김경종
338 lines
21 KiB
Markdown
338 lines
21 KiB
Markdown
<!-- source-page: 261 -->
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# Predefined fields
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Predefined temperature fields are not allowed in mode-based steady-state dynamic analysis. Other predefined fields are ignored.
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# Material options
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As in any dynamic analysis procedure, mass or density (“Density,” Section 21.2.1) must be assigned to some regions of any separate parts of the model where dynamic response is required. The following material properties are not active during mode-based steady-state dynamic analyses: plasticity and other inelastic effects, viscoelastic effects, thermal properties, mass diffusion properties, electrical properties (except for the electrical potential, , in piezoelectric analysis), and pore fluid flow properties—see “General and linear perturbation procedures,” Section 6.1.3.
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# Elements
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Any of the following elements available in Abaqus/Standard can be used in a steady-state dynamics procedure:
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• stress/displacement elements (other than generalized axisymmetric elements with twist);
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• acoustic elements;
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• piezoelectric elements; or
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• hydrostatic fluid elements.
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See “Choosing the appropriate element for an analysis type,” Section 27.1.3.
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# Output
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In mode-based steady-state dynamic analysis the value of an output variable such as strain (E) or stress (S) is a complex number with real and imaginary components. In the case of data file output the first printed line gives the real components while the second lists the imaginary components. Results and data file output variables are also provided to obtain the magnitude and phase of many variables (see “Abaqus/Standard output variable identifiers,” Section 4.2.1). In this case the first printed line in the data file gives the magnitude while the second gives the phase angle.
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The following variables are provided specifically for steady-state dynamic analysis:
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Element integration point variables:
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<table><tr><td>PHS</td><td>Magnitude and phase angle of all stress components.</td></tr><tr><td>PHE</td><td>Magnitude and phase angle of all strain components.</td></tr><tr><td>PHEPG</td><td>Magnitude and phase angles of the electrical potential gradient vector.</td></tr><tr><td>PHEFL</td><td>Magnitude and phase angles of the electrical flux vector.</td></tr><tr><td>PHMFL</td><td>Magnitude and phase angle of the mass flow rate in fluid link elements.</td></tr><tr><td>PHMFT</td><td>Magnitude and phase angle of the total mass flow in fluid link elements.</td></tr></table>
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For connector elements, the following element output variables are available:
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<table><tr><td>PHCTF</td><td>Magnitude and phase angle of connector total forces.</td></tr><tr><td>PHCEF</td><td>Magnitude and phase angle of connector elastic forces.</td></tr><tr><td>PHCVF</td><td>Magnitude and phase angle of connector viscous forces.</td></tr><tr><td>PHCRF</td><td>Magnitude and phase angle of connector reaction forces.</td></tr><tr><td>PHCSF</td><td>Magnitude and phase angle of connector friction forces.</td></tr><tr><td>PHCU</td><td>Magnitude and phase angle of connector relative displacements.</td></tr><tr><td>PHCCU</td><td>Magnitude and phase angle of connector constitutive displacements.</td></tr></table>
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Nodal variables:
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<table><tr><td>PU</td><td>Magnitude and phase angle of all displacement/rotation components at a node.</td></tr><tr><td>PPOR</td><td>Magnitude and phase angle of the fluid or acoustic pressure at a node.</td></tr><tr><td>PHPOT</td><td>Magnitude and phase angle of the electrical potential at a node.</td></tr><tr><td>PRF</td><td>Magnitude and phase angle of all reaction forces/moments at a node.</td></tr><tr><td>PHCHG</td><td>Magnitude and phase angle of the reactive charge at a node.</td></tr></table>
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Element energy densities (such as the elastic strain energy density, SENER) and whole element energies (such as the total kinetic energy of an element, ELKE) are not available for output in a modebased steady-state dynamic analysis.
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The standard output variables U, V, A, and the variable PU listed above correspond to motions relative to the motion of the primary base in a mode-based analysis. Total values, which include the motion of the primary base, are also available:
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<table><tr><td>TU</td><td>Magnitude of all components of total displacement/rotation at a node.</td></tr><tr><td>TV</td><td>Magnitude of all components of total velocity at a node.</td></tr><tr><td>TA</td><td>Magnitude of all components of total acceleration at a node.</td></tr><tr><td>PTU</td><td>Magnitude and phase angle of all total displacement/rotation components at a node.</td></tr></table>
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The following modal variables are also available for mode-based steady-state dynamic analysis and can be output to the data, results, and/or output database files (see “Output to the data and results files,” Section 4.1.2, and “Output to the output database,” Section 4.1.3):
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<table><tr><td>GU</td><td>Generalized displacements for all modes.</td></tr><tr><td>GV</td><td>Generalized velocities for all modes.</td></tr><tr><td>GA</td><td>Generalized accelerations for all modes.</td></tr><tr><td>GPU</td><td>Phase angle of generalized displacements for all modes.</td></tr><tr><td>GPV</td><td>Phase angle of generalized velocities for all modes.</td></tr><tr><td>GPA</td><td>Phase angle of generalized acceleration for all modes.</td></tr><tr><td>SNE</td><td>Elastic strain energy for the entire model per mode.</td></tr></table>
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KE Kinetic energy for the entire model per mode.
