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6.3.4 DIRECT-SOLUTION STEADY-STATE DYNAMIC ANALYSIS
Products: Abaqus/Standard Abaqus/CAE
References
• “Dynamic analysis procedures: overview,” Section 6.3.1
• “Mode-based steady-state dynamic analysis,” Section 6.3.8
• “Subspace-based steady-state dynamic analysis,” Section 6.3.9
• “Defining an analysis,” Section 6.1.2
• “General and linear perturbation procedures,” Section 6.1.3
• *STEADY STATE DYNAMICS
• “Configuring a direct-solution steady-state dynamic procedure” in “Configuring linear perturbation analysis procedures,” Section 14.11.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
• “Creating and modifying prescribed conditions,” Section 16.4 of the Abaqus/CAE User’s Guide
Overview
A direct-solution steady-state dynamic analysis:
• is used to calculate the steady-state dynamic linearized response of a system to harmonic excitation;
• is a linear perturbation procedure;
• calculates the response directly in terms of the physical degrees of freedom of the model;
• is an alternative to mode-based steady-state dynamic analysis, in which the response of the system is calculated on the basis of the eigenmodes;
• is more expensive computationally than mode-based or subspace-based steady-state dynamics;
• is more accurate than mode-based or subspace-based steady-state dynamics, in particular if significant frequency-dependent material damping or viscoelastic material behavior is present in the structure; and
• is able to bias the excitation frequencies toward the approximate values that generate a response peak.
Introduction
Steady-state dynamic analysis provides the steady-state amplitude and phase of the response of a system due 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 direct-solution steady-state dynamic procedure conducts this frequency sweep. In a direct-solution steady-state analysis the steady-state harmonic response is calculated directly in terms of the physical degrees of freedom of the model using the mass, damping, and stiffness matrices of the system.
When defining a direct-solution 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. Frequencies are given in cycles/time.
Those 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 (described below).
The direct-solution steady-state analysis procedure can be used in the following cases for which the eigenvalues cannot be extracted (and, thus, the mode-based steady-state dynamics procedures are not applicable):
• for nonsymmetric stiffness;
• when any form of damping other than modal damping must be included; and
• when viscoelastic material properties must be taken into account.
While the response in this procedure is linear, the prior response can be nonlinear. Initial stress effects (stress stiffening) as well as load stiffness effects will be included in the steady-state dynamics response if nonlinear geometric effects (“General and linear perturbation procedures,” Section 6.1.3) were included in any general analysis step prior to the direct-solution steady-state dynamic procedure.
Input File Usage: *STEADY STATE DYNAMICS, DIRECT
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Direct
Ignoring damping
If damping terms can be ignored, you can specify that a real, rather than a complex, system matrix be factored, which can significantly reduce computational time. Damping is discussed below.
Input File Usage: *STEADY STATE DYNAMICS, DIRECT, REAL ONLY
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Direct: Compute real response only
Selecting the type of frequency interval for which output is requested
Three types of frequency intervals are permitted for output from a direct-solution steady-state dynamic step. If an eigenvalue extraction step precedes the direct-solution steady-state dynamic step, you can select either the range or the eigenfrequency type of frequency interval; otherwise, only the range type can be used.
Dividing the specified frequency range using the user-defined number of points and the optional bias function
For the range type of frequency interval (the default), the specified frequency range of interest is divided using the user-defined number of points and the optional bias function.
Input File Usage: *STEADY STATE DYNAMICS, DIRECT, INTERVAL=RANGE
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Direct: toggle off Use eigenfrequencies to subdivide each frequency range
Specifying the frequency ranges by using the system’s eigenfrequencies
If the direct-solution steady-state dynamic analysis is preceded by an eigenfrequency extraction step, you can select the eigenfrequency type of frequency interval. The following intervals then exist in each frequency range:
• First interval: extends from the lower limit of the frequency range given to the first eigenfrequency in the range.
• Intermediate intervals: extend from eigenfrequency to eigenfrequency.
• Last interval: extends from the highest eigenfrequency in the range to the upper limit of the frequency range.
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. Figure 6.3.4–1 illustrates the division of the frequency range for 5 calculation points and a bias parameter equal to 1.
Input File Usage: *STEADY STATE DYNAMICS, DIRECT, INTERVAL=EIGENFREQUENCY
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Direct: Use eigenfrequencies to subdivide each frequency range
flowchart
graph TD
A["frequency points"] --> B["lower end of the range"]
A --> C["mode n"]
A --> D["mode n +1"]
A --> E["mode n +2"]
A --> F["upper end of the range"]
Figure 6.3.4–1 Division of range for the eigenfrequency type of interval and 5 calculation points.
