Default. To find many modes (about 40+) of large models. Recommen-ded when the model consists of poorly shaped solid and shell ele-ments. This solver performs well when the model consists of shells or a combination of shells and solids. Works faster but requires about 50% more memory than subspace.
Block Lanczos
High Low
To find few modes (up to about 40) of large models. Recommended when the model consists of well-shaped solid and shell elements.
Works well if memory availability is limited.
Subspace
Low High
To find few modes (up to about 20) of large models. Recommended for fast computation of eigenvalues of over 100K DOF models. On coarse mesh models, the frequencies are approximate. Missed modes are possible when repeated frequencies are present.
Power Dynamics
Low Low
To find all modes of small to medium models (less than 10K DOF). Can be used to find few modes (up to about 40) of large models with proper selection of master DOF, but accuracy of frequencies depends on the master DOF selected.
Reduced
3.13.1. Block Lanczos Method
The Block Lanczos eigenvalue solver is the default. It uses the Lanczos algorithm where the Lanczos recursion is performed with a block of vectors. This method is as accurate as the subspace method, but faster. The Block Lanczos method uses the sparse matrix solver, overriding any solver specified via the EQSLV command.
The Block Lanczos method is especially powerful when searching for eigenfrequencies in a given part of the ei-genvalue spectrum of a given system. The convergence rate of the eigenfrequencies will be about the same when extracting modes in the midrange and higher end of the spectrum as when extracting the lowest modes.
Therefore, when you use a shift frequency (FREQB) to extract n modes beyond the starting value of FREQB, the algorithm extracts the n modes beyond FREQB at about the same speed as it extracts the lowest n modes.
3.13.2. Subspace Method
The subspace method uses the subspace iteration technique, which internally uses the generalized Jacobi iteration algorithm. It is highly accurate because it uses the full [K] and [M] matrices. For the same reason, however, the subspace method is slower than the reduced method. This method is typically used in cases where high accuracy is required or where selecting master DOF is not practical.
When doing a modal analysis with a large number of constraint equations, use the subspace method with the frontal solver instead of the JCG solver, or use the Block Lanczos mode-extraction method. Using the JCG solver when your analysis has many constraint equations could result in an internal element stiffness assembly that requires large amounts of memory.
3.13.3. PowerDynamics Method
The PowerDynamics method internally uses the subspace iterations, but uses the PCG iterative solver. This method may be significantly faster than either the subspace or the Block Lanczos methods, but may not converge if the model contains poorly-shaped elements, or if the matrix is ill-conditioned. This method is especially useful in very large models (100,000+ DOFs) to obtain a solution for the first few modes. Do not use this method if you will be running a subsequent spectrum analysis.
Section 3.13: Comparing Mode-Extraction Methods
The PowerDynamics method does not perform a Sturm sequence check (that is, it does not check for missing modes), which might affect problems with multiple repeated frequencies. This method always uses lumped mass approximation.
Note — If you use PowerDynamics to solve a model that includes rigid body modes, be sure to issue the RIGID command or choose one of its GUI equivalents (Main Menu> Solution> Analysis Options or Main Menu> Preprocessor> Loads> Analysis Options).
3.13.4. Reduced Method
The reduced method uses the HBI algorithm (Householder-Bisection-Inverse iteration) to calculate the eigenvalues and eigenvectors. It is relatively fast because it works with a small subset of degrees of freedom called master DOF. Using master DOF leads to an exact [K] matrix but an approximate [M] matrix (usually with some loss in mass). The accuracy of the results, therefore, depends on how well [M] is approximated, which in turn depends on the number and location of masters. Section 3.14: Matrix Reduction presents guidelines to select master DOFs.
3.13.5. Unsymmetric Method
The unsymmetric method, which also uses the full [K] and [M] matrices, is meant for problems where the stiffness and mass matrices are unsymmetric (for example, acoustic fluid-structure interaction problems). It uses the Lanczos algorithm which calculates complex eigenvalues and eigenvectors if the system is non-conservative (for example, a shaft mounted on bearings). The real part of the eigenvalue represents the natural frequency and the imaginary part is a measure of the stability of the system - a negative value means the system is stable, whereas a positive value means the system is unstable. Sturm sequence checking is not available for this method.
Therefore, missed modes are a possibility at the higher end of the frequencies extracted.
3.13.6. Damped Method
The damped method is meant for problems where damping cannot be ignored, such as rotor dynamics applica-tions. It uses full matrices ([K], [M], and the damping matrix [C]). It uses the Lanczos algorithm and calculates complex eigenvalues and eigenvectors (as described below). Sturm sequence checking is not available for this method. Therefore, missed modes are a possibility at the higher end of the frequencies extracted.
