1、Abaqus 固有频率提取6.3.5 Natural frequency extractionProducts: Abaqus/Standard Abaqus/CAE Abaqus/AMS References “Procedures: overview,” Section 6.1.1 “General and linear perturbation procedures,” Section 6.1.2 “Dynamic analysis procedures: overview,” Section 6.3.1 *FREQUENCY “Configuring a frequency proce
2、dure” in “Configuring linear perturbation analysis procedures,” Section 14.11.2 of the Abaqus/CAE Users ManualOverviewThe frequency extraction procedure: performs eigenvalue extraction to calculate the natural frequencies and the corresponding mode shapes of a system; will include initial stress and
3、 load stiffness effects due to preloads and initial conditions if geometric nonlinearity is accounted for in the base state, so that small vibrations of a preloaded structure can be modeled; will compute residual modes if requested; is a linear perturbation procedure; can be performed using the trad
4、itional Abaqus software architecture or, if appropriate, the high-performance SIM architecture (see “Using the SIM architecture for modal superposition dynamic analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1); and solves the eigenfrequency problem only for symmetric mass and stif
5、fness matrices; the complex eigenfrequency solver must be used if unsymmetric contributions, such as the load stiffness, are needed.Eigenvalue extractionThe eigenvalue problem for the natural frequencies of an undamped finite element model iswhere is the mass matrix (which is symmetric and positive
6、definite);is the stiffness matrix (which includes initial stiffness effects if the base state included the effects of nonlinear geometry);is the eigenvector (the mode of vibration); andM and Nare degrees of freedom.When is positive definite, all eigenvalues are positive. Rigid body modes and instabi
7、lities cause to be indefinite. Rigid body modes produce zero eigenvalues. Instabilities produce negative eigenvalues and occur when you include initial stress effects. Abaqus/Standard solves the eigenfrequency problem only for symmetric matrices. Selecting the eigenvalue extraction methodAbaqus/Stan
8、dard provides three eigenvalue extraction methods: Lanczos Automatic multi-level substructuring (AMS), an add-on analysis capability for Abaqus/Standard Subspace iterationIn addition, you must consider the software architecture that will be used for the subsequent modal superposition procedures. The
9、 choice of architecture has minimal impact on the frequency extraction procedure, but the SIM architecture can offer significant performance improvements over the traditional architecture for subsequent mode-based steady-state or transient dynamic procedures (see “Using the SIM architecture for moda
10、l superposition dynamic analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1). The architecture that you use for the frequency extraction procedure is used for all subsequent mode-based linear dynamic procedures; you cannot switch architectures during an analysis. The software archite
11、ctures used by the different eigensolvers are outlined in Table 6.3.51.Table 6.3.51 Software architectures available with different eigensolvers.EigensolverSoftware ArchitectureLanczos AMS Subspace IterationTraditional SIM The Lanczos solver with the traditional architecture is the default eigenvalu
12、e extraction method because it has the most general capabilities. However, the Lanczos method is generally slower than the AMS method. The increased speed of the AMS eigensolver is particularly evident when you require a large number of eigenmodes for a system with many degrees of freedom. However,
13、the AMS method has the following limitations: All restrictions imposed on SIM-based linear dynamic procedures also apply to mode-based linear dynamic analyses based on mode shapes computed by the AMS eigensolver. See “Using the SIM architecture for modal superposition dynamic analyses” in “Dynamic a
14、nalysis procedures: overview,” Section 6.3.1, for details. The AMS eigensolver does not compute composite modal damping factors, participation factors, or modal effective masses. However, if participation factors are needed for primary base motions, they will be computed but are not written to the p
15、rinted data (.dat) file. You cannot use the AMS eigensolver in an analysis that contains piezoelectric elements. You cannot request output to the results (.fil) file in an AMS frequency extraction step.If your model has many degrees of freedom and these limitations are acceptable, you should use the
16、 AMS eigensolver. Otherwise, you should use the Lanczos eigensolver. The Lanczos eigensolver and the subspace iteration method are described in“Eigenvalue extraction,” Section 2.5.1 of the Abaqus Theory Manual. Lanczos eigensolverFor the Lanczos method you need to provide the maximum frequency of in
17、terest or the number of eigenvalues required; Abaqus/Standard will determine a suitable block size (although you can override this choice, if needed). If you specify both the maximum frequency of interest and the number of eigenvalues required and the actual number of eigenvalues is underestimated,
