Frequency is an important aspect for mechanical systems, which can be used to predict the behavior of systems under different dynamic conditions. The reliability and performance of the mechanical systems can be optimized, and potential design issues can be identified by analyzing its frequency response. Every physical system has natural frequencies associated with it, which depend on the system’s mass, damping and stiffness properties. When the external excitation matches the natural frequency of the system, there will be amplified response known as resonance. It is characterized by a huge increase in amplitude and energy transfer leading to vibrations or oscillations. All the accumulated energy from the system is released during resonance resulting in an amplified response. This phenomenon can yield both advantageous and adverse consequences.
For example, it is useful in medical imaging, radio communication and musical instruments. Whereas it can cause excessive vibrations, increased stress, and potential failure in engineering scenarios. Hence, frequency analysis is incorporated in the design phases by engineers to avoid issues related to resonance.
This blog post examines the Eigenvalue problem and the extraction methods available in the Abaqus solver.
Eigenvalue Problems
Frequency extraction is a linear perturbation procedure to calculate the natural frequencies and corresponding mode shapes of a multi-body systems. The general equation of motion without external excitation is given below,
where [M] is the symmetric and positive definite mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, mu is the eigen value and {o} is the eigenvector.
The above equation system yields complex eigenvalues and eigenvectors. This equation can be simplified by ignoring [C] and assuming symmetric [K]. This symmetric system has real squared eigenvalues mu2, and real eigenvectors.
Expressing mu = iω, where ω is the circular frequency, the above eigen value problem becomes,
In the above equation, all the eigen values are positive when [K] is positive definite. Instabilities and rigid body modes cause [K] to be indefinite leading to negative and zero eigenvalues. There are different eigenvalue extraction methods available in Abaqus. Let’s briefly discuss these methods in this article:
Eigenvalue Extraction Methods Available In Abaqus
1. Lanczos Eigensolver
This is the default method of extracting frequencies in Abaqus because of its more general capabilities, especially while dealing with symmetric sparse matrices. This is an iterative numerical method used to extract approximate eigenvalues and eigenvectors. In each Lanczos run, a set of iterations called steps are performed. In each of these steps, the size of vector subspace grows allowing for a better approximation of the eigenvectors. The size at which the subspace grows is determined by the block size at each Lanczos step. By using this sequence of subspaces, a tridiagonal matrix [T] that approximates the eigenvalues and eigenvectors of a large, sparse symmetric matrix is constructed iteratively in this solver.
The eigenvalues extracted from this tridiagonal matrix [T] are the approximate eigenvalues of the original symmetric matrix.
Termination of Lanczos run:
The run will be terminated when one of the following conditions is met.
All the required eigen values are extracted.
The tridiagonal matrix is ill-conditioned or singular.
The number of Lanczos steps exceed the specified maximum value.
The cost per solving an eigenvalue is increasing.
2. Automatic Multi-Level Substructuring (AMS)
This method is faster than Lanczos and a powerful tool, which can be used when a large number of eigenvalues are needed to be extracted for huge models with many degrees of freedom. AMS solves for the eigenvectors at every node in the model by default, but we can also request the eigenvectors only at specific nodes by creating a node set. This improves the efficiency of the model by reducing the amount of stored data.
This AMS method consists of three phases which are briefly described below.
Reduction Phase
The full physical system is reduced using a multilevel substructuing technique to help in a highly efficient eigenvalue extraction. A multilevel supernode elimination tree is created as shown in the figure below. Starting from the lowest level of supernodes in the elimination tree and moving upwards, the substructure reduction technique is used to reduce the size of the matrix system. Local eigenvalues are extracted at each of these supernodes by fixing the DOF attached to the next highest level supernode. By the end of the reduction phase, the original matrix system is reduced to a unit diagonal matrix with only off diagonal non-zero values representing the coupling between supernodes.
Reduced Eigen Solution Phase
In this phase, the eigenvalues and vectors are computed for the reduced matrix system. This reduced system will usually be too large to solve directly. Hence, a single subspace iteration step is used for solving this reduced problem.
Recovery Phase
Using the eigenvectors of the reduced problem and local substructure modes, the eigenvectors of the original system are extracted at the specified nodes.
3. Subspace Iteration
This method is faster when only a few (<20) eigenvalues are needed. In this method, the eigenvectors of the symmetric, undamped system are extracted first. A subspace is created using these extracted eigenvectors. The difference between Lanczos and subspace iteration methods lies in the fact that the dimension of subspace used to approximate the eigenvectors is fixed in the subspace iteration method. To extract the original complex eigenvalues of the actual system, the original mass, damping, and stiffness matrices are projected onto the created subspace as shown below.
Substituting these projected matrices into the original equation of motion gives,
The transformation will reduce the size of the matrix system to be solved. The above small complex eigenvalue problem is solved using the standard QZ method to calculate the approximate eigenvalues of the original system. The eigenvectors of the original system can then be recovered from the projected eigenvectors using the equation below.
Thus, the subspace iteration method can be used effectively to extract the eigenvalues and eigenvectors of a complex system by reducing its size. The lower the number of eigenvalues required, the smaller the size of the matrix system.
Final Thoughts
Frequency analysis is an integral part of the design process. It helps in predicting the behavior of the system under different dynamic loads. Hopefully this article has provided a summary of eigenvalue extraction methods available in Abaqus. Typically for a small model, any of the three extraction methods described here can be used. But while solving large models, the proper choice of extraction method should be made for better computational efficiency and reducing stored data.
We’re always here to help, so if you have questions about contact in your models, or just FEA in general, don’t hesitate to reach out!