How is a mode shape calculated

Modal analysis

From ESOCAETWIKIPLUS

engl: modal analysis Category: Level 2 theory mechanics


General information on this can be found, for example, at wikipedia: Modal Analysis

simulation

A modal analysis is used to determine the Natural frequencies (eigenvalues) and the Eigenforms (natural modes of vibration) used in structural dynamics. Modal analysis will too Eigenvalue Analysis or Eigenvalue problem called.

The results of the modal analysis - the eigenvalues, eigenfrequencies and eigenmodes - are important parameters for the design of a structure with regard to dynamic loads. They are also required if a spectrum analysis or a frequency response analysis or transient dynamic simulation with the help of modal superposition (superposition of the natural oscillation forms) is to be carried out afterwards.

As a result, the participation factors are usually output for the natural frequencies found. This makes it easier to evaluate the respective eigenvalue or the associated eigenmode with regard to its significance for the dynamics of the component.

In coordinate directions in which no restraints are defined, rigid body modes (zero frequency modes), but also higher-order natural oscillation forms of the elastic body (higher free body modes, frequency not equal to zero) are calculated.

Note: a modal analysis always applies to the linear simulation model.

Basics

The basics of modal analysis are presented in Modalanalyse_Theorie.

With the modal analysis, the Natural frequencies and also the Eigenforms be determined. The natural frequencies are system or component parameters, i.e. frequencies characteristic of this component. There is an associated eigenmode for each of these natural frequencies. This eigenmode is the deformation that the component would show if it vibrated at this frequency. The expression "... would show" is intended to make it clear that vibration and deformation only really occur when there is a stimulus. Depending on this excitation, an overall oscillation of the component then results, which is essentially composed of the individual oscillation forms. The natural frequencies and natural forms provide an indication of how a system behaves under dynamic loading. The amplitudes of the eigenmodes are NOT suitable for a technical quantitative evaluation of the component (they are scaled appropriately in the solution), only the shape is suitable for a qualitative assessment of the dynamics of the component. As an example, the figure on the right shows the first 3 eigenmodes of a rectangular plate.

You can refresh your basic knowledge of dynamic simulations of structural mechanics with the excerpt from the book FEM for Practitioners, Volume 2: Structural Dynamics, Part 1.

What is required as input?

NO loads are required as input. Storage can only be specified as retention.

What can be expected as a result?

The result of the modal analysis are eigenvalues ​​and eigenfrequencies. For each eigenvalue or each eigenfrequency, an eigenmode is calculated, that is, displacements, strains, stresses for a state. With these results one can "understand" the component in terms of structural dynamics and recognize its principal properties.

An eigenmode can only be understood qualitatively, it is the deformation that the component would show if it vibrated at this frequency. The numerical values ​​are standardized, so they are only suitable for comparison with one another. So: a shift of xxx m will not occur, but should show that it is larger or smaller than a shift at another point in the model.

What is NOT to be expected as a result?

The modal analysis does not provide any information about actual displacements, strains, stresses under certain loads.

Nonlinearities, contact

The modal analysis is a linear analysis. This means that no change in the component behavior that depends on the displacements or rotations can be taken into account. Any non-linearities such as plasticity and contact elements are not taken into account, even if they are present in practice.

In the case of non-linearities such as plasticity, the modal analysis is based on an initial state, for example the modulus of elasticity or the initial gradient of the stress-strain function.

Contact elements are based on a certain state (open, closed). This state remains independent of the calculated eigenmode, so the contact does not change due to the displacements. You can find further details under Contact: Basics.

Solution method

During the simulation, you can choose from several calculation methods to determine the natural vibration forms:

  • Householder method (reduced),
  • Subspace method,
  • Procedure for asymmetric matrices (asymmetric) or for damped systems (damped) and
  • Block Lanczos Method.

A useful aid in idealization is cyclic symmetry, in which the analysis of a cyclically symmetric structure only needs to be carried out on the basis of one sector.

The only "loads" that can be defined in a modal analysis are the setting of displacements to the value zero. Shifts with specified values ​​not equal to zero are set to zero. Other loads can be entered, but will not be taken into account.

Modal analyzes can also be carried out on prestressed structures such as a rotating turbine blade. In this case, a static simulation is carried out first. This provides the numerical values ​​that represent the prestress (stress matrix). The modal analysis of the prestressed structure builds on this. The eigenvalues ​​are therefore dependent on the mass, stiffness and stress matrix.

tips and tricks

Models in which rigid-body modes occur deliver natural frequencies of zero as a result (or values ​​close to zero because of the approximations when using the solution method).

Duplicate eigenvalues ​​occur when a component has the same properties in two spatial directions or when symmetries are present. What is meant by "the same properties in two spatial directions"? For example a vertical mast like in the figure on the right. The properties should be the same in both horizontal directions (blue and red). Then there are two equal eigenvalues ​​for the oscillations in these two directions (blue and red).

The modal analysis can also be based on a "free floating" component. In everyday technical life, this can apply to components with very soft bearings. Even a floating ship or an airplane in flight behave in almost the same way. Of course, this also applies to a satellite in weightlessness. This is also referred to as a free-free vibration.

If the results of the modal analysis deviate by orders of magnitude (i.e. 10, 100 or 1000 times) from the expected results, then there are mostly errors in the use of the units of the input data. Especially if there are deviations by a factor of around 103 occur, this indicates the units. In a modal analysis, a factor of around 30 is typical because mass and stiffness are "below the root" and therefore √103 acts.

When creating simulation models - especially when based on CAD data - all separate parts of the component have to be connected to one another (for example via contact conditions) or statically sufficient restraints. To check this, a modal analysis can be carried out. Parts of the model that are not adequately held produce zero eigenvalues ​​/ eigenfrequencies.

Other terms

If damping is to be taken into account, the modal analysis results in complex eigenvalues. In this case, more numerical values ​​have to be processed in the numerical solution, the numerical effort becomes higher. In addition, the combination of the numerical values ​​leads to asymmetrical matrices, so that the solution of the modal analysis requires special numerical algorithms (asymmetrical solver).

An accompanying modal analysis can be used as a solution support for buckling analyzes. The accompanying modal analysis uses the property that in the event of a stability case (i.e. when a dent occurs) in the structural mechanics, the eigenvalue of the current matrix values ​​reaches the value zero. This procedure is an example of a perturbation.

The Coriolis forces can be taken into account in a modal analysis.

When using a modal analysis as part of an optimization, attention must be paid to the assignment of eigenvalues ​​and eigenmodes using modal tracking.

Example: wind turbine tower

Wind turbine tower modal analysis

Self study

You can find a particularly clear example here: the idealization and modal analysis of a ruler on the edge of a desk.

Here you can find some (not at all easy) comprehension questions and put your knowledge to the test:

Additional information

Further information on Ansys Mechanical can be found on the CADFEM website. Get to know the CADFEM seminar and eLearning offer and deepen your knowledge of structural dynamics.

Practical procedure as a video

You can find a description of the practical procedure on the CADFEM YouTube channel. The CADFEM tutorial no. 8 - Modal analysis offered there shows the implementation of a modal analysis.