mech dynamics 14.5 l04 modal
TRANSCRIPT
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Lecture 4
Modal Analysis
ANSYS MechanicalLinear and Nonlinear Dynamics
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Modal Analysis
Topics Covered
A. Definition and Purpose
B. Theory and Terminology
C. Eigen-frequencies, mode shapes
D. Participation factors, Effective Mass
E. Mode extraction methods- undamped
F. Contact
G. Analysis settings Workshop 4A
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Modal Analysis
Topics Covered
H. Damped Modal Analysis
I. Mode extraction methods- damped
Workshop 4B
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A. What is Modal Analysis
The modal analysis technique is used to determine the vibrationcharacteristics (i.e., natural frequencies and mode shapes) of linear
elastic structures.
The most fundamental of all dynamic analysis types.
Allows the design to avoid resonant vibrations or to vibrate at a specified
frequency.
Gives engineers an idea of how the design will respond to different types
of dynamic loads.
Helps in calculating solution controls for other dynamic analyses.
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... What is Modal Analysis
Assumptions and restrictions
The structure is linear (i.e. [M] and [K] matrices are constant).
No loads (forces, displacements, pressures, or temperatures) areallowed; i.e., free vibration.
Recommendation: Because a structures vibration
characteristics determine how it responds to any type of
dynamic load, it is generally recommended to perform a
modal analysis first before trying any other dynamic
analysis.
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B. Theory and Terminology
The linear equation of motion for free, un-damped vibration is
Assume harmonic motion:
Substituting{}
and{} In the governing equation gives an eigenvalue
equation:
0 uKuM
iiii
iii
tu
tu
sin
sin
2
02 i
MK
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... Theory and Terminology
This equality is satisfied if
i= 0
(trivial, implies no vibration)
2. or if =
This is an eigenvalue problem which may be solved for up to nroots
(12
,22
, ..,n2
).
These roots are the eigenvaluesof the equation
For each root (eigenvalue), there is a corresponding eigenvector
(1,2,.,n)
02
iMK
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In Modal Analysis:
The eigenvalues the square of the natural circular frequency of the
structure i
The eigenvectors the corresponding mode shapes i
Mode shapes can be normalized either to the mass matrix
or to unity, where the largest component of the vector {}iis set to 1.
Workbench displays results normalized to the mass matrix.
Because of this normalization, only the shape of the DOF solution hasreal meaning.
1i
T
i M
C. Eigen-frequencies and mode shapes.
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The square roots of the eigenvaluesare i, the structures naturalcircular frequencies (rad/s).
Natural frequenciesfican then
calculated asfi= i/2p(cycles/s).
It is the natural frequencies,fiin Hz,
that are input by the user and
output by Workbench.
The eigenvectors {
}irepresent themode shapes, i.e.the shape
assumed by the structure whenvibrating at frequencyfi.
... Eigen-frequencies and mode shapes.
mode 1
{
}1f1= 109 Hz
mode 2
{
}2f2= 202 Hz
mode 3
{
}3f3= 249 Hz
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The participation factors are calculated by
where {D} is an assumed unit displacement spectrum in each of the global
Cartesian directions and rotation about each of these axes.
Measures the amount of mass moving in each direction for each mode.
A high value in a direction indicates that the mode will be excited by
forces in that direction.
The Ratio is simply another list of participation factors, normalized to
the largest.
The concept of participation factors will be important in later chapters.
D. Participation Factor, Effective Mass
DMTii
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... Participation Factor, Effective Mass
DMTii
The participation factors are calculated by
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The effective mass is calculated by
Ideally, the sum of the effective masses in each direction should equaltotal mass of structure, but will depend on the number of modesextracted.
The ratio of effective mass to total mass can be useful for determining
whether or not a sufficient number of modes have been extracted.
... Participation Factor, Effective Mass
1if,2
2
, i
T
ii
i
T
i
iieff M
MM
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The effective mass is calculated by
... Participation Factor, Effective Mass
1if,2
2
, i
T
ii
i
T
i
iieff M
MM
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E. Mode Extraction Methods - Undamped
Procedure Usages Applications
Block Lanczos
(Direct)
Symmetric Many modes (about 40+) of large models.
