrees of freedom

2. Coupled mass matrix

Also contains off-diagonal terms coupling translational degrees of freedom and rotational degrees of freedom. (Note: for a rod element, only translational DOFs are coupled.)

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MASS MATRIX (Cont.)Example of Mass Matrix

where = mass density

A = cross section

Lumped Mass Matrix

Coupled Mass Matrix

L

2 1 3 4

M AL

1 2 0 0 00 0 0 0

0 0 1 2 0

0 0 0 0

=

M AL

5 12 0 1 12 00 0 0 0

1 12 0 5 12 0

0 0 0 0

=

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MASS MATRIX (Cont.)

Coupled versus Lumped Mass Coupled mass is generally more accurate than lumped mass. Lumped mass is preferred for computational speed in dynamic

analysis. User-selectable coupled mass matrix for elements

PARAM,COUPMASS,1 to select coupled mass matrices for all BAR, ROD, and PLATE elements that include bending stiffness

Default is lumped mass.

Elements that have either lumped or coupled mass BAR, BEAM, CONROD, HEXA, PENTA, QUAD4, QUAD8, ROD,

TETRA, TRIA3, TRIA6, TRIAX6, TUBE

NAS101, Page 6 - NAS101, Page 6 - 99

MASS MATRIX (Cont.)

Elements that have lumped mass only CONEAX, SHEAR

Elements that have coupled mass only BEND, HEX20, TRAPRG, TRIARG

Lumped mass contains only diagonal, translational components (no rotational ones).

Coupled mass contains off-diagonal translational components as well as rotations for BAR (though no torsion), BEAM, and BEND elements.

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THEORETICAL RESULTSConsider (6-1)

Assume a harmonic solution of the form

(6-2)

(Physically, this means that all the coordinates perform synchronous motions and the system configuration does not change its shape during motion only its amplitude.)

From Equation 6-2 (6-3)

M x·· K x 0=+

x eit

=

x·· –2 e

it=

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THEORETICAL RESULTS (cont.)

Substituting Eqs. 6-2 and 6-3 into Equation 6-1, we get

which simplifies to

This is an eigenvalue problem.

2M e

itK e

it0=+–

K 2M – 0=

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THEORETICAL RESULTS(Cont.)

Therefore, there are two cases:

1. If det ( [ K ] – 2 [ M ] ) = 0 , the only possibility (from Eq.

6-4) is {

which is the so-called trivial solution and is not interesting from a physical point of view.

2. We need det ( [ K ] – 2 [ M ] ) = 0 in order to have a nontrivial solution for { }.

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THEORETICAL RESULTS(Cont.)

The eigenvalue problem reduces to

det ( [ K ] – 2 [ M ] ) = 0

or

det ( [ K ] – [ M ] ) = 0

where =2

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THEORETICAL RESULTS (Cont.)

If the structure has N dynamic degrees of freedom (degrees of freedom with mass), there are N number of ’s that are solution of the eigenvalue problem. These ’s (1, 2, ..., N) are the natural frequencies of the structure, also known as normal frequencies, characteristic frequencies, fundamental frequencies, or resonant frequencies.

The eigenvector associated with the natural frequency j is called normal mode or mode shape. The normal mode corresponds to deflected shape patterns of the structure.

When a structure is vibrating, its shape at any given time is a linear combination of its normal modes.

j

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THEORETICAL RESULTS (Cont.)

Example

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REASONS TO COMPUTE NATURAL FREQUENCIES

AND NORMAL MODES Assess the dynamic characteristics of the structure. For

example, if rotating machinery is going to be installed on a certain structure, it might be necessary to see if the frequency of the rotating mass is close to one of the natural frequencies of the structure to avoid excessive vibrations.

Assess possible dynamic amplification of loads. Use natural frequencies and normal modes to guide subsequent

dynamic analysis (transient response, response spectrum analysis) i.e., what should be the appropriate t for integrating the equation of motion in transient analysis?

