김경종
377 lines
16 KiB
Markdown
377 lines
16 KiB
Markdown
<!-- source-page: 941 -->
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<details>
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<summary>text_image</summary>
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-2d
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a
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-d
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d
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2d
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n = 2
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b
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y
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x
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</details>
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Figure 41.1.1–4 Two-dimensional periodic symmetry.
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The repeated images are bounded by lines parallel to line ab. The distance vector must be defined so that it points away from line ab and into the domain of the model. This type of periodic symmetry can be used only with two-dimensional cavities.
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Input File Usage: \*PERIODIC, TYPE=2D, NR=n
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Abaqus/CAE Usage: Interaction module: Create Interaction: Cavity radiation: Symmetry: Periodic: Number of periodic symmetries: n
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Periodic symmetry of three-dimensional cavities
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You can create a cavity that is composed of a series of similar images generated by repetition along a three-dimensional distance vector, as shown in Figure 41.1.1–5. The repeated images are bounded by planes that are parallel to plane abc. The distance vector must be defined so that it points away from plane abc and into the domain of the model. This type of periodic symmetry can be used only with three-dimensional cavities.
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<!-- source-page: 942 -->
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<details>
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<summary>text_image</summary>
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2d
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d
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-d
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-2d
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z
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y
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n = 2
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c
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a
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b
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</details>
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Figure 41.1.1–5 Three-dimensional periodic symmetry.
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Input File Usage: \*PERIODIC, TYPE=3D, NR=n
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Abaqus/CAE Usage: Interaction module: Create Interaction: Cavity radiation: Symmetry:
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Periodic: Number of periodic symmetries: n
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Periodic symmetry of axisymmetric cavities
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You can create a cavity that is composed of a series of similar images generated by repetition in the z-direction, as shown in Figure 41.1.1–6. The repeated images are bounded by lines of constant zcoordinate. The z-distance vector must be defined so that it points away from the z-constant periodic symmetry reference line and into the domain of the model. This type of periodic symmetry can be used only with axisymmetric cavities.
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Input File Usage: \*PERIODIC, TYPE=ZDIR, NR=n
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Abaqus/CAE Usage: Interaction module: Create Interaction: Cavity radiation: Symmetry:
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Periodic: Number of periodic symmetries: n
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# Cyclic symmetry
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You can define cavity symmetry by cyclic repetition of the user-defined cavity surface about a point or an axis. The cavity defined by cyclic repetition must cover 360°.
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You must define the number of cyclically similar images that compose the cavity, n. The angle of rotation about a point or axis used to create cyclically similar images is equal to 360°/n.
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You must identify the dimensionality of the cavity when you define cyclic symmetry.
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Cyclic symmetry of two-dimensional cavities
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You can define the cavity symmetry by rotating the cavity about a point, l, as shown in Figure 41.1.1–7. The cavity surface defined in the model must be bounded by the line lk and a line passing through l at an
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<!-- source-page: 943 -->
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<details>
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<summary>text_image</summary>
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z
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2d
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-n
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-2d
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r
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n = 2
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z = const periodic
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symm reference line
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</details>
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Figure 41.1.1–6 Axisymmetric periodic symmetry.
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<details>
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<summary>text_image</summary>
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n = 4
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l
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k
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</details>
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Figure 41.1.1–7 Cyclic symmetry about a point.
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angle, measured counterclockwise when looking into the plane of the model, of $3 6 0 \%$ to lk. This type of cyclic symmetry can be used only for two-dimensional cavities.
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<!-- source-page: 944 -->
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Input File Usage: \*CYCLIC, TYPE=POINT, NC=n
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Abaqus/CAE Usage: Interaction module: Create Interaction: Cavity radiation:
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Symmetry: Cyclic: toggle on Use cyclic symmetric,
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Total number of sectors: n
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Cyclic symmetry of three-dimensional cavities
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You can define the cavity symmetry by rotating the cavity about an axis, lm, as shown in Figure 41.1.1–8. The cavity surface defined in the model must be bounded by the plane lmk and a plane passing through the line lm at an angle, measured clockwise when looking from l to m, of 360°/n to lmk. Line lk must be normal to line lm. This type of cyclic symmetry can be used only for three-dimensional cavities.