3.13.6.1. Damped Method-Real and Imaginary Parts of the Eigenvalue
The imaginary part of the eigenvalue, Ω, represents the steady-state circular frequency of the system. The real part of the eigenvalue, σ, represents the stability of the system. If σ is less than zero, then the displacement amplitude will decay exponentially, in accordance with EXP(σ). If σ is greater than zero, then the amplitude will increase exponentially. (Or, in other words, negative σ gives an exponentially decreasing, or stable, response;
and positive σ gives an exponentially increasing, or unstable, response.) If there is no damping, the real component of the eigenvalue will be zero.
Note — The eigenvalue results reported by ANSYS are actually divided by 2* π. This gives the frequency in Hz (cycles/second). In other words:
Imaginary part of eigenvalue, as reported = Ω/(2* π) Real part of eigenvalue, as reported = σ/(2* π)
3.13.6.2. Damped Method-Real and Imaginary Parts of the Eigenvector
In a damped system, the response at different nodes can be out of phase. At any given node, the amplitude will be the vector sum of the real and imaginary components of the eigenvector.
3.13.7. QR Damped Method
The QR damped method combines the advantages of the Block Lanczos method with the complex Hessenberg method. The key concept is to approximately represent the first few complex damped eigenvalues by a linear combination of 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. Using the QR algorithm, a smaller eigenvalue problem is then solved in the modal subspace. This approach gives good results for lightly damped systems and can also be applicable to any arbitrary damping type (proportional or non-proportional symmetric damping or unsymmetric gyroscopic damping matrix). Because the accuracy of this method is dependent on the number of modes used in the calculations, a sufficient number of fundamental modes are recommended, especially for highly damped systems to provide good results. This method is not recommended for critically damped or overdamped systems. This method outputs both the real and imaginary eigenvalues (frequencies), but outputs only the real eigenvectors (mode shapes).
3.14. Matrix Reduction
Matrix reduction is a way to reduce the size of the matrices of a model and perform a quicker and cheaper ana-lysis. It is mainly used in dynamic analyses such as modal, harmonic, and transient analyses. Matrix reduction is also used in substructure analyses to generate a superelement.
Matrix reduction allows you to build a detailed model, as you would for a static stress analysis, and use only a
"dynamic" portion of it for a dynamic analysis. You choose the "dynamic" portion by identifying key degrees of freedom, called master degrees of freedom, that characterize the dynamic behavior of the model. The ANSYS program then calculates reduced matrices and the reduced DOF solution in terms of the master DOF. You can then expand the solution to the full DOF set by performing an expansion pass. The main advantage of this pro-cedure is the savings in CPU time to obtain the reduced solution, especially for dynamic analyses of large problems.
3.14.1. Theoretical Basis of Matrix Reduction
The ANSYS program uses the Guyan Reduction procedure to calculate the reduced matrices. The key assumption in this procedure is that for the lower frequencies, inertia forces on the slave DOF (those DOF being reduced out) are negligible compared to elastic forces transmitted by the master DOF. Therefore, the total mass of the structure is apportioned among only the master DOF. The net result is that the reduced stiffness matrix is exact, whereas the reduced mass and damping matrices are approximate. For details about how the reduced matrices are cal-culated, refer to the ANSYS, Inc. Theory Reference.
3.14.1.1. Guidelines for Selecting Master DOF
Choosing master DOF is an important step in a reduced analysis. The accuracy of the reduced mass matrix (and hence the accuracy of the solution) depends on the number and location of masters. For a given problem, you can choose many different sets of master DOF and will probably obtain acceptable results in all cases.
You can choose masters using M and MGEN commands, or you can have the program choose masters during solution using the TOTAL command. We recommend that you do both: choose a few masters yourself, and also have the ANSYS program choose masters. This way, the program can pick up any modes that you may have missed. The following list summarizes the guidelines for selecting master DOF:
• The total number of master DOF should be at least twice the number of modes of interest.
• Choose master DOF in directions in which you expect the structure or component to vibrate. For a flat plate, for example, you should choose at least a few masters in the out-of-plane direction (see
Fig-Section 3.14: Matrix Reduction
ure 3.2: “Choose Master DOF” (a)). In cases where motion in one direction induces a significant motion in another direction, choose master DOF in both directions (see Figure 3.2: “Choose Master DOF” (b)).