김경종
277 lines
17 KiB
Markdown
277 lines
17 KiB
Markdown
<!-- source-page: 51 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Time | a |
|
||
|------|------|
|
||
| 0.0 | 0.0 |
|
||
| 0.1 | 1.0 |
|
||
</details>
|
||
|
||
$$
|
||
t _ {0} = 0. 0 \quad A _ {0} = 0. 0 \quad t _ {1} = 0. 1 \quad A _ {1} = 1. 0
|
||
$$
|
||
|
||
$$
|
||
a = A _ {0} \text { for } t \leq t _ {0}
|
||
$$
|
||
|
||
$$
|
||
= A _ {0} + \left(A _ {1} - A _ {0}\right) \xi^ {3} (1 0 - 1 5 \xi + 6 \xi^ {2}) \text {for} t _ {0} < t < t _ {1}
|
||
$$
|
||
|
||
$$
|
||
= A _ {1} \text { for } t \geq t _ {1}
|
||
$$
|
||
|
||
$$
|
||
\text { where } \xi = \frac {t - t _ {0}}{t _ {1} - t _ {0}}
|
||
$$
|
||
|
||
Figure 34.1.2–5 Smooth step amplitude definition example with two data points.
|
||
|
||
The preferred interface for incident wave loading due to an underwater explosion specifies bubble dynamics using the UNDEX charge property definition (see “Defining bubble loading for spherical incident wave loading” in “Acoustic and shock loads,” Section 34.4.6). The alternative interface for incident wave loading uses the bubble definition described in this section to define bubble load amplitude curves.
|
||
|
||
An example of the bubble amplitude definition with the following input data is shown in Figure 34.1.2–7.
|
||
|
||
$$
|
||
K = 5. 2 1 \times 1 0 ^ {7}, \quad k = 9. 0 \times 1 0 ^ {- 5}, \quad A = 0. 1 8, \quad B = 0. 1 8 5,
|
||
$$
|
||
|
||
$$
|
||
K _ {c} = 8. 3 9 6 \times 1 0 ^ {8}, \quad \gamma = 1. 2 7, \quad \rho_ {c} = 1. 5 \times 1 0 ^ {3}, \quad m _ {c} = 2 2 6. 8,
|
||
$$
|
||
|
||
$$
|
||
d _ {I} = 1 3 7. 1 6, \quad \rho_ {f} = 1. 0 \times 1 0 ^ {3}, \quad c _ {f} = 1. 5 \times 1 0 ^ {3}, \quad \mathbf {n} _ {X} = 0. 0,
|
||
$$
|
||
|
||
$$
|
||
\mathbf {n} _ {Y} = 0. 0, \qquad \mathbf {n} _ {Z} = 1. 0, \qquad g = 9. 8, \qquad p _ {a t m} = 9. 8 \times 1 0 ^ {4},
|
||
$$
|
||
|
||
$$
|
||
T _ {f i n a l} = 0. 5 5 5.
|
||
$$
|
||
|
||
<!-- source-page: 52 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Time Point | Value |
|
||
| ---------- | ----- |
|
||
| (t₀, A₀) | (t₀, A₀) |
|
||
| (t₁, A₁) | (t₁, A₁) |
|
||
| (t₂, A₂) | (t₂, A₂) |
|
||
| (t₃, A₃) | (t₃, A₃) |
|
||
| (t₄, A₄) | (t₄, A₄) |
|
||
| (t₅, A₅) | (t₅, A₅) |
|
||
| (t₆, A₆) | (t₆, A₆) |
|
||
</details>
|
||
|
||
$$
|
||
t _ {0} = 0. 0 \qquad A _ {0} = 0. 1 \qquad t _ {1} = 0. 1 \qquad A _ {1} = 0. 1 \qquad t _ {2} = 0. 2 \qquad A _ {2} = 0. 3 \qquad t _ {3} = 0. 3 \qquad A _ {3} = 0. 5
|
||
$$
|
||
|
||
$$
|
||
t _ {4} = 0. 4 \qquad A _ {4} = 0. 5 \qquad t _ {5} = 0. 5 \qquad A _ {5} = 0. 2 \qquad t _ {6} = 0. 8 \qquad A _ {6} = 0. 2
|
||
$$
|
||
|
||
$$
|
||
a = A _ {0} \text { for } t \leq t _ {0}
|
||
$$
|
||
|
||
$$
|
||
= A _ {6} \text { for } t \geq t _ {6}
|
||
$$
|
||
|
||
Amplitude, a, between any two consecutive data points
|
||
|
||
$$
|
||
\left(t _ {i}, A _ {i}\right) \text { and } \left(t _ {i + 1}, A _ {i + 1}\right) \text { is }
|
||
$$
|
||
|
||
$$
|
||
a = A _ {i} + \left(A _ {i + 1} - A _ {i}\right) \xi^ {3} (1 0 - 1 5 \xi + 6 \xi^ {2})
|
||
$$
|
||
|
||
$$
|
||
\text { where } \xi = \frac {t - t _ {i}}{t _ {i + 1} - t _ {i}}
|
||
$$
|
||
|
||
Figure 34.1.2–6 Smooth step amplitude definition example with multiple data points.