Specifying the frequency ranges by the frequency spread
If the direct-solution steady-state dynamic analysis is preceded by an eigenfrequency extraction step, you can select the spread type of frequency interval. In this case 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 6.3.4–2 illustrates the division of the frequency range for 5 calculation points.
The bias parameter is not supported with the spread type of frequency interval.
text_image
Frequency points (1 - spread) · fₙ (1 + spread) · fₙ Frequency points (1 - spread) · fₙ₊₁ (1 + spread) · fₙ₊₁ fₙ₊₁
Figure 6.3.4–2 Division of range for the spread type of interval and 5 calculation points. f _ { n } and f _ { n + 1 } are eigenfrequencies of the system.
Input File Usage: *STEADY STATE DYNAMICS, DIRECT, INTERVAL=SPREAD lwr_freq, upr_freq, numpts, bias_param, freq_scale_factor, spread
Abaqus/CAE Usage: You cannot specify frequency ranges by frequency spread in Abaqus/CAE.
Selecting the frequency spacing
Two types of frequency spacing are permitted for a direct-solution 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.
Input File Usage: Use either of the following options:
*STEADY STATE DYNAMICS, DIRECT, FREQUENCY SCALE=LOGARITHMIC
*STEADY STATE DYNAMICS, DIRECT, FREQUENCY SCALE=LINEAR
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Direct: Scale: Logarithmic or Linear
Requesting multiple frequency ranges
You can request multiple frequency ranges or multiple single frequency points for a direct-solution steady-state dynamic step.
| Input File Usage: | *STEADY STATE DYNAMICS, DIRECTlwr_freq1, upr_freq1, numpts1, bias_param1, freq_scale_factor1lwr_freq2, upr_freq2, numpts2, bias_param2, freq_scale_factor2...single_freq1single_freq2...Repeat the data lines as often as necessary. |
| Abaqus/CAE Usage: | Step module: Create Step: Linear perturbation: Steady-state dynamics,Direct: Data: enter data in table, and add rows as necessary |
The bias parameter
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 6.3.4–3 shows a few examples of the effect of the bias parameter on the frequency spacing.
flowchart
graph TD
A["frequency points"] --> B["f̂₁"]
A --> C["f̂₂"]
B --> D["1"]
C --> E["2"]
D --> F["3"]
E --> G["5"]
style A fill:#f9f,stroke:#333
style B fill:#ccf,stroke:#333
style C fill:#ccf,stroke:#333
style D fill:#cfc,stroke:#333
style E fill:#cfc,stroke:#333
style F fill:#fcc,stroke:#333
style G fill:#fcc,stroke:#333
Figure 6.3.4–3 Effect of the bias parameter on the frequency spacing for a number of points .
The bias formula used in direct-solution steady-state dynamics is
\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),
where
y = -1 + 2(k - 1)/(n - 1); n is the number of frequency points at which results are to be given; k is one such frequency point ( k = 1, 2, \ldots, n ); \hat{f}_1 is the lower limit of the frequency range; \hat{f}_2 is the upper limit of the range; \hat{f}_k is the frequency at which the k th results are given; 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.
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 1.0 for a range frequency interval and 3.0 for an eigenfrequency interval.
The frequency scale factor
The frequency scale factor can be used to scale frequency points. All the frequency points, except the lower and upper limit of the frequency range, are multiplied by this factor. This scale factor can be used only when the frequency interval is specified by using the system’s eigenfrequencies (see “Specifying the frequency ranges by using the system’s eigenfrequencies,” above).
Damping
If damping is absent, the response of a structure will be unbounded if the forcing frequency is equal to an eigenfrequency of the structure. To get quantitatively accurate results, especially near natural frequencies, accurate specification of damping properties is essential. The various damping options available are discussed in “Material damping,” Section 26.1.1.
In direct-solution steady-state dynamics damping can be created by the following:
• dashpots (see “Dashpots,” Section 32.2.1),
• “Rayleigh” damping associated with materials and elements (see “Material damping,” Section 26.1.1),
• damping associated with acoustic elements (see “Acoustic medium,” Section 26.3.1; “Infinite elements,” Section 28.3.1; and “Acoustic and shock loads,” Section 34.4.6),
• structural damping (see “Damping in dynamic analysis” in “Dynamic analysis procedures: overview,” Section 6.3.1), and
• viscoelasticity included in the material definitions (see “Frequency domain viscoelasticity,” Section 22.7.2).