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REASONS TO COMPUTE NATURAL FREQUENCIES

AND NORMAL MODES (cont.) Use natural frequencies and mode shapes for subsequent

dynamic analysis i.e., transient analysis of the structure using modal expansion.

Guide the experimental analysis of the structure, i.e., the location of accelerometers, etc.

Your boss told you to

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IMPORTANT FACTS AND RESULTS REGARDING NORMAL MODES AND

NATURAL FREQUENCIESIf a structure is not totally constrained, i.e., if it admits a rigid body

mode (stress-free mode) or a mechanism, at least one natural frequency will be zero.

Example: The following unconstrained structure has a rigid body mode.

mm

k

x1 x2

1 0

1 1

1

==

NAS101, Page 6 - NAS101, Page 6 - 1919


NATURAL FREQUENCIES (Cont.)

The natural frequencies (1, 2, ...,) are expressed in radians/seconds. They can also be expressed in hertz (cycles/seconds) using

f j hertz

j radians second

2-------------------------------------------------=

NAS101, Page 6 - NAS101, Page 6 - 2020



Scaling of normal modes is arbitrary. For example

Represent the same “mode of vibration”

1

1

0.5

1

300

150

= = 1

.66

.33

=

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Determination of the natural frequencies, i.e., solution of

Requires the use of a numerical approach.

det K 2M – 0=

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METHODS OF COMPUTATIONMSC/NASTRAN provides the user with the following three types of

methods for eigenvalue extraction.

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. This approach is more convenient when few natural frequencies are to be determined. In general, SINV is more reliable than INV.

Transformation Methods

The original eigenvalue problemK M – 0=

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METHODS OF COMPUTATION (Cont.)

is transformed to the form

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 technique and two variations of the Householder technique are provided: GIV, MGIV, HOU, and MHOU. These methods are more efficient for small models when a large proportion of eigenvalues are needed.

A =

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METHODS OF COMPUTATION (Cont.)

Lanczos Method

This is the recommended method and is a combined tracking transformation method. This method is most efficient for computing a few eigenvalues of large, sparse problems (most structural models fit into this category).

NAS101, Page 6 - NAS101, Page 6 - 2525

NORMAL MODES ANALYSIS ENTRIES

Executive SOLs 103

Case Control METHOD = x Where x is the SID Number associated with the EIGR or

EIGRL entry that is included in the Bulk Data. Multiple subcases may be used with additional qualifier.

Bulk Data EIGR entry - Eigenvalue extraction entry

or EIGRL entry for Lanczos Method Mass properties are required.

NAS101, Page 6 - NAS101, Page 6 - 2626

Mass PropertiesMass Properties

Structural Mass Adds mass of the elements (example - usedfor calculating gravity effects)

Density on MATi entries,

units = (“mass”/volume)

Nonstructural Mass Adds mass (example - building floor loads, ship cargo loads)

1 2 3 4 5 6 7 8 9 10

MAT1 MID E G NU RHO

MAT1 1 10.+7 0.3 0.1

NAS101, Page 6 - NAS101, Page 6 - 2727

Mass Properties (Cont.) Mass per unit dimension (mass per unit area in this case)

Concentrated Mass Explicit mass properties at a point (CONM2) (i.e., center of gravity of the

concentrated mass offset from the grid point, moments, and products of inertia

1 2 3 4 5 6 7 8 9 10

PSHELL PID MID1 T MID2 121/T3 MID3 TS/T NSM

PSHELL 2 1 0.1 1 1 0.15

NAS101, Page 6 - NAS101, Page 6 - 2828

Mass Properties (Cont.)

Mass Units Program assumes inertial units:

PARAM,WTMASS multiplies the input data to obtain inertial units. This is commonly used to change from weight units to mass units.

Example: The weight density (RHO) of steel is specified as 490.0 lb/cu ft on a MAT1.