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<details>
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<summary>text_image</summary>
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k
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m
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l
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n = 8
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</details>
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Figure 41.1.1–8 Cyclic symmetry about an axis.
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Input File Usage: \*CYCLIC, TYPE=AXIS, NC=n
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Abaqus/CAE Usage: Interaction module: Create Interaction: Cavity radiation: Symmetry:
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Cyclic: toggle on Use cyclic symmetric,
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Total number of sectors: n
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# Combining symmetries
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Reflection, periodic, and cyclic symmetries can be combined as shown in Table 41.1.1–1.
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Figure 41.1.1–9 through Figure 41.1.1–12 illustrate some possible symmetry combinations.
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<!-- source-page: 945 -->
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Table 41.1.1–1 Permissible number of symmetry definitions used in combination.
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<table><tr><td>Reflection</td><td>Periodic</td><td>Cyclic</td><td>2D</td><td>3D</td><td>Axi</td><td>Restrictions</td></tr><tr><td>1</td><td>0</td><td>0</td><td>•</td><td>•</td><td>•</td><td></td></tr><tr><td>2</td><td>0</td><td>0</td><td>•</td><td>•</td><td></td><td> $n_{1} \perp n_{2}$ </td></tr><tr><td>3</td><td>0</td><td>0</td><td></td><td>•</td><td></td><td> $n_{1} \perp n_{2} \perp n_{3} \perp n_{1}$ </td></tr><tr><td>0</td><td>1</td><td>0</td><td>•</td><td>•</td><td>•</td><td></td></tr><tr><td>0</td><td>2</td><td>0</td><td>•</td><td>•</td><td></td><td></td></tr><tr><td>0</td><td>3</td><td>0</td><td></td><td>•</td><td></td><td></td></tr><tr><td>1</td><td>1</td><td>0</td><td>•</td><td>•</td><td></td><td> $n \perp d$ </td></tr><tr><td>1</td><td>2</td><td>0</td><td></td><td>•</td><td></td><td> $d_{1} \perp n \perp d_{2}$ </td></tr><tr><td>2</td><td>1</td><td>0</td><td></td><td>•</td><td></td><td> $d \perp n_{1} \perp n_{2} \perp d$ </td></tr><tr><td>0</td><td>0</td><td>1</td><td>•</td><td>•</td><td></td><td></td></tr><tr><td>1</td><td>0</td><td>1</td><td></td><td>•</td><td></td><td> $n \parallel l - m$ </td></tr><tr><td>0</td><td>1</td><td>1</td><td></td><td>•</td><td></td><td> $d \parallel l - m$ </td></tr></table>
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, , , $\mathbf { n } _ { 3 }$ are normals to lines or planes of reflection symmetry.
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$\mathbf { d } , \mathbf { d } _ { 1 } , \mathbf { d } _ { 2 }$ are distance vectors used to define periodic symmetry.
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$1 - \mathbf { m }$ is the direction of the axis of cyclic symmetry in three-dimensional cases.
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<details>
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<summary>text_image</summary>
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a₂
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a₁
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n₁
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n₂
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</details>
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Figure 41.1.1–9 Combination of two reflection symmetries in two dimensions.
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<!-- source-page: 946 -->
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<details>
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<summary>text_image</summary>
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a₁
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d₂ (n=2)
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d₁
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(n=3)
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</details>
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Figure 41.1.1–10 Combination of two periodic symmetries in two dimensions.
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<details>
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<summary>text_image</summary>
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a₂
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</details>
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Figure 41.1.1–11 Combination of one reflection symmetry and one periodic symmetry in two dimensions.
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<!-- source-page: 947 -->
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<details>
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<summary>text_image</summary>
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10 d
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-10 d
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n = 4 (cyclic)
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n = 10 (periodic)
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</details>
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Figure 41.1.1–12 Combination of one cyclic symmetry and one periodic symmetry in three dimensions.
