General Flexible Plate

Thin plate with elastic properties for deformation

• Library:
• Simscape / Multibody / Body Elements / Flexible Bodies / Plates and Shells

Description

The General Flexible Plate block models thin, flat structure capable of elastic deformations, including stretch, bending, and shear effects. The block applies the shear deformation Mindlin plate theory [1][2][3] and uses the finite element method [4] for its solution. You can use this block to model thin, flat structures, such as linkages and satellite solar panels.

To specify the geometry of a plate, use the Midsurface and Thickness parameters. The midsurface of the plate is in the xy plane and the thickness is along the z axis. The thickness must be much smaller than the width and length of the plate, and the plate is symmetric about the midsurface. See the Geometry section for more details.

The block models a flexible plate made of homogeneous, isotropic, and linearly elastic materials. You can specify the density, Young's modulus, and Poisson's ratio or shear modulus of the plate in the Stiffness and Inertia section. Additionally, the block supports two damping methods to control the performance of the modeling. To add custom frames to the plate, specify parameters in the Frames section.

Ports

Frame

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Custom body-attached frames. Specified the name of the port using the New Frame parameter. If you do not specify the name of the custom frame, the block names the frame FN, where `N` is an identifying number.

Dependencies

To enable a custom frame port, create a frame by clicking New Frame.

Parameters

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Geometry

Coordinates used to specify the midsurface boundaries of the plate on the local xy plane. You can specify the midsurface by using:

• An N-by-2 matrix of xy coordinates to specify a midsurface. Each row gives the [x,y] coordinates of a point on the mid-surface boundaries. The points connect in the specified order to form a closed polyline. To ensure that the polyline is closed, the block inserts a line segment between the last and first points.

• An M-by-1 or 1-by-M cell array of N-by-2 matrices of xy coordinates to specify a midsurface with holes. The first element in the array represents the outer boundary and subsequent elements specify the hole boundaries.

Note

Ensure that any two boundaries do not intersect, overlap, or touch.

Additionally, each individual boundary must have:

• No repeated vertices

• No self-intersections

• At least three non-collinear points

Thickness of the plate. The block models the plate by extruding the specified midsurface along the local z axis of the plate. The extrusion is symmetric about the local xy plane of the pate. The thickness should be much smaller than the overall midsurface dimensions for plate theory to apply.

Stiffness and Inertia

Mass per unit volume of material. The default value corresponds to aluminum.

Elastic properties used to parameterize the plate. You can specify either `Young's Modulus and Poisson's Ratio` or ```Young's and Shear Modulus```. These properties are commonly available in materials databases.

Young's modulus of the elasticity of the plate. The greater the value of this parameter, the stronger the resistance to bending and in-plane normal deformation. The default value corresponds to aluminum.

Poisson's ratio of the plate. The value specified must be greater than or equal to 0 and smaller than 0.5. The default value corresponds to aluminum.

Dependencies

To enable this parameter, set Specify to ```Young's Modulus and Poisson's Ratio```.

Shear modulus, also known as the modulus of rigidity, of the plate. The larger value correlates to a stronger resistance to shearing and twisting. The default value corresponds to aluminum.

Dependencies

To enable this parameter, set Specify to ```Young's and Shear Modulus```.

Damping

Damping method for the plate:

• Select `None` to model undamped plates.

• Select `Proportional` to apply the proportional (or Rayleigh) damping method. This method defines the damping matrix [C] as a linear combination of the mass matrix [M] and stiffness matrix [K]:

$\left[C\right]=\alpha \left[M\right]+\beta \left[K\right]$,

where α and β are the scalar coefficients.

• Select `Uniform Modal` to apply the uniform modal damping method. This method applies a single damping ratio to all the vibration modes of the plate. The larger the value, the faster vibrations decay.

Coefficient α of the mass matrix. This parameter defines damping proportional to the mass matrix [M].

Dependencies

To enable this parameter, set Type to `Proportional`.

Coefficient β of the stiffness matrix. This parameter defines damping proportional to the stiffness matrix [K].

Dependencies

To enable this parameter, set Type to `Proportional`.

Damping ratio ζ applied to all vibration modes of a plate. The larger the value, the faster vibrations decay. The vibration modes are underdamped if ζ < 1 and overdamped if ζ > 1.

Dependencies

To enable this parameter, set Type to ```Uniform Modal```.

Data Types: `double`

Fidelity

Method to use to model flexible bodies, specified as `None` or `Modally Reduced`. Set the parameter to `None` to use full nodal elastic coordinates generated by the finite-element method or set the parameter to ```Modally Reduced``` to use the modal transformation method to reduce the elastic coordinates of the body. For both settings, the block uses the floating frame of the reference formulation [5-6] to couple the body with its elastic deformation.

Retained modes, specified as an integer in range [0, n], where n is the number of elastic degrees of freedom of the body. If you set the number to 0 the flexible body is treated as a rigid body.

Dependencies

To enable this parameter, set Reduction to `Modally Reduced`.

Graphic

Type of the visual representation of the plate, specified as ```From Geometry``` or `None`. Set the parameter to `From Geometry` to show the visual representation of the plate. Set the parameter to `None` to hide the plate in the model visualization.

Parameterization for specifying visual properties. Select `Simple` to specify color and opacity. Select `Advanced` to specify more visual properties, such as Specular Color, Ambient Color, Emissive Color, and Shininess.

Dependencies

To enable this parameter, set Type to ```From Geometry```.

Color of the graphic under direct white light, specified as an [R G B] or [R G B A] vector on a 0–1 scale. An optional fourth element (A) specifies the color opacity on a scale of 0–1. Omitting the opacity element is equivalent to specifying a value of 1.

