Normal Modes Analysis
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Normal Modes Analysis Normal modes analysis computes the natural freque ncies and mode shapes of a structure. The natural frequencies are the frequencies at which a structure will tend to vibrate if subjected to a disturbance. For example, the strings of a piano are each tuned to vibrate at a specific frequency. The deformed shape at a specific natural frequency is called the mode shape. Normal modes analysis is also called real eigenvalue analysis. Normal modes analysis forms the foundation for a thorough un derstanding of the dynamic characteristics of the structure. Normal modes analysis is perfor med for many reasons, among them: m Assessing the dynamic interaction between a component (such as a piece of rotating machinery) and its supporting structure; if the natural frequency of the supporting structure is close to an operating frequency of the component then there can be significant dynamic amplification of the loads. m Assessing the effects of design changes on the dynamic characteristics. m Using the modes in a subsequent forced response analysis. m Using the natural frequencies as a guide to selecting the proper time or frequency step for transient and frequency response analyses, respectively. m Assessing the degree of correlation between modal test data and analytical results. In normal modes analysis there is no applied load and the structure has no damping properties. The equation of motion is of the form: [M]{u} + [K]{u} = 0 where [K] and [M] are the stiffness a nd mass matrices representing the elastic and inertial properties of the structure respectively. These system matrices are g enerated automatically by MSC/NASTRAN, based on the geometry and properties of the FE model. Assuming a harmonic solution, the above reduces to an eigenvalue problem. where } is the eigenvector (or mode shape) corresponding to the eigenvalue (the natural or characteristic frequency). For each eigenvalue, which is proportional to a natural frequency, there is a corresponding eigenvector, or mode shape. The eigenvalues are related to the natural frequencies as follows: Each mode shape is similar to a static displaced shape in that there are displacements and rotations for each node. However, there is one important difference between the mode shape and the static displacements: the scaling. In static analysis the displacements are the true physical displacements due to the applied loads. Since there is no applied load in normal mode analysis, the mode shape components can all be scaled by an arbitrary factor. With MSC/NASTRAN this scaling can be done so that the maximum displacement in any K – i M i = 0 f i = i 2 4-26 4 Basic Analysis mode is 1.0. Another option is to allow any user s pecified degree of freedom have a modal displacement of 1.0. The first option — unit modal mass — is generally preferred, though the scaling of a maximum displacement to 1.0 is useful for comparison to modal test data. Element forces and stresses and reaction forces are computed in the same manner as for static analysis, with each mode shape treated the same as a set o f static displacements. Due to the scaling of each mode, the resulting element stresses and forces are on a per mode basis and cannot necessarily be compa red from one mode to another. Because no single eigenvalue extraction method is perfect for all models, we have incorporated several methods in MSC/NASTRAN. These eigenvalue extraction methods are: m Lanczos method m Givens method m Householder method m Modified Givens method m Modified Householder method m Inverse power method m Sturm modified inverse power method The Lanczos method is the best overall method due to its robustness, but the other methods (particularly the modified Givens method and the Sturm modified inverse power method) have applicability for particular cases.
Transcript of Normal Modes Analysis
Normal Modes Analysis Normal modes analysis computes the natural frequencies and mode shapes of a structure.
The natural frequencies are the frequencies at which a structure will tend to vibrate if
subjected to a disturbance. For example, the strings of a piano are each tuned to vibrate at
a specific frequency. The deformed shape at a specific natural frequency is called the mode
shape. Normal modes analysis is also called real eigenvalue analysis.
Normal modes analysis forms the foundation for a thorough understanding of the dynamic
characteristics of the structure. Normal modes analysis is performed for many reasons,
among them:
m Assessing the dynamic interaction between a component (such as a piece of rotatingmachinery) and its supporting structure; if the natural frequency of the supporting
structure is close to an operating frequency of the component then there can be
significant dynamic amplification of the loads.
m Assessing the effects of design changes on the dynamic characteristics.
m Using the modes in a subsequent forced response analysis.
m Using the natural frequencies as a guide to selecting the proper time or frequency step
for transient and frequency response analyses, respectively.
m Assessing the degree of correlation between modal test data and analytical results.
In normal modes analysis there is no applied load and the structure has no damping
properties. The equation of motion is of the form:
[M]{u} + [K]{u} = 0
where [K] and [M] are the stiffness and mass matrices representing the elastic and inertial
properties of the structure respectively. These system matrices are generated automatically
by MSC/NASTRAN, based on the geometry and properties of the FE model.
Assuming a harmonic solution, the above reduces to an eigenvalue problem.
where } is the eigenvector (or mode shape) corresponding to the eigenvalue (thenatural or characteristic frequency). For each eigenvalue, which is proportional to a natural
frequency, there is a corresponding eigenvector, or mode shape. The eigenvalues are
related to the natural frequencies as follows:Each mode shape is similar to a static displaced shape in that there are displacements and
rotations for each node. However, there is one important difference between the mode
shape and the static displacements: the scaling. In static analysis the displacements are the
true physical displacements due to the applied loads. Since there is no applied load in
normal mode analysis, the mode shape components can all be scaled by an arbitrary factor.With MSC/NASTRAN this scaling can be done so that the maximum displacement in any
K – i M i = 0
f i = i 2
4-26 4 Basic Analysis
mode is 1.0. Another option is to allow any user specified degree of freedom have a modaldisplacement of 1.0. The first option — unit modal mass — is generally preferred, though
the scaling of a maximum displacement to 1.0 is useful for comparison to modal test data.
Element forces and stresses and reaction forces are computed in the same manner as for
static analysis, with each mode shape treated the same as a set of static displacements. Due
to the scaling of each mode, the resulting element stresses and forces are on a per mode
basis and cannot necessarily be compared from one mode to another.Because no single eigenvalue extraction method is perfect for all models, we have
incorporated several methods in MSC/NASTRAN. These eigenvalue extraction methods
are:
m Lanczos method
m Givens method
m Householder method
m Modified Givens method
m Modified Householder method
m Inverse power method
m Sturm modified inverse power methodThe Lanczos method is the best overall method due to its robustness, but the other
methods (particularly the modified Givens method and the Sturm modified inverse power
method) have applicability for particular cases.
![Fitting Normal Modes to HF Radial and Total Surface ...codar.com/images/about/2003Aguilar_Thesis.pdfThis research will show how by using Normal Mode Analysis (NMA) (Eremeev [1996],](https://static.fdocuments.us/doc/165x107/5f5df0914773eb35d80a90b6/fitting-normal-modes-to-hf-radial-and-total-surface-codarcomimagesabout2003aguilar.jpg)
