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Abaqus固有頻率提取 6.3.5Natural frequency extraction Products:Abaqus/StandardAbaqus/CAEAbaqus/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 procedure” in “Configuring linear perturbation analysis procedures,”Section 14.11.2 of the Abaqus/CAE Users Manual Overview The frequency extraction procedure: performs eigenvalue extraction to calculate the natural frequencies and the corresponding mode shapes of a system; will include initial stress and 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 traditional 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 stiffness matrices; the complex eigenfrequency solver must be used if unsymmetric contributions, such as the load stiffness, are needed. Eigenvalue extraction The eigenvalue problem for the natural frequencies of an undamped finite element model is where is the mass matrix (which is symmetric and positive 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); and MandN are degrees of freedom. Whenis positive definite, all eigenvalues are positive. Rigid body modes and instabilities causeto 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 method Abaqus/Standard provides three eigenvalue extraction methods: Lanczos Automatic multi-level substructuring (AMS), an add-on analysis capability for Abaqus/Standard Subspace iteration In addition, you must consider the software architecture that will be used for the subsequent modal superposition procedures. The 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 modal 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 architectures used by the different eigensolvers are outlined inTable 6.3.5–1. Table 6.3.5–1Software architectures available with different eigensolvers. Software Architecture Eigensolver Lanczos AMS Subspace Iteration Traditional SIM The Lanczos solver with the traditional architecture is the default eigenvalue 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, 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 analysis 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 printed 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 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 eigensolver For the Lanczos method you need to provide the maximum frequency of interest 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, 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 extracted 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. InputFileUsage: *FREQUENCY, EIGENSOLVER=LANCZOS Abaqus/CAEUsage: Step module:StepCreate:Frequency:Basic:Eigensolver:Lanczos Choosing a block size for the Lanczos method In general, the block size for the Lanczos method should be as large as the largest expected multiplicity of eigenvalues (that is, the largest number of modes with the same frequency). A block size larger than 10 is not recommended. If the number of eigenvalues requested isn, 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 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 of problem converges quickly, providing fewer block Lanczos steps will reduce the amount of in-core memory used. The default values are Block size Maximum number of block Lanczos steps 1 80 2 50 3 45 ≥ 4 35 Automatic multi-level substructuring (AMS) eigensolver For the AMS method you need only specify the maximum 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 the 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 cut-off frequencies.(default value of 5) controls the cutoff frequency for substructure eigenproblems in the reduction phase, whileand(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, increasing the value ofandimproves the accuracy of the results but may affect the performance of the analysis. Requesting eigenvectors at all nodes By default, the AMS eigensolver computes eigenvectors at every node of the model. InputFileUsage: *FREQUENCY, EIGENSOLVER=AMS Abaqus/CAEUsage: Step module:StepCreate:Frequency:Basic:Eigensolver:AMS Requesting eigenvectors only at specified nodes Alternatively, 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 are 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 nodes improves performance and reduces the amount of stored data. Therefore, it is recommended that you use this option for large problems. InputFileUsage: *FREQUENCY, EIGENSOLVER=AMS, NSET=name Abaqus/CAEUsage: Step module:StepCreate:Frequency:Basic:Eigensolver:AMS:Limit region of saved eigenvectors Controlling the AMS eigensolver The 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 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 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 that 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 (the 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 theparameter. This parameter multiplied by the highest eigenfrequency specified for the full model yields the highest eigenfrequency that is extracted in the local supernode eigensolutions. Increasing the value ofincreases 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, 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 too 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 carefully 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 eigenvectors 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 method For the subspace iteration procedure you need only specify the number of eigenvalues required; Abaqus/Standard chooses 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 the maximum frequency of interest. InputFileUsage: *FREQUENCY, EIGENSOLVER=SUBSPACE Abaqus/CAEUsage: Step module:StepCreate:Frequency:Basic:Eigensolver:Subspace Structural-acoustic coupling Structural-acoustic