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# Prescribing motion during a cavity radiation analysis
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In many cavity radiation problems such as simulations of manufacturing sequences, radiation view factors change because surfaces are moved during the analysis. You can specify surface motions during heat transfer or coupled thermal-electrical analysis.
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The prescribed motions affect only the calculation of view factors (and, therefore, radiation fluxes) in heat transfer due to cavity radiation. They do not affect heat conduction, storage, or distributed flux contributions.
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You can define both the translational and rotational components of the motion within a step independently. For example, you can prescribe the translational motion of a node set according to a certain amplitude function and then prescribe the rotational motion of the node set according to a different amplitude function. In each step, each component of motion can be specified only once for any particular node.
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Motions can also be prescribed during steps in which the cavity radiation is turned off, as described below.
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<!-- source-page: 948 -->
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# Translational motion
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Translations, , are specified in terms of global x-, y-, and z-components unless a local coordinate system is defined at the nodes for which motion is specified; then translations are specified in terms of local x-, y-, and z-components (see “Transformed coordinate systems,” Section 2.1.5).
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Translational displacements are always specified as total values of translational motion. This treatment of translations is consistent with that used for displacement boundary conditions (“Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1) in stress/displacement analyses. The default is to apply translational motion.
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Translational velocities can also be specified. Translational velocities always refer to the current step; therefore, the rate of translational motion specified as a velocity is in effect only during the step for which it is defined. This behavior is different from velocity boundary conditions, where velocities stay in effect in subsequent steps if they are not redefined.
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Input File Usage: Use either of the following options to prescribe translational motion:
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\*MOTION, TRANSLATION, TYPE=DISPLACEMENT
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\*MOTION, TRANSLATION, TYPE=VELOCITY
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Abaqus/CAE Usage: Surface motion is not supported with cavity radiation in Abaqus/CAE.
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# Rotational motion
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Displacements due to a rigid body rotation, , can be defined by specifying the magnitude of the rotation and the rotation axis. In three dimensions the rotation axis is defined by specifying two points, and , on the axis of rotation. In two dimensions the rotation axis is assumed to be normal to the plane of the model and is defined by specifying one point, .
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The coordinates of the points defining the axis of rotation must be defined in the configuration at the beginning of the step for which rigid body rotation is being defined.
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Motion due to rigid body rotation during a step is specified as the amount of rotation that takes place during that step only. Therefore, the rigid body rotation specified during a step is local to that step; if no rigid body rotation is specified in the following step, no further rotation occurs.
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The treatment of rigid body rotations is different from that of translations: rigid body rotations are specified incrementally from step to step while translations are specified as total values.
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Input File Usage: Use either of the following options to prescribe rotational motion:
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\*MOTION, ROTATION, TYPE=DISPLACEMENT
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\*MOTION, ROTATION, TYPE=VELOCITY
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Abaqus/CAE Usage: Surface motion is not supported with cavity radiation in Abaqus/CAE.
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# Prescribing large rotational motions
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Prescribed rotational motions of more than radians or complex sequences of rotations about different directions in three-dimensional models are most simply defined by specifying rotational velocities, which allows the definition to be given in terms of the angular velocity instead of the total rotation. Abaqus/Standard calculates the increment of rotation as the average of the angular velocities
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<!-- source-page: 949 -->
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at the beginning and end of each increment multiplied by the time increment. (See “Conventions,” Section 1.2.2.)
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# Example
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For example, if a rotation of about the z-axis is required, with no rotation about the x- and y-axes, and assuming a step time of 1.0, specify a constant angular velocity of as follows:
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```txt
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*MOTION, TYPE=VELOCITY, ROTATION
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node (node set), 18.84955592, 0., 0., 0., 0., 0., 1.
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```
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The angular velocity will be constant since the default variation for motions prescribed using a predefined velocity field in a heat transfer or coupled thermal-electrical step (both steady-state and transient) is a step function (see “Defining an analysis,” Section 6.1.2). An amplitude reference could be used to specify other variations of the angular velocity.