|
||
|
||
Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=BUBBLE
|
||
|
||
Abaqus/CAE Usage: Bubble amplitudes are not supported in Abaqus/CAE. However, bubble loading for an underwater explosion is supported in the Interaction module using the UNDEX charge property definition.
|
||
|
||
<!-- source-page: 53 -->
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Time | Radius of bubble |
|
||
|-------|------------------|
|
||
| 0.0 | 0.0 |
|
||
| 0.1 | 3.8 |
|
||
| 0.2 | 0.7 |
|
||
| 0.3 | 2.8 |
|
||
| 0.4 | 1.0 |
|
||
| 0.5 | 2.5 |
|
||
| 0.6 | 1.3 |
|
||
| 0.7 | 1.9 |
|
||
</details>
|
||
|
||
(a)
|
||
|
||
|
||
|
||
<details>
|
||
<summary>line</summary>
|
||
|
||
| Time | Depth of bubble |
|
||
|------|-----------------|
|
||
| 0.0 | -138.0 |
|
||
| 0.1 | -137.5 |
|
||
| 0.2 | -136.0 |
|
||
| 0.3 | -134.0 |
|
||
| 0.4 | -132.0 |
|
||
| 0.5 | -130.0 |
|
||
| 0.6 | -128.0 |
|
||
</details>
|
||
|
||
(b)
|
||
Figure 34.1.2–7 Bubble amplitude definition example: (a) radius of bubble and (b) depth of bubble center under fluid surface.
|
||
|
||
# Defining an amplitude via a user subroutine
|
||
|
||
Choose the user definition method to define the amplitude curve via coding in user subroutine UAMP (Abaqus/Standard) or VUAMP (Abaqus/Explicit). You define the value of the amplitude function in time and, optionally, the values of the derivatives and integrals for the function sought to be implemented as outlined in “UAMP,” Section 1.1.19 of the Abaqus User Subroutines Reference Guide, and “VUAMP,” Section 1.2.9 of the Abaqus User Subroutines Reference Guide.
|
||
|
||
You can use an arbitrary number of properties to calculate the amplitude, and you can use an arbitrary number of state variables that can be updated independently for each amplitude definition.
|
||
|
||
In Abaqus/Standard user-defined amplitudes are not supported for complex eigenvalue extraction, linear dynamic procedures, and steady-state dynamic analysis with the response computed directly in terms of the physical degrees of freedom.
|
||
|
||
Moreover, solution-dependent sensors can be used to define the user-customized amplitude. The sensors can be identified via their name, and two utilities allow for the extraction of the current sensor value inside the user subroutine (see “Obtaining sensor information,” Section 2.1.16 of the Abaqus User Subroutines Reference Guide). Simple control/logical models can be implemented using this feature as illustrated in “Crank mechanism,” Section 4.1.2 of the Abaqus Example Problems Guide.
|
||
|
||
Input File Usage: $\scriptstyle * \mathrm { A M P L I T U D E , ~ N A M E = } n a m e , \ \mathrm { D E F I N I T I O N = U S E R } ,$ ${ \mathrm { P R O P E R T I E S } } { = } m { \mathrm { , ~ } } { \mathrm { V A R I A B L E S } } { = } n$
|
||
|
||
<!-- source-page: 54 -->
|
||
|
||
Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: User: Number of variables: n
|
||
|
||
User-defined amplitude properties are not supported in Abaqus/CAE.
|
||
|
||
# Defining an actuator amplitude via co-simulation
|
||
|
||
The current value of an actuator amplitude can be imported at any given time from a co-simulation with a logical modeling program (see “Co-simulation: overview,” Section 17.1.1). The name specified on the actuator amplitude definition is used as the actuator name for co-simulation purposes. Therefore, at a given time each actuator is associated with one real number—the current value of the amplitude. As with any amplitude definition, the user-specified name can be used in conjunction with any Abaqus feature that can reference an amplitude.
|
||
|
||
Input File Usage: \*AMPLITUDE, NAME=name, DEFINITION=ACTUATOR
|
||
|
||
Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: Actuator
|
||
|
||
# Using an amplitude definition with boundary conditions
|
||
|
||
When an amplitude curve is used to prescribe a variable of the model as a boundary condition (by referring to the amplitude from the boundary condition definition), the first and second time derivatives of the variable may also be needed. For example, the time history of a displacement can be defined for a direct integration dynamic analysis step by an amplitude variation; in this case Abaqus must compute the corresponding velocity and acceleration.