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T External work for the entire model per mode.
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BM Base motion.
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Whole model variables such as ALLIE (total strain energy) are available for mode-based steadystate dynamics as output to the data, results, and/or output database files (see “Output to the data and results files,” Section 4.1.2).
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# Acoustic contribution factors
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Computation of the acoustic contribution factors helps you determine the major noise sources. The procedure for computing the acoustic contribution factors is based on the modal analysis formulation of acoustic-structural problems with uncoupled modes. For more information, see “Acoustic contribution factors in mode-based and subspace-based steady-state dynamic analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1.
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Input file template
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```txt
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*HEADING
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...
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*AMPLITUDE, NAME=loadamp
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Data lines to define an amplitude curve as a function of frequency (cycles/time)
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*AMPLITUDE, NAME=base
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Data lines to define an amplitude curve to be used to prescribe base motion
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**
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*STEP, NLGEOM
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Include the NLGEOM parameter so that stress stiffening effects will
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be included in the steady-state dynamics step
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*STATIC
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**Any general analysis procedure can be used to preload the structure
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...
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*CLOAD and/or *DLOAD
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Data lines to prescribe preloads
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*TEMPERATURE and/or *FIELD
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Data lines to define values of predefined fields for preloading the structure
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*BOUNDARY
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Data lines to specify boundary conditions to preload the structure
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*END STEP
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**
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*STEP
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*FREQUENCY
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Data line to control eigenvalue extraction
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*BOUNDARY
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Data lines to assign degrees of freedom to the primary base
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```
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<!-- source-page: 264 -->
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\*BOUNDARY, BASE NAME=base2
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Data lines to assign degrees of freedom to a secondary base
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\*END STEP
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\*\*
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\*STEP
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\*STEADY STATE DYNAMICS
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Data lines to specify frequency ranges and bias parameters
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\*SELECT EIGENMODES
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Data lines to define the applicable mode ranges
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\*ACOUSTIC CONTRIBUTION
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\*MODAL DAMPING
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Data lines to define the modal damping factors
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\*BASE MOTION, DOF=dof, AMPLITUDE=base
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\*BASE MOTION, DOF=dof, AMPLITUDE=base, BASE NAME=base2
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\*CLOAD and/or \*DLOAD, AMPLITUDE=loadamp
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Data lines to specify sinusoidally varying, frequency-dependent loads
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\*END STEP
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<!-- source-page: 265 -->
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# 6.3.9 SUBSPACE-BASED STEADY-STATE DYNAMIC ANALYSIS
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Products: Abaqus/Standard Abaqus/CAE
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# References
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• “Defining an analysis,” Section 6.1.2
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• “General and linear perturbation procedures,” Section 6.1.3
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• “Dynamic analysis procedures: overview,” Section 6.3.1
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• “Direct-solution steady-state dynamic analysis,” Section 6.3.4
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• “Natural frequency extraction,” Section 6.3.5
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• “Mode-based steady-state dynamic analysis,” Section 6.3.8
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• \*STEADY STATE DYNAMICS
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• “Configuring a subspace-based steady-state dynamic analysis” in “Configuring linear perturbation analysis procedures,” Section 14.11.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
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# Overview
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A subspace-based steady-state dynamic analysis:
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• is used to calculate the steady-state dynamic linearized response of a system to harmonic excitation;
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• is based on projection of the steady-state dynamic equations on a subspace of selected modes of the undamped system;
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• is a linear perturbation procedure;
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• provides a cost-effective way to include frequency-dependent effects (such as frequency-dependent damping and viscoelastic effects) in the model;
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• allows for nonsymmetric stiffness;
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• requires that an eigenfrequency extraction procedure be performed prior to the steady-state dynamic analysis;
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• can use the high-performance SIM software architecture (see “Using the SIM architecture for modal superposition dynamic analyses” in “Dynamic analysis procedures: overview,” Section 6.3.1);
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• is an alternative to direct-solution steady-state dynamic analysis, in which the system’s response is calculated in terms of the physical degrees of freedom of the model;
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• can include computation of acoustic contribution factors to help determine the major contributors to acoustic noise;
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• is computationally cheaper than direct-solution steady-state dynamics but more expensive than mode-based steady-state dynamics;
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<!-- source-page: 266 -->
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• is less accurate than direct-solution steady-state analysis, in particular if significant material damping or viscoelasticity with a high loss modulus is present; and
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• is able to bias the excitation frequencies toward the values that generate a response peak.
