1. Real Eigenvalue Analysis
Section 6
Real Eigenvalue Analysis

2. Real Eigenvalue Analysis
PAGE
Governing Equations
Mass Matrix
Theoretical Results
Reasons to Compute Natural Frequencies
and Normal Modes
Important Facts and Results Regarding Normal Modes
and Natural Frequencies
Methods of Computation
Normal Modes Analysis Entries
Mass Properties

3. Real Eigenvalue Analysis (cont.)
PAGE
Output from Grid Point Weight Generator
SUPORT Entry
Normal Modes Analysis Entries
Workshop
Normal Mode Analysis of Stiffened Plate
Partial Input File for Workshop #
F06 Output for Workshop #
Mode # 1 for Workshop
Mode # 2 for Workshop
Solution for Workshop #

4. 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:

5. GOVERNING EQUATIONS (Cont.)
For a multi-degree-of-freedom system, this equation becomes
where [K] = the stiffness matrix of the structure (the same as in 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.

6. MASS MATRIX
The mass matrix represents the inertia properties of the structure. MSC/NASTRAN provides the user with two choices:
Lumped mass matrix (default)
Contains only diagonal terms associated with translational degrees of freedom
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.)

7. MASS MATRIX (Cont.) Example of Mass Matrix where r = mass density
A = cross section
Lumped Mass Matrix
Coupled Mass Matrix

8. 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

9. 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.

10. THEORETICAL RESULTS Consider Assume a harmonic solution of the form
(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)

11. THEORETICAL RESULTS (cont.)
Substituting Eqs. 6-2 and 6-3 into Equation 6-1, we get
which simplifies to
This is an eigenvalue problem.

12. THEORETICAL RESULTS(Cont.)
Therefore, there are two cases:
If det ( [ K ] – w2 [ M ] ) = 0 , the only possibility (from Eq.
6-4) is { f } = 0
which is the so-called trivial solution and is not interesting from a physical point of view.
We need det ( [ K ] – w2 [ M ] ) = in order to have a nontrivial solution for { f }.

13. THEORETICAL RESULTS(Cont.)
The eigenvalue problem reduces to
det ( [ K ] – w2 [ M ] ) = 0 or
det ( [ K ] – l [ M ] ) = 0
where l = w2

14. THEORETICAL RESULTS (Cont.)
If the structure has N dynamic degrees of freedom (degrees of freedom with mass), there are N number of w’s that are solution of the eigenvalue problem. These w’s (w1, w2, ..., wN) 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 wj 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.

15. THEORETICAL RESULTS (Cont.)
Example

16. 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 dt for integrating the equation of motion in transient analysis?

17. 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

18. IMPORTANT FACTS AND RESULTS REGARDING NORMAL MODES AND NATURAL FREQUENCIES
If 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.

19. IMPORTANT FACTS AND RESULTS REGARDING NORMAL MODES AND NATURAL FREQUENCIES (Cont.)
The natural frequencies (w1, w2, ...,) are expressed in radians/seconds. They can also be expressed in hertz (cycles/seconds) using

20. IMPORTANT FACTS AND RESULTS REGARDING NORMAL MODES AND NATURAL FREQUENCIES (Cont.)
Scaling of normal modes is arbitrary. For example
Represent the same “mode of vibration”

21. IMPORTANT FACTS AND RESULTS REGARDING NORMAL MODES AND NATURAL FREQUENCIES (Cont.)
Determination of the natural frequencies, i.e., solution of
Requires the use of a numerical approach.

22. METHODS OF COMPUTATION
MSC/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 problem

23. 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.

24. 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).

25. 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.

26. Mass Properties
Mass Properties
Structural Mass Adds mass of the elements (example - used for 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
10.+7
0.3
0.1

27. 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
0.1
0.15

28. 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 lb/cu ft on a MAT1.
Include PARAM,WTMASS, which multiplies the terms of the structural mass matrix by 1/g (= 1/ ft/sec2) to change the density to proper inertial units.
lb-sec2/ft (ft-lb-sec system)
kg-sec2/m

29. Output from Grid Point Weight Generator

30. 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 16
125

31. 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.

32. 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
0.1
3.2

33. NORMAL MODES ANALYSIS ENTRIES (Cont.)
V1, V2 Vibration analysis: Frequency range of interest
Buckling analysis: l 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)
MSGLVL Diagnostic level (integer 0 through 3 or blank)
MAXSET Number of vectors in block (integer 1 through 15 or blank)

34. NORMAL MODES ANALYSIS ENTRIES (Cont.)
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

35. NORMAL MODES ANALYSIS ENTRIES (Cont.)
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)

36. Normal Mode Analysis of Stiffened Plate
Workshop 8
Normal Mode Analysis of Stiffened Plate

37. Normal Mode Analysis of Stiffened Plates (cont.)

38. 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.

39. Partial Input File for Workshop # 8

40. Partial Input File for Workshop # 8 (cont.)

41. F06 Output for Workshop # 8

42. Mode # 1 for Workshop 8

43. Mode # 2 for Workshop 8

44. Solution for Workshop # 8

45. Solution for Workshop # 8 (cont.)

46. Blank Page