|
||
|
||
When the displacement time history is defined by a piecewise linear amplitude variation (tabular or equally spaced amplitude definition), the corresponding velocity is piecewise constant and the acceleration may be infinite at the end of each time interval given in the amplitude definition table, as shown in Figure 34.1.2–8(a). This behavior is unreasonable. (In Abaqus/Explicit time derivatives of amplitude curves are typically based on finite differences, such as , so there is some $\frac { [ A ( t _ { i + 1 } ^ { \bullet } ) - A ^ { \bullet } ( t _ { i } ) ] } { \Delta t }$ △t inherent smoothing associated with the time discretization.)
|
||
|
||
You can modify the piecewise linear displacement variation into a combination of piecewise linear and piecewise quadratic variations through smoothing. Smoothing ensures that the velocity varies continuously during the time period of the amplitude definition and that the acceleration no longer has singularity points, as illustrated in Figure 34.1.2–8(b).
|
||
|
||
When the velocity time history is defined by a piecewise linear amplitude variation, the corresponding acceleration is piecewise constant. Smoothing can be used to modify the piecewise linear velocity variation into a combination of piecewise linear and piecewise quadratic variations. Smoothing ensures that the acceleration varies continuously during the time period of the amplitude definition.
|
||
|
||
You specify t, the fraction of the time interval before and after each time point during which the piecewise linear time variation is to be replaced by a smooth quadratic time variation. The default in Abaqus/Standard is t=0.25; the default in Abaqus/Explicit is t=0.0. The allowable range is $0 . 0 < t \leq 0 . 5 .$ . A value of 0.05 is suggested for amplitude definitions that contain large time intervals to avoid severe deviation from the specified definition.
|
||
|
||
<!-- source-page: 55 -->
|
||
|
||
|
||
Figure 34.1.2–8 Piecewise linear displacement definitions.
|
||
|
||
In Abaqus/Explicit if a displacement jump is specified using an amplitude curve (i.e., the beginning displacement defined using the amplitude function does not correspond to the displacement at that time), this displacement jump will be ignored. Displacement boundary conditions are enforced in
|
||
|
||
<!-- source-page: 56 -->
|
||
|
||
Abaqus/Explicit in an incremental manner using the slope of the amplitude curve. To avoid the “noisy” solution that may result in Abaqus/Explicit when smoothing is not used, it is better to specify the velocity history of a node rather than the displacement history (see “Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1).
|
||
|
||
When an amplitude definition is used with prescribed conditions that do not require the evaluation of time derivatives (for example, concentrated loads, distributed loads, temperature fields, etc., or a static analysis), the use of smoothing is ignored.
|
||
|
||
When the displacement time history is defined using a smooth-step amplitude curve, the velocity and acceleration will be zero at every data point specified, although the average velocity and acceleration may well be nonzero. Hence, this amplitude definition should be used only to define a (smooth) step function.
|
||
|
||
<table><tr><td>Input File Usage:</td><td>Use either of the following options:*AMPLITUDE, NAME=name, DEFINITION=TABULAR, SMOOTH=t*AMPLITUDE, NAME=name, DEFINITION=EQUALLY SPACED, SMOOTH=t</td></tr></table>
|
||
|
||
<table><tr><td>Abaqus/CAE Usage:</td><td>Load or Interaction module: Create Amplitude: choose Tabular or Equally spaced: Smoothing: Specify: t</td></tr></table>
|
||
|
||
# Using an amplitude definition with secondary base motion in modal dynamics
|
||
|
||
When an amplitude curve is used to prescribe a variable of the model as a secondary base motion in a modal dynamics procedure (by referring to the amplitude from the base motion definition during a modal dynamic procedure), the first or second time derivatives of the variable may also be needed. For example, the time history of a displacement can be defined for secondary base motion in a modal dynamics procedure. In this case Abaqus must compute the corresponding acceleration.
|
||
|
||
The modal dynamics procedure uses an exact solution for the response to a piecewise linear force. Accordingly, secondary base motion definitions are applied as piecewise linear acceleration histories. When displacement-type or velocity-type base motions are used to define displacement or velocity time histories and an amplitude variation using the tabular, equally spaced, periodic, modulated, or exponential decay definitions is used, an algorithmic acceleration is computed based on the tabular data (the amplitude data evaluated at the time values used in the modal dynamics procedure). At the end of any time increment where the amplitude curve is linear over that increment, linear over the previous increment, and the slopes of the amplitude variations over the two increments are equal, this algorithmic acceleration reproduces the exact displacement and velocity for displacement time histories or the exact velocity for velocity time histories.