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# Introduction
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Steady-state dynamic analysis provides the steady-state amplitude and phase of the response of a system subjected to harmonic excitation at a given frequency. Usually such analysis is done as a frequency sweep, by applying the loading at a series of different frequencies and recording the response. In Abaqus/Standard the subspace-based steady-state dynamic analysis procedure is used to conduct the frequency sweep.
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In a subspace-based steady-state dynamic analysis the response is based on direct solution of the steady-state dynamic equations projected onto a subspace of modes. The modes of the undamped, symmetric system must first be extracted using the eigenfrequency extraction procedure. The modes will include eigenmodes and, if activated in the eigenfrequency extraction step, residual modes. The procedure is based on the assumption that the forced steady-state vibration can be represented accurately by a number of modes of the undamped system that are in the range of the excitation frequencies of interest. The number of modes extracted must be sufficient to model the dynamic response of the system adequately, which is a matter of judgment on your part. The projection of the dynamic equilibrium equations onto a subspace of selected modes leads to a small system of complex equations that is solved for modal amplitudes, which are then used to compute nodal displacements, stresses, etc.
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When defining a subspace-based steady-state dynamic step, you specify the frequency ranges of interest and the number of frequencies at which results are required in each range (including the bounding frequencies of the range). In addition, you can specify the type of frequency spacing (linear or logarithmic) to be used, as described below (“Selecting the frequency spacing”). Logarithmic frequency spacing is the default if the frequency ranges are specified directly or by eigenfrequencies. If the frequency ranges are specified by the frequency spread, only linear spacing can be used. Frequencies should be given in cycles/time.
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The frequency points for which results are required can be spaced equally along the frequency axis (on a linear or a logarithmic scale), or they can be biased toward the ends of the user-defined frequency range by introducing a bias parameter (see “The bias parameter” below).
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The subspace-based steady-state dynamic analysis procedure can be used:
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• for nonsymmetric stiffness;
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• when any form of damping (except modal damping) is included; and
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• when viscoelastic material properties must be taken into account.
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While the response in this procedure is for linear vibrations, the prior response can be nonlinear. Initial stress effects (stress stiffening) will be included in the steady-state dynamic response if nonlinear geometric effects (“General and linear perturbation procedures,” Section 6.1.3) were included in any general analysis step prior to the eigenfrequency extraction step preceding the subspace-based steadystate dynamic procedure.
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Input File Usage: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION
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<!-- source-page: 267 -->
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# Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace
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# Ignoring damping
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If damping terms can be ignored, you can specify that a real, rather than a complex, system matrix be generated and projected, which can significantly reduce computational time, at the cost of ignoring the damping effects.
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Input File Usage: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION, REAL ONLY
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace: Compute real response only
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# Selecting the type of frequency interval for which output is requested
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Three types of frequency intervals are permitted for output from a subspace-based steady-state dynamic step.
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# Specifying the frequency ranges by using the system’s eigenfrequencies
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By default, the eigenfrequency type of frequency interval is used; in this case the following intervals exist in each frequency range:
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• First interval: extends from the lower limit of the frequency range given to the first eigenfrequency in the range.
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• Intermediate intervals: extend from eigenfrequency to eigenfrequency.
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• Last interval: extends from the highest eigenfrequency in the range to the upper limit of the frequency range.
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For each of these intervals the frequencies at which results are calculated are determined using the userdefined number of points (which includes the bounding frequencies for the interval) and the optional bias function (which is discussed below and allows the sampling points on the frequency scale to be spaced closer together at eigenfrequencies in the frequency range). Thus, detailed definition of the response close to resonance frequencies is allowed. Figure 6.3.9–1 illustrates the division of the frequency range for 5 calculation points and a bias parameter equal to 1.
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["frequency points"] --> B["lower end of the range"]
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A --> C["mode n"]
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A --> D["mode n + 1"]
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A --> E["mode n + 2"]
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A --> F["upper end of the range"]
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```
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</details>
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Figure 6.3.9–1 Division of range for the eigenfrequency type of interval and 5 calculation points.
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<!-- source-page: 268 -->
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Input File Usage: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION, INTERVAL=EIGENFREQUENCY
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace: Use eigenfrequencies to subdivide each frequency range
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Specifying the frequency ranges by the frequency spread
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If the spread type of frequency interval is selected, intervals exist around each eigenfrequency in the frequency range. For each of the intervals the equally spaced frequencies at which results are calculated are determined using the user-defined number of points (which includes the bounding frequencies for the interval). The minimum number of frequency points is 3. If the user-defined value is less than 3 (or omitted), the default value of 3 points is assumed. Figure 6.3.9–2 illustrates the division of the frequency range for 5 calculation points.