coupling affects the natural frequency response of systems. In Abaqus only the Lanczos 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 of 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 the SIM architecture If 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 subsequent 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 interaction 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 boundary is treated as traction-free on the structural side and rigid on the acoustic side. InputFileUsage: 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 used) Use the following option to ignore structural-acoustic coupling during the frequency extraction: *FREQUENCY, EIGENSOLVER=LANCZOS, ACOUSTIC COUPLING=OFF Abaqus/CAEUsage: Step module:StepCreate:Frequency:Basic:Eigensolver: Lanczos, toggleInclude acoustic-structural coupling where applicable Structural-acoustic coupling using the AMS and Lanczos eigensolver based on the SIM architecture For frequency extractions that use the AMS eigensolver or the Lanczos eigensolver based on the SIM architecture, the modes are computed using traction-free boundary conditions on the structural side of the coupling boundary and rigid boundary conditions on the acoustic side. Structural-acoustic coupling operators (see“Acoustic, shock, and coupled acoustic-structural analysis,”Section 6.10.1) are projected by default onto the subspace of eigenvectors. Contributions to these global operators, which come from surface-based tie constraints defined between structural and acoustic surfaces, are assembled into global matrices that are projected onto the mode shapes and used in subsequent SIM-based modal dynamic procedures. User-defined acoustic-structural interaction elements (see“Acoustic interface elements,”Section 29.14.1) cannot be used in an AMS eigenvalue extraction analysis. InputFileUsage: Use either of the following options to project structural-acoustic coupling operators onto the subspace of eigenvectors: *FREQUENCY, EIGENSOLVER=AMS, ACOUSTIC COUPLING=PROJECTION (default for the AMS eigensolver) or *FREQUENCY, EIGENSOLVER=LANCZOS, SIM, ACOUSTIC COUPLING=PROJECTION (default in SIM-based analysis) Use the following option to disable the projection of structural-acoustic coupling operators: *FREQUENCY, ACOUSTIC COUPLING=OFF Abaqus/CAEUsage: Use the following option to project structural-acoustic coupling operators onto the subspace of eigenvectors: Step module:StepCreate:Frequency:Basic:Eigensolver: AMS, toggle onProject acoustic-structural coupling where applicable Use the following option to disable the projection of structural-acoustic coupling operators: Step module:StepCreate:Frequency:Basic:Eigensolver: AMS, toggle offProject acoustic-structural coupling where applicable Projection of structural-acoustic coupling operators using the Lanczos eigensolver based on the SIM architecture is not supported in Abaqus/CAE. Specifying a frequency range for the acoustic modes Because structural-acoustic coupling is ignored during the AMS and SIM-based Lanczos eigenanalysis, the computed resonances will, in principle, be higher than those of the fully coupled system. This may be understood as a consequence of neglecting the mass of the fluid in the structural phase and vice versa. For the common metal and air case, the structural resonances may be relatively unaffected; however, some acoustic modes that are significant in the coupled response may be omitted due to the airs upward frequency shift during eigenanalysis. Therefore, Abaqus allows you to specify a multiplier, so that the maximum acoustic frequency in the analysis is taken to be higher than the structural maximum. InputFileUsage: Use either of the following options: *FREQUENCY, EIGENSOLVER=AMS , , , , , , acoustic range factor or *FREQUENCY, EIGENSOLVER=LANCZOS, SIM , , , , , , acoustic range factor Abaqus/CAEUsage: Step module:StepCreate:Frequency:Basic:Eigensolver: AMS,Acoustic range factor:acoustic range factor Specifying a frequency range for the acoustic modes when using the SIM-based Lanczos eigenanalysis is not supported in Abaqus/CAE. Effects of fluid motion on natural frequency analysis of acoustic systems To extract natural frequencies from an acoustic-only or coupled structural-acoustic system in which fluid motion is prescribed using an acoustic flow velocity, either the Lanczos method or the complex eigenvalue extraction procedure can be used. In the former case Abaqus extracts real-only eigenvalues and considers the fluid motions effects only on the acoustic stiffness matrix. Thus, these results are of primary interest as a basis for subsequent linear perturbation procedures. When the complex eigenvalue extraction procedure is used, the fluid motion effects are included in their entirety; that is, the acoustic stiffness and damping matrices are included in the analysis. Frequency shift For the Lanczos and subspace iteration eigensolvers you can specify a positive or negative shifted squared frequency,S. This feature is useful when a particular frequency is of concern or when the natural frequencies of an unrestrained structure or a structure that uses secondary base motions (large mass approach) are needed. In the latter case a shift from zero (the frequency of the rigid body modes) will avoid singularity problems or round-off errors for the large mass approach; a negative frequency shift is normally used. The default is no shift. If the Lanczos eigensolver is in use and the user-specified shift is outside the requested frequency range, the shift will be adjusted automatically to a value close to the requested range. Normalization For the Lanczos and subspace iteration e