|
||
|
||
When the displacement time history is defined using a smooth-step amplitude curve, the velocity and acceleration will be zero at every data point specified, although the average velocity and acceleration may well be nonzero. Hence, this amplitude definition should be used only to define a (smooth) step function.
|
||
|
||
<!-- source-page: 57 -->
|
||
|
||
# Defining multiple amplitude curves
|
||
|
||
You can define any number of amplitude curves and refer to them from any load, boundary condition, or predefined field definition. For example, one amplitude curve can be used to specify the velocity of a set of nodes, while another amplitude curve can be used to specify the magnitude of a pressure load on the body. If the velocity and the pressure both follow the same time history, however, they can both refer to the same amplitude curve. There is one exception in Abaqus/Standard: only one solution-dependent amplitude (used for superplastic forming) can be active during each step.
|
||
|
||
# Scaling and shifting amplitude curves
|
||
|
||
You can scale and shift both time and magnitude when defining an amplitude. This can be helpful for example when your amplitude data need to be converted to a different unit system or when you reuse existing amplitude data to define similar amplitude curves. If both scaling and shifting are applied at the same time, the amplitude values are first scaled and then shifted. The amplitude shifting and scaling can be applied to all amplitude definition types except for solution dependent, bubble, and user; for the actuator amplitude definition type, only scaling and shifting of the amplitude magnitude is supported.
|
||
|
||
Input File Usage: \*AMPLITUDE, NAME=name, SHIFTX=shiftx\_value, SHIFTY=shifty\_value, $\mathrm { S C A L E X } { = } s c a l e x \_ { \nu } a l u e , \mathrm { S C A L E Y } { = } s c a l e y \_ { \nu } a l u e$
|
||
|
||
Abaqus/CAE Usage: The scaling and shifting of amplitude curves is not supported in Abaqus/CAE.
|
||
|
||
# Reading the data from an alternate file
|
||
|
||
The data for an amplitude curve can be contained in a separate file.
|
||
|
||
Input File Usage: \*AMPLITUDE, NAME=name, INPUT=file\_name
|
||
|
||
If the INPUT parameter is omitted, it is assumed that the data lines follow the keyword line.
|
||
|
||
Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: any type: click mouse button 3 while holding the cursor over the data table, and select Read from File
|
||
|
||
# Baseline correction in Abaqus/Standard
|
||
|
||
When an amplitude definition is used to define an acceleration history in the time domain (a seismic record of an earthquake, for example), the integration of the acceleration record through time may result in a relatively large displacement at the end of the event. This behavior typically occurs because of instrumentation errors or a sampling frequency that is not sufficient to capture the actual acceleration history. In Abaqus/Standard it is possible to compensate for it by using “baseline correction.”
|
||
|
||
The baseline correction method allows an acceleration history to be modified to minimize the overall drift of the displacement obtained from the time integration of the given acceleration. It is relevant only with tabular or equally spaced amplitude definitions.
|
||
|
||
<!-- source-page: 58 -->
|
||
|
||
Baseline correction can be defined only when the amplitude is referenced as an acceleration boundary condition during a direct-integration dynamic analysis or as an acceleration base motion in modal dynamics.
|
||
|
||
Input File Usage: Use both of the following options to include baseline correction:
|
||
|
||
\*AMPLITUDE, DEFINITION=TABULAR or EQUALLY SPACED \*BASELINE CORRECTION
|
||
|
||
The \*BASELINE CORRECTION option must appear immediately following the data lines of the \*AMPLITUDE option.
|
||
|
||
Abaqus/CAE Usage: Load or Interaction module: Create Amplitude: choose Tabular or Equally spaced: Baseline Correction
|
||
|
||
# Effects of baseline correction
|
||
|
||
The acceleration is modified by adding a quadratic variation of acceleration in time to the acceleration definition. The quadratic variation is chosen to minimize the mean squared velocity during each correction interval. Separate quadratic variations can be added for different correction intervals within the amplitude definition by defining the correction intervals. Alternatively, the entire amplitude history can be used as a single correction interval.
|
||
|
||
The use of more correction intervals provides tighter control over any “drift” in the displacement at the expense of more modification of the given acceleration trace. In either case, the modification begins with the start of the amplitude variation and with the assumption that the initial velocity at that time is zero.
|
||
|
||
The baseline correction technique is described in detail in “Baseline correction of accelerograms,” Section 6.1.2 of the Abaqus Theory Guide.
|
||
|
||
<!-- source-page: 59 -->
|
||
|
||
# 34.2 Initial conditions
|
||
|
||
• “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.2.1
|
||
• “Initial conditions in Abaqus/CFD,” Section 34.2.2
|
||
|
||
<!-- source-page: 60 -->
|