When a real-only system matrix is factored, all forms of damping are ignored, including quiet boundaries on infinite elements and nonreflecting boundaries on acoustic elements.
Contact conditions with sliding friction
Abaqus/Standard automatically detects the contact nodes that are slipping due to velocity differences imposed by the motion of the reference frame or the transport velocity in prior steps. At those nodes the tangential degrees of freedom are not constrained and the effect of friction results in an unsymmetric contribution to the stiffness matrix. At other contact nodes the tangential degrees of freedom are constrained.
Friction at contact nodes at which a velocity differential is imposed can give rise to damping terms. There are two kinds of friction-induced damping effects. The first effect is caused by the friction forces stabilizing the vibrations in the direction perpendicular to the slip direction. This effect exists only in three-dimensional analysis. The second effect is caused by a velocity-dependent friction coefficient. If the friction coefficient decreases with velocity (which is usually the case), the effect is destabilizing and is also known as “negative damping.” For more details, see “Coulomb friction,” Section 5.2.3 of the Abaqus Theory Guide. Direct-solution steady-state dynamics analysis allows you to include these friction-induced contributions to the damping matrix.
Input File Usage: *STEADY STATE DYNAMICS, DIRECT, FRICTION DAMPING=YES
Abaqus/CAE Usage: Step module: Create Step: Linear perturbation: Steady-state dynamics, Direct: Include friction-induced damping effects
Initial conditions
The base state is the current state of the model at the end of the last general analysis step prior to the steady-state dynamic step. If the first step of an analysis is a perturbation step, the base state is determined from the initial conditions (“Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1). Initial condition definitions that directly define solution variables, such as velocity, cannot be used in a steady-state dynamic analysis.
Boundary conditions
In a steady-state dynamic analysis the real and imaginary parts of any degree of freedom are either restrained or unrestrained simultaneously; it is physically impossible to have one part restrained and the other part unrestrained. Abaqus/Standard will automatically restrain both the real and imaginary parts of a degree of freedom even if only one part is prescribed specifically. The unspecified part will be assumed to have a perturbation magnitude of zero.
Boundary conditions can be applied to any of the displacement or rotation degrees of freedom (1–6) in a direct-solution steady-state analysis. See “Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1. These boundary conditions will vary sinusoidally with time. You specify the real (in-phase) part of a boundary condition and the imaginary (out-of-phase) part of a boundary condition separately.
Input File Usage: Use either of the following options to define the real (in-phase) part of the boundary condition:
*BOUNDARY
*BOUNDARY, REAL
Use the following option to define the imaginary (out-of-phase) part of the boundary condition:
*BOUNDARY, IMAGINARY
Abaqus/CAE Usage: Load module: boundary condition editor: real (in-phase) part + imaginary (out-of-phase) part i
Frequency-dependent boundary conditions
An amplitude definition can be used to specify the amplitude of a boundary condition as a function of frequency (“Amplitude curves,” Section 34.1.2).
Input File Usage: Use both of the following options:
*AMPLITUDE, NAME=name
*BOUNDARY, REAL or IMAGINARY, AMPLITUDE=name
Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Name: name
Load module: boundary condition editor: real (in-phase) part + imaginary (out-of-phase) part i: Amplitude: name
Loads
The following loads can be prescribed in a steady-state dynamic analysis:
• Concentrated nodal forces can be applied to the displacement degrees of freedom (1–6); see “Concentrated loads,” Section 34.4.2.
• Distributed pressure forces or body forces can be applied; see “Distributed loads,” Section 34.4.3. The distributed load types available with particular elements are described in Part VI, “Elements.”
• Incident wave loads can be applied; see “Acoustic and shock loads,” Section 34.4.6.
These loads are assumed to vary sinusoidally with time over a user-specified range of frequencies. Loads are given in terms of their real and imaginary components.
Coriolis distributed loading adds an imaginary antisymmetric contribution to the overall system of equations. This contribution is currently accounted for in solid and truss elements only and is activated by using the unsymmetric matrix storage and solution scheme for the step (“Defining an analysis,” Section 6.1.2).
Incident wave loads can be used to model sound waves from distinct planar or spherical sources or from diffuse fields.
Fluid flux loading cannot be used in a steady-state dynamic analysis.
Input File Usage: Use any of the following options to define the real (in-phase) part of the load:
*CLOAD or *DLOAD
*CLOAD or *DLOAD, REAL
Use either of the following options to define the imaginary (out-of-phase) part of the load:
*CLOAD or *DLOAD, IMAGINARY