Eigenfrequency Analysis
An Eigenfrequency study solves for the eigenfrequencies (natural frequencies) and the shapes of the corresponding eigenmodes.
When performing an eigenfrequency analysis, you can specify whether to look at the mathematically more fundamental eigenvalue, λ, or the eigenfrequency, f, which is more commonly used in a structural mechanics context. The relation between the two is
where i is the imaginary unit.
The undamped eigenvalue problem is commonly written as
where K is the stiffness matrix, M is the mass matrix, u is the eigenmode displacement vector, and ω = 2πf is the angular frequency. If damping is present, the eigenvalue equation is expanded to
(2-1)
where C is the viscous damping matrix, and K can be complex-valued.
Because only the shape, and not the size, of the modes (eigenvectors) have physical significance, the computed modes can be scaled arbitrarily. You can select the method for scaling in the Eigenvalue Solver node of the solver sequence. If Scaling of eigenvectors is set to Mass matrix, the eigenmodes u are orthogonalized with respect to the mass matrix M so that
(2-2)
This is a common choice for the scaling of eigenvectors within the structural mechanics field. The choice of eigenvector scaling does not affect for example the results of a subsequent mode superposition analysis, but it will affect the interpretation of an exported modal representation of the system.
•
There are some situations where nonlinear expressions are used for evaluation of result quantities. Such expressions will only be correct if the scale of the eigenmodes is small enough so that any nonlinearities are negligible. For this reason, the default scaling uses a small number. The maximum value of an eigenmode is set to 10−6 times the size of the bounding box of the geometry.
•
The maximum value scaling checks each degree of freedom separately. This means that in a pure solid mechanics problem, the norm of the displacement of a mode shape can exceed the given scale somewhat.
•
In a multiphysics problem, it is possible that the largest values in the eigenmode vector represent another field, for example, electric potential. In that case, the largest displacement components will be smaller than the specified number.
Modal Participation Factors
Modal (or mass) participation factors are useful tools when working with the modal representation of a structure. Through them, you can get the following information:
•
The fraction of the total mass of a structure that a certain number of modes represent is one result. This can be important when judging if a set of modes forms a good enough base for a mode superposition.
•
The main direction of vibration for a certain mode can be seen from the relation between the participation factors.
•
When you have a large set of modes, an examination of the participation factors can give information about the dominant modes. This may then be used for a reduction of the size of subsequent modal-based studies.
To compute modal participation factors, a Participation Factors node must be present under Definitions in the current component. When you add an Eigenfrequency study from the Add Study window, such a node is automatically created.
You can also add a Participation Factors node manually under Definitions > Physics Utilities. If you do that after an eigenfrequency study has been run, you need to do an Update Solution in order to get access to the variables containing the participation factors.
The modal participation factors are available as global variables, and these can, for example, be displayed in a table using a Global Evaluation node under Derived Values in the Results branch. The participation factor results are available as predefined variables in the Definitions submenu for the component. In Table 2-1, the variables created from a Participation Factors node is listed (assuming the default tag mpf1).
Table 2-1: Participation Factor Variables.
Variable
Description
mpf1.mass
Total mass
mpf1.CMJ
Center of mass, J coordinate
mpf1.mEffLJ
Effective mass, translation along J-axis
mpf1.mEffRJ
Effective mass, rotation around J-axis
mpf.pfLJ
Participation factor, translation along J-axis
mpf.pfRJ
Participation factor, rotation around J-axis
mpf.pfLnormJ
Normalized participation factor, translation along J-axis
mpf.pfRnormJ
Normalized participation factor, rotation around J-axis
The normalized participation factors are those that would be obtained if mass matrix scaled eigenmodes would have been used.
If a Participation Factors node is present in the model when an eigenfrequency study is run, an evaluation group named Participation Factors is automatically generated. It contains a table with the translational and rotational modal participation factors for all computed eigenfrequencies.
•
If you would compute all eigenmodes of a structure, and sum all modal masses, they will usually not exactly match the total mass of the structure. The reason is that any mass which is associated with constrained degrees of freedom is lost. This effect is discretization dependent. The mass lost is a fraction of the mass of the elements having constrained nodes.
•
The computation of participation factors is a feature at the component level. For a case where there are couplings between two or more components that affect the eigenvalue problem, an evaluation of the participation factors and modal masses may give unpredictable results.