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The bias parameter is not supported with the spread type of frequency interval.
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<details>
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<summary>text_image</summary>
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Frequency points
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(1 - spread) · fₙ
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(1 + spread) · fₙ
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Frequency points
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(1 - spread) · fₙ₊₁
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(1 + spread) · fₙ₊₁
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fₙ₊₁
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</details>
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Figure 6.3.9–2 Division of range for the spread type of interval and 5 calculation points. $f _ { n }$ and $f _ { n + 1 }$ are eigenfrequencies of the system.
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Input File Usage: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION, INTERVAL=SPREAD lwr\_freq, upr\_freq, numpts, bias\_param, freq\_scale\_factor, spread
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Abaqus/CAE Usage: You cannot specify frequency ranges by frequency spread in Abaqus/CAE.
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Specifying the frequency ranges directly
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If the alternative range type of frequency interval is chosen, there is only one interval in the specified frequency range spanning from the lower to the upper limit of the range. This interval is divided using the user-defined number of points and the optional bias function, which can be used to space the sampling frequency points closer to the range limits. For the range type of frequency interval, the peak responses
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<!-- source-page: 269 -->
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around the system’s eigenfrequencies may be missed since the sampling frequencies at which output will be reported will not be biased toward the eigenfrequencies.
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Input File Usage: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION, INTERVAL=RANGE
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace: toggle off Use eigenfrequencies to subdivide each frequency range
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# Selecting the frequency spacing
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Two types of frequency spacing are permitted for a subspace-based steady-state dynamic step. For the logarithmic frequency spacing (the default), the specified frequency ranges of interest are divided using a logarithmic scale. Alternatively, a linear frequency spacing can be used if a linear scale is desired.
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Input File Usage: Use the following option to specify logarithmic frequency spacing: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION, FREQUENCY SCALE=LOGARITHMIC (default) Use the following option to specify linear frequency spacing: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION, FREQUENCY SCALE=LINEAR
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace: Scale: Logarithmic or Linear
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# Requesting multiple frequency ranges
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You can request multiple frequency ranges for a subspace-based steady-state dynamic step. When both frequency ranges and additional single frequency points are requested, the frequency ranges must be specified first.
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Input File Usage: Repeat the data lines as often as necessary to request multiple frequency ranges or multiple single frequency points: \*STEADY STATE DYNAMICS, SUBSPACE PROJECTION lwr\_freq1, upr\_freq1, numpts1, bias\_param1, freq\_scale\_factor1 lwr\_freq2, upr\_freq2, numpts2, bias\_param2, freq\_scale\_factor2 single\_freq1 single\_freq2
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Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Subspace: Data: enter data in table, and add rows as necessary
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# The bias parameter
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The bias parameter can be used to provide closer spacing of the results points either toward the middle or toward the ends of each frequency interval. Figure 6.3.9–3 shows a few examples of the effect of the bias parameter on the frequency spacing.
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<!-- source-page: 270 -->
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["frequency points"] --> B["f̂₁"]
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A --> C["f̂₂"]
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B --> D["1"]
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C --> E["2"]
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D --> F["3"]
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E --> G["5"]
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style A fill:#f9f,stroke:#333
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style B fill:#ccf,stroke:#333
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style C fill:#ccf,stroke:#333
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style D fill:#cfc,stroke:#333
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style E fill:#cfc,stroke:#333
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style F fill:#fcc,stroke:#333
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style G fill:#fcc,stroke:#333
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```
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</details>
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Figure 6.3.9–3 Effect of the bias parameter on the frequency spacing for a number of points .
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The bias formula used in subspace-based steady-state dynamics is
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$$
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\hat {f} _ {k} = \frac {1}{2} (\hat {f} _ {1} + \hat {f} _ {2}) + \frac {1}{2} (\hat {f} _ {2} - \hat {f} _ {1}) | y | ^ {1 / p} \mathrm{sign} (y),
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$$
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where
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y = -1 + 2(k - 1)/(n - 1);
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n is the number of frequency points at which results are to be given within a frequency interval (discussed above);
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k is one such frequency point (k = 1, 2, ..., n); $\hat{f}_{1}$ is the lower limit of the frequency interval; $\hat{f}_{2}$ is the upper limit of the frequency interval; $\hat{f}_{k}$ is the frequency at which the kth results are given;
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p is the bias parameter value; and $\hat{f}$ is the frequency or the logarithm of the frequency, depending on the value chosen for the frequency scale.
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A bias parameter, p, that is greater than 1.0 provides closer spacing of the results points toward the ends of the frequency interval, while values of p that are less than 1.0 provide closer spacing toward the middle of the frequency interval. The default bias parameter is 3.0 for an eigenfrequency interval and 1.0 for a range frequency interval.
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