18、Abaqus/Standard will issue a corresponding warning message; the remaining eigenmodes can be found by restarting the frequency extraction.You can also specify the minimum frequencies of interest; Abaqus/Standard will extract eigenvalues until either the requested number of eigenvalues has been extrac
19、ted in the given range or all the frequencies in the given range have been extracted.See “Using the SIM architecture for modal superposition dynamic analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1, for information on using the SIM architecture with the Lanczos eigensolver.Input F
20、ile Usage: *FREQUENCY, EIGENSOLVER=LANCZOS Abaqus/CAE Usage: Step module: Step Create: Frequency: Basic: Eigensolver: LanczosChoosing a block size for the Lanczos methodIn general, the block size for the Lanczos method should be as large as the largest expected multiplicity of eigenvalues (that is,
21、the largest number of modes with the same frequency). A block size larger than 10 is not recommended. If the number of eigenvalues requested is n, the default block size is the minimum of (7, n). The choice of 7 for block size proves to be efficient for problems with rigid body modes. The number of
22、block Lanczos steps within each Lanczos run is usually determined by Abaqus/Standard but can be changed by you. In general, if a particular type of eigenproblem converges slowly, providing more block Lanczos steps will reduce the analysis cost. On the other hand, if you know that a particular type o
23、f problem converges quickly, providing fewer block Lanczos steps will reduce the amount of in-core memory used. The default values areBlock size Maximum number of block Lanczos steps1 802 503 45 4 35Automatic multi-level substructuring (AMS) eigensolverFor the AMS method you need only specify the ma
24、ximum frequency of interest (the global frequency), and Abaqus/Standard will extract all the modes up to this frequency. You can also specify the minimum frequencies of interest and/or the number of requested modes. However, specifying these values will not affect the number of modes extracted by th
25、e eigensolver; it will affect only the number of modes that are stored for output or for a subsequent modal analysis.The execution of the AMS eigensolver can be controlled by specifying three parameters: , , and . These three parameters multiplied by the maximum frequency of interest define three cu
26、t-off frequencies. (default value of 5) controls the cutoff frequency for substructure eigenproblems in the reduction phase, while and (default values of 1.7 and 1.1, respectively) control the cutoff frequencies used to define a starting subspace in the reduced eigensolution phase. Generally, increa
27、sing the value of and improves the accuracy of the results but may affect the performance of the analysis.Requesting eigenvectors at all nodesBy default, the AMS eigensolver computes eigenvectors at every node of the model.Input File Usage: *FREQUENCY, EIGENSOLVER=AMS Abaqus/CAE Usage: Step module:
28、Step Create: Frequency: Basic: Eigensolver: AMSRequesting eigenvectors only at specified nodesAlternatively, you can specify a node set, and eigenvectors will be computed and stored only at the nodes that belong to that node set. The node set that you specify must include all nodes at which loads ar
29、e applied or output is requested in any subsequent modal analysis (this includes any restarted analysis). If element output is requested or element-based loading is applied, the nodes attached to the associated elements must also be included in this node set. Computing eigenvectors at only selected
30、nodes improves performance and reduces the amount of stored data. Therefore, it is recommended that you use this option for large problems.Input File Usage: *FREQUENCY, EIGENSOLVER=AMS, NSET=name Abaqus/CAE Usage: Step module: Step Create: Frequency: Basic: Eigensolver: AMS: Limit region of saved ei
31、genvectorsControlling the AMS eigensolverThe AMS method consists of the following three phases:Reduction phase: In this phase Abaqus/Standard uses a multi-level substructuring technique to reduce the full system in a way that allows a very efficient eigensolution of the reduced system. The approach
32、combines a sparse factorization based on a multi-level supernode elimination tree and a local eigensolution at each supernode. Starting from the lowest level supernodes, we use a Craig-Bampton substructure reduction technique to successively reduce the size of the system as we progress upward in the
33、 elimination tree. At each supernode a local eigensolution is obtained based on fixing the degrees of freedom connected to the next higher level supernode (these are the local retained or “fixed-interface” degrees of freedom). At the end of the reduction phase the full system has been reduced such t