For an example showing how to compute modal participation factors and modal mass, see In-Plane Framework with Discrete Mass and Mass Moment of Inertia: Application Library path Structural_Mechanics_Module/Verification_Examples/inplane_framework_freq.
In the COMSOL Multiphysics Reference Manual:
•
Eigenvalue Solver
•
Studies and Solvers
•
Postprocessing of Eigenmodes
•
Derived Values, Evaluation Groups, and Tables
In the theory chapter of the Structural Mechanics User’s Guide:
•
Modal Participation Factors
It is possible to compute eigenfrequencies and eigenmodes for structures which are not fully constrained. Such eigenmodes are sometimes referred to as free-free modes. For each possible rigid body mode, there is one eigenvalue which in theory is zero. The number of possible rigid body modes for different geometrical dimensions is shown in the table below.
Table 2-2: Number of Possible Rigid Body Modes.
Dimension
Number of rigid body modes
3D
6 (3 translations + 3 rotations)
2D axisymmetric
1 (Z-direction translation)
2D (solid, beam, truss)
3 (2 translations + 1 rotation)
2D (plate)
3 (1 translation + 2 rotations)
The computed rigid body modes will in general not be recognizable as having pure translation or rotation. Rather, they will contain linear combinations of all the fundamental rigid body motions.
In a piezoelectric model, one more zero eigenfrequency could appear if you have not set a reference value for the electric potential.
In practice, the natural frequencies of the rigid body modes are not computed as exactly zero, but can appear as small numbers which may even be negative or complex. If rigid body modes are present in the model, then it is important to use a nonzero value in the Search for eigenfrequencies around text field in the settings for the Eigenfrequency study step. The value should reflect the order of magnitude of the first important nonzero eigenfrequency.
For an example showing an eigenfrequency computation in a model having a rigid body mode, see Eigenfrequency Analysis of a Free Cylinder: Application Library path Structural_Mechanics_Module/Verification_Examples/free_cylinder.
Damping
If any type of damping is included in the model, an eigenfrequency solution automatically returns the damped eigenvalues. The eigenfrequencies and, in general, also the mode shapes are complex in this case. A complex-valued eigenfrequency can be interpreted so that the real part represents the actual frequency, and the imaginary part represents the damping. The damping ratio of the corresponding eigenmode can be defined as
where the approximate expression is valid with high accuracy (within 2%) as long as the damping is less than 0.2.
If the damping ratio is high, more than 0.6 or so, then the default ARPACK eigenvalue solver will show better convergence if you set Search method around shift to Larger real part. In particular, Rayleigh damping can cause high damping ratios for higher eigenmodes, since the damping asymptotically is proportional to the frequency.
In a complex mode shape, there are phase shifts between different parts of the structure, so that not all points reach the maximum at the same time under free vibration.
Some damping types will still give real-valued eigenmodes; this is the case for Rayleigh damping and loss factor damping.
If an eigenfrequency study is performed as a first step of a mode superposition analysis, then all features that supply damping should be disabled in this step. This can be done in the Physics and Variables Selection section in the settings for the Eigenfrequency study step.
The damping is taken into account in the mode superposition studies.
See also the discussion under Mode Superposition.
Prestressed Analysis
In a loaded structure, the natural frequencies may be shifted due to stress stiffening.
To do a prestressed analysis, Include geometric nonlinearity must be selected in the Eigenfrequency study step. This is automatic when you add the Eigenfrequency, Prestressed study type.
The prestress loading can include a contact analysis, in which case the subsequent eigenfrequency analysis provide a linearization around the current contact state.
Prestressed Structures
Frequency Dependent Material Properties
If the material data (stiffness or damping) is frequency dependent, the eigenvalue problem will become nonlinear. This can, for example, occur for some viscoelastic materials. In this case, the eigenvalue equation Equation 2-1 becomes
(2-3)
The eigenvalue solver as such assumes that the matrices involved are constant, so they must be evaluated at a certain frequency, the linearization point.
(2-4)
In order to get a correct solution to Equation 2-3, the linearization point ωL must be close to the actual eigenvalue ω. This is in general possible only for one single eigenfrequency at a time. You must solve this problem either by manual iteration, or by using some type of scripting, for example through a model method.
Eigenmodes of a Viscoelastic Structural Damper: Application Library path Structural_Mechanics_Module/Dynamics_and_Vibration/viscoelastic_damper_eigenmodes.
The built-in viscoelastic materials use a special formulation that do not generate a nonlinear eigenvalue problem, as long fractional derivatives are not used.