Recommended when poorly shaped solid and shell elements exist.
Shells or a combination of shells and solids.
PCG Lanczos
(Iterative)
Symmetric
(but not applicable
for buckling)
Few modes (up to about 100) of very large models (500,000+
degrees of freedom).
Well-shaped 3-D solid elements.
Unsymmetric Unsymmetric
matrices
Uses the full [K] and [M] matrices.
When K and M are unsymmetric (i.e., acoustic fluid-structure
interaction problems).
Supernode Symmetric
(but not applicable
to buckling)
Many modes (up to 10,000).
Used for 2-D plane or shell/beam structures (100 modes or more)
and for 3-D solid structures (250 modes or more).
In most cases, the Program Controlledoption selects the optimal solverautomatically.
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Contact regions are available in modal analysis; however, since this is apurely linear analysis, contact behavior will differ for the nonlinear
contact types, as shown below:
Contact behavior will reduce to its linear counterparts.
Contact Type Static Analysis
Linear Dynamic Analysis
Initially TouchingInside Pinball
RegionOutside Pinball Region
Bonded Bonded Bonded Bonded Free
No Separation No Separation No Separation No Separation Free
Rough Rough Bonded Free Free
Frictionless Frictionless No Separation Free Free
Frictional Frictional
= 0, No Separation
> 0, Bonded
Free Free
F. Contact in Modal Analysis
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Number of modes You need to specify the number of frequencies of interest. The default is to
extract thefirst 6 natural frequencies.
The number of frequencies can be specified in two ways:
1. The first N frequencies (N > 0), or
2. The first N frequencies in a selected range of frequencies.
G. Analysis Settings
1 2
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Solver Control Two settings are available in this controlDamped and Solver Type.
Damped >> No (Default; i.e., undamped system)
Yes.
Depending on the selection made for Damped, different solver options are
provided accordingly.
... Analysis Settings
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Output Control By default only mode shapes are calculated.
Stress and Strain results can be requested to be calculated.
stress results only show the relative distribution of stress in the structure
and are not real stress values. You can also choose whether or not to have
these results stored for faster result calculations in linked systems.
... Analysis Settings
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ANSYS MechanicalLinear and Nonlinear Dynamics
Workshop 4A
Modal Analysis(Plate with a Hole)
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The linear equation of motion for free, damped vibration is
The eigenvalues are complex
The imaginary part of the eigenvalue is the natural frequency The real part of eigenvalue is a measure of stability
* Positive = unstable * Negative = stable
H. Damped Modal Analysis
0 uKuCuM
Undamped Damped =
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The complete expression for the structural damping matrix, [C], is
dampingStructural
1
dampingMass
1
mb
ma
N
j
j
m
j
N
ii
m
i
KK
MMC
... Damped Modal Analysis
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The value of and can be input using the following:
[1] Mater ial-dependent damping v alue
(Mass-Matrix Damping Multiplier, and k-Matrix Damping
Multiplier)
. Structural Damping Matrix [C]
mbma N
j j
m
j
N
i i
m
i
KMC11
22
i
i
i
Equivalent damping
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[2] Direct ly as glo bal damp ing v alue
(Details section of Analysis Settings)
. Structural Damping Matrix [C]
KMC 22
i
i
i
Equivalent damping
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I. Mode Extraction Methods - Damped
Procedure Usages Applications
Damped Symmetric or
unsymmetric
damped systems
To find all modes of small to medium models (less than 10,000
degrees of freedom).
Uses full matrices ([K], [M], and the damping matrix [C])Reduced
Damping
Symmetric or
unsymmetric
damped systems
Approximately represent the first few complex damped eigenvalues by
modal transformation using a small number of eigenvectors of the
undamped system. After the undamped mode shapes are evaluated by
using the real eigensolution (Block Lanczos method), the equations of
motion are transformed to these modal coordinates.
In most cases, the Program Controlledoption selects the optimal solverautomatically.
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ANSYS MechanicalLinear and Nonlinear Dynamics
Workshop 4B
Modal Analysis
(Model Airplane Wing)