34、hat the reduced stiffness matrix is diagonal and the reduced mass matrix has unit diagonal values but contains off-diagonal blocks of nonzero values representing the coupling between the supernodes.The cost of the reduction phase depends on the system size and the number of eigenvalues extracted (th
35、e number of eigenvalues extracted is controlled indirectly by specifying the highest eigenfrequency desired). You can make trade-offs between cost and accuracy during the reduction phase through the parameter. This parameter multiplied by the highest eigenfrequency specified for the full model yield
36、s the highest eigenfrequency that is extracted in the local supernode eigensolutions. Increasing the value of increases the accuracy of the reduction since more local eigenmodes are retained. However, increasing the number of retained modes also increases the cost of the reduced eigensolution phase,
37、 which is discussed next.Reduced eigensolution phase: In this phase Abaqus/Standard computes the eigensolution of the reduced system that comes from the previous phase. Although the reduced system typically is two orders of magnitude smaller in size than the original system, generally it still is to
38、o large to solve directly. Thus, the system is further reduced mainly by truncating the retained eigenmodes and then solved using a single subspace iteration step. The two AMS parameters, and , define a starting subspace of the subspace iteration step. The default values of these parameters are care
39、fully chosen and provide accurate results in most cases. When a more accurate solution is needed, the recommended procedure is to increase both parameters proportionally from their respective default values.Recovery phase: In this phase the eigenvectors of the original system are recovered using eig
40、envectors of the reduced problem and local substructure modes. If you request recovery at specified nodes, the eigenvectors are computed only at those nodes.Subspace iteration methodFor the subspace iteration procedure you need only specify the number of eigenvalues required; Abaqus/Standard chooses
41、 a suitable number of vectors for the iteration. If the subspace iteration technique is requested, you can also specify the maximum frequency of interest; Abaqus/Standard extracts eigenvalues until either the requested number of eigenvalues has been extracted or the last frequency extracted exceeds
42、the maximum frequency of interest.Input File Usage: *FREQUENCY, EIGENSOLVER=SUBSPACE Abaqus/CAE Usage: Step module: Step Create: Frequency: Basic: Eigensolver: SubspaceStructural-acoustic couplingStructural-acoustic coupling affects the natural frequency response of systems. In Abaqus only the Lancz
43、os eigensolver fully includes this effect. In Abaqus/AMS and the subspace eigensolver the effect of coupling is neglected for the purpose of computing the modes and frequencies; these are computed using natural boundary conditions at the structural-acoustic coupling surface. An intermediate degree o
44、f consideration of the structural-acoustic coupling operator is the default in Abaqus/AMS and the Lanczos eigensolver, which is based on the SIM architecture: the coupling is projected onto the modal space and stored for later use.Structural-acoustic coupling using the Lanczos eigensolver without th
45、e SIM architectureIf structural-acoustic coupling is present in the model and the Lanczos method not based on the SIM architecture is used, Abaqus/Standard extracts the coupled modes by default. Because these modes fully account for coupling, they represent the mathematically optimal basis for subse
46、quent modal procedures. The effect is most noticeable in strongly coupled systems such as steel shells and water. However, coupled structural-acoustic modes cannot be used in subsequent random response or response spectrum analyses. You can define the coupling using either acoustic-structural intera
47、ction elements (see “Acoustic interface elements,” Section 29.14.1) or the surface-based tie constraint (see “Acoustic, shock, and coupled acoustic-structural analysis,” Section 6.10.1). It is possible to ignore coupling when extracting acoustic and structural modes; in this case the coupling bounda
48、ry is treated as traction-free on the structural side and rigid on the acoustic side.Input File Usage: Use the following option to account for structural-acoustic coupling during the frequency extraction:*FREQUENCY, EIGENSOLVER=LANCZOS, ACOUSTIC COUPLING=ON (default if the SIM architecture is not us
49、ed)Use the following option to ignore structural-acoustic coupling during the frequency extraction:*FREQUENCY, EIGENSOLVER=LANCZOS, ACOUSTIC COUPLING=OFF Abaqus/CAE Usage: Step module: Step Create: Frequency: Basic: Eigensolver: Lanczos, toggle Include acoustic-structural coupling where applicableStructural-acoustic coupling using the AMS and Lanczos eigensolver based on the SIM architectureFor frequency extractions that use the AMS eigensolver or the Lanczos eigensolver based on the SIM architecture, the modes are computed using traction-free