Include PARAM,WTMASS,.031056 which multiplies the terms of the structural mass matrix by 1/g (= 1/32.174 ft/sec2) to change the density to proper inertial units.

lb-sec2/ft (ft-lb-sec system)

kg-sec2/m

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Output from Grid Point Weight Generator

NAS101, Page 6 - NAS101, Page 6 - 3030

SUPORT Entry

SUPORT Bulk Data entry

A program aid used in computing rigid body modes

Esthetics Absolute zero eigenvalues instead of computed zeros (for all but Lanczos, where the program will "judge" whether the eigenvalues should be 0.0 or not)

Cost Separate subroutine used to compute rigid body modes can significantly increase cpu requirement

SUPORT ID C ID C ID C ID C

SUPORT 16 125

NAS101, Page 6 - NAS101, Page 6 - 3131

SUPORT Entry (Cont.)

Notes: 1. Statically determinate set of constraints

2. Sufficient number of constraints to support all rigid body modes

3. The Lanczos method uses the computed eigenvectors.

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NORMAL MODES ANALYSIS ENTRIES (Cont.)

EIGRL Entry - recommended eigenvalue solution method

Defines data needed to perform real eigenvalue or buckling analysis with the Lanczos Method.

Field Contents

SID Set identification number (unique integer > 0)

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

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V1, V2 Vibration analysis: Frequency range of interest

Buckling analysis: range of interest (V1 < V2, real). If all modes below a frequency are desired , set V2 to the

desired frequency and leave V1 blank. It is not recommended to put 0.0 for V1, it is more efficient to use a small negative number or to leave it blank.

ND Number of roots desired (integer > 0 or blank)

MSGLVLDiagnostic level (integer 0 through 3 or blank)

MAXSETNumber of vectors in block (integer 1 through 15 or blank)

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EIGRL Entry - recommended eigenvalue solution method

SHFSCL Estimate of the first flexible mode natural frequency (real or blank)

NORM Method for normalizing eigenvectors, either "MASS" or "MAX"

MASS Normalize to unit value of the generalized mass (default)

MAX Normalize to unit value of the largest component in the analysis set

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Based on the input, the program will either:

Calculate all modes below V2 (V1 = blank, V2 = highest frequency of interest, ND = blank)

Calculate a maximum of ND roots between V1 and V2 (V1, V1, ND not blank)

Calculate ND roots above V1 (V1 = lowest frequency of interest, V2 = blank, ND = number of roots desired)

Calculate the first ND roots (V1 and V2 blank, ND = number of roots desired).

Calculate all roots between V1 and V2 (V1 = lowest frequency of interest, V2 = highest frequency of interest, ND = blank)

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Workshop 8

Normal Mode Analysis of Stiffened Plate

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Normal Mode Analysis of Stiffened Plates (cont.)

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Normal Mode Analysis of Stiffened Plates (cont.)

Model description Same stiffened plate model used in workshop # 5.

Calculate the first 6 modes.

Make sure that the masses are in the proper units.

NAS101, Page 6 - NAS101, Page 6 - 3939

Partial Input File for Workshop # 8

NAS101, Page 6 - NAS101, Page 6 - 4040

Partial Input File for Workshop # 8 (cont.)

NAS101, Page 6 - NAS101, Page 6 - 4141

F06 Output for Workshop # 8

NAS101, Page 6 - NAS101, Page 6 - 4242

Mode # 1 for Workshop 8

NAS101, Page 6 - NAS101, Page 6 - 4343

Mode # 2 for Workshop 8

NAS101, Page 6 - NAS101, Page 6 - 4444

Solution for Workshop # 8

NAS101, Page 6 - NAS101, Page 6 - 4545

Solution for Workshop # 8 (cont.)

NAS101, Page 6 - NAS101, Page 6 - 4646

Blank Page

nas101, page 6 - 1 section 6 real eigenvalue analysis

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