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If, in the next step, the same node (or node set) should have an additional rotation of radians about the global x-axis, assuming again a step time of 1.0, prescribe a constant angular velocity as follows:
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```txt
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*MOTION, TYPE=VELOCITY, ROTATION
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node (node set), 1.570796327, 0., 0., 0., 1., 0., 0.
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```
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# Prescribing simultaneous rigid body rotations
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Motions involving two or more simultaneous rigid body rotations about different axes cannot be specified directly. An example of simultaneous rigid body rotations is a satellite rotating about its own axis while orbiting the earth. Such complex motions can be defined with user subroutine UMOTION. This subroutine allows specification of the time variation of the magnitude of the translational components of the motion (degrees of freedom 1–3) at each node.
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If you specify the magnitude of the translation as part of the prescribed motion definition, it will be modified by the amplitude curve (if any) and passed into subroutine UMOTION, where it can be redefined.
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When user subroutine UMOTION is used to define the motion of a certain node set in a step, only one prescribed motion can be defined in that step for that node set. The complete motion of all nodes in the node set during the step must be defined in the user subroutine.
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Input File Usage: \*MOTION, USER
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Abaqus/CAE Usage: Surface motion is not supported with cavity radiation in Abaqus/CAE.
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# Simultaneous translational and rotational motion
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Whenever simultaneous translational and rotational motion is specified, the total motion of a node during step k is defined as
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$$
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\mathbf {x} = \mathbf {X} + \mathbf {u} ^ {t} + \sum_ {i = 1} ^ {k} \mathbf {u} _ {i} ^ {r},
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$$
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<!-- source-page: 950 -->
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where ${ \bf x } ( t )$ is the current location of the node due to the specified motion history, is the original location of the node, ${ \bf u } ^ { t } ( t )$ is the displacement of the node due to the translational motion specified in the step, and $\mathbf { u } _ { i } ^ { r }$ is the displacement of the node due to rigid body rotation during step i.
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In these cases the translation is applied first and the rotation is then assumed to be about the translated (material) axis. In other words, the displacement $\mathbf { u } _ { i } ^ { r }$ due to rigid body rotation during step i is computed as the rotation about an axis defined by points $\mathbf { a } _ { i } ^ { t }$ and $\mathbf { b } _ { i } ^ { t }$ where
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$$
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\mathbf {a} _ {i} ^ {t} = \mathbf {a} + \mathbf {u} _ {i} ^ {t}, \qquad \mathbf {b} _ {i} ^ {t} = \mathbf {b} + \mathbf {u} _ {i} ^ {t}.
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$$
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In the preceding equations and are the locations of the points used to define the axis of rotation for the prescribed rotational motion (they refer to the configuration at the beginning of step i) and ${ \bf \delta u } _ { i } ^ { t } ( t )$ is the displacement due to translational motion during the step $( \mathbf { u } _ { i } ^ { t } ( t ) = \mathbf { u } ^ { t } ( t ) - \mathbf { u } _ { i - 1 } ^ { t } ( T _ { i - 1 } )$ , where $T _ { i - 1 }$ is the time at the end of step ).
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# Example
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As an example, consider a three-dimensional problem with x–y planar motion as shown in Figure 41.1.1–13.
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<details>
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<summary>text_image</summary>
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4
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A
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B
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C
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53.13°
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D
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</details>
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Figure 41.1.1–13 Planar motion example.
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The centroid of the object of interest is initially located at $x = 0 , y = 3 , z = 0$ . In the first step the object is translated 4 length units in the x-direction while at the same time it rotates clockwise $1 8 0 ^ { \circ }$ ( radians) about the z-axis at constant angular velocity. This motion moves the object from position A to position C in Figure 41.1.1–13. Halfway through this motion, at position $B ,$ the displacements due to the rigid body rotation are calculated by applying the translation to the z-axis (the axis of rotation) and then applying a $9 0 ^ { \circ }$ rotation about this translated axis.
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