Dependencies

To enable this parameter, set:

1. Type to ```From Geometry```

2. Visual Properties to `Simple`

Graphic opacity, specified as a scalar in the range of 0 to 1. A scalar of 0 corresponds to completely transparent, and a scalar of 1 corresponds to completely opaque.

Dependencies

To enable this parameter, set:

1. Type to ```From Geometry```

2. Visual Properties to `Simple`

Color of the graphic under direct white light, specified as an [R G B] or [R G B A] vector on a 0–1 scale. The optional fourth element specifies the color opacity on a scale of 0–1. Omitting the opacity element is equivalent to specifying a value of 1.

Dependencies

To enable this parameter, set:

1. Type to `From Geometry`

2. Visual Properties to `Advanced`

Color of the light due to specular reflection, specified as an [R,G,B] or [R,G,B,A] vector with values in the range of 0 to 1. The vector can be a row or column vector. The optional fourth element specifies the color opacity. Omitting the opacity element is equivalent to specifying a value of 1. This parameter changes the color of the specular highlight, which is the bright spot on the rendered beam due to the reflection of the light from the light source.

Dependencies

To enable this parameter, set:

1. Type to ```From Geometry```

2. Visual Properties to `Advanced`

Color of the ambient light, specified as an [R,G,B] or [R,G,B,A] vector with values in the range of 0 to 1. The vector can be a row or column vector. The optional fourth element specifies the color opacity. Omitting the opacity element is equivalent to specifying a value of 1.

Ambient light refers to a general level of illumination that does not come directly from a light source. The Ambient light consists of light that has been reflected and re-reflected so many times that it is no longer coming from any particular direction. You can adjust this parameter to change the shadow color of the rendered beam.

Dependencies

To enable this parameter, set:

1. Type to ```From Geometry```

2. Visual Properties to `Advanced`

Color due to self illumination, specified as an [R,G,B] or [R,G,B,A] vector in the range of 0 to 1. The vector can be a row or column vector. The optional fourth element specifies the color opacity. Omitting the opacity element is equivalent to specifying a value of 1.

The emission color is color that does not come from any external source, and therefore seems to be emitted by the beam itself. When a beam has a emissive color, the beam can be seen even if there is no external light source.

Dependencies

To enable this parameter, set:

1. Type to ```From Geometry```

2. Visual Properties to `Advanced`

Sharpness of specular light reflections, specified as a scalar number on a 0–128 scale. Increase the shininess value for smaller but sharper highlights. Decrease the value for larger but smoother highlights.

Dependencies

To enable this parameter, set:

1. Type to ```From Geometry```

2. Visual Properties to `Advanced`

Frames

Click the Create button to open a pane for creating a new body-attached frame. In this pane, you can specify the name, origin, and orientation for the frame.

• To name the custom frame, enter a name in the Frame Name parameter. The name identifies the port on the block and in the tree view pane of the Mechanics Explorer.

• Select a frame origin in the Frame Origin section:

• At Reference Frame Origin: Make the new frame origin coincident with the origin of the reference frame of the undeformed plate.

• Based on Geometric Feature: Make the new frame origin coincident with the center of the selected undeformed geometry feature. Valid features include surfaces, lines, and points. Select a feature from the visualization pane, then click button to confirm the location of the origin. The name of the origin location appears in the field below this option.

• To define the orientation of the custom frame, under the Frame Axes section, select the Primary Axis and Secondary Axis of the custom frame and then specify their directions.

Use the following methods to select a vector for specifying the directions of the primary and secondary axes. The primary axis is parallel to the selected vector and constrains the remaining two axes to its normal plane. The secondary axis is parallel to the projection of the selected vector onto the normal plane.

• Along Reference Frame Axis: Selects an axis of the reference frame of the undeformed plate.

• Based on Geometric Feature: Selects the vector associated with the chosen geometry feature of the undeformed plate. Valid features include surfaces and lines. The corresponding vector is indicated by a white arrow in the visualization pane. You can select a feature from the visualization pane and then click button to confirm the selection. The name of the selected feature appears in the field below this option.

Frames that you created. Specified the name of the port using the New Frame parameter. If you do not specify the name of the custom frame, the block names the frame FN, where `N` is an identifying number.

• Click the text field to edit the name of an existing custom frame.

• Click the Edit button to edit other aspects of the custom frame, such as the origin and axes.

• Click the Delete button to delete the custom frame.

Dependencies

To enable this parameter, create a frame by clicking New Frame.

References

[1] Bathe, Klaus-Jürgen. Finite Element Procedures. 2nd ed. Englewood Cliffs, N.J: Prentice-Hall, 2014.

[2] Cook, Robert Davis. Concepts and Applications of Finite Element Analysis. 4th ed. New York, NY: Wiley, 2001.

[3] Dvorkin, Eduardo N., and Klaus‐Jürgen Bathe. “A Continuum Mechanics Based Four‐node Shell Element for General Non‐linear Analysis.” Engineering Computations 1, no. 1 (January 1984): 77–88. https://doi.org/10.1108/eb023562.

[4] Bucalem, M. L., and K. J. Bathe. “Finite Element Analysis of Shell Structures.” Archives of Computational Methods in Engineering 4, no. 1 (March 1997): 3–61. https://doi.org/10.1007/BF02818930.

[5] Shabana, Ahmed A. Dynamics of Multibody Systems. Fourth edition. New York: Cambridge University Press, 2014.

[6] Agrawal, Om P., and Ahmed A. Shabana. “Dynamic Analysis of Multibody Systems Using Component Modes.” Computers & Structures 21, no. 6 (January 1985): 1303–12. https://doi.org/10.1016/0045-7949(85)90184-1.

Version History

Introduced in R2021b