General Flexible Plate
Libraries:
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
Parameters
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
Noneto model undamped plates.Select
Proportionalto 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]:,
where α and β are the scalar coefficients.
Select
Uniform Modalto 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.The chosen reference conditions impact the types of modes to which the damping ratio applies. For details on reference conditions, see Reference Conditions. When you use modal reduction for the flexible body, only a subset of modes is retained. For details on modal reduction, see Reduce the Degrees of Freedom for a Flexible Body.
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. If the boundary conditions applied to the plate exactly match the chosen reference conditions, ζ < 1 yields underdamped modal decay, while ζ > 1 yields overdamped dissipation. When the boundary conditions and reference conditions do not match, the effect of the chosen damping ratio varies. For details of reference condition, see Reference Conditions.
Dependencies
To enable this parameter, set Type to Uniform
Modal.
Data Types: double
Reference Conditions
Specify the type of reference conditions to apply. For details, see Reference Conditions.
Linearized Mean Axes: The body frame of reference follows the instantaneous center of mass and the flexible body exhibits unconstrained vibration modes of deformation.Body-Fixed: Choose a body-attached frame as the body frame of reference. The deformation of the flexible body exhibits free vibration mode shapes with zero translational and rotational deformation at the location of the body frame of reference.
Enter the name of the connection frame to use as the body frame of reference. The frame name is case-sensitive. Ensure the selected frame is exposed as a port on the block. You cannot use the R frame, which is the reference frame of the fictitious undeformed body.
Dependencies
To enable this parameter, set Type to
Body-Fixed.
Select to expose the R port, which represents the reference frame of the fictitious undeformed body. Note that the R frame is not physically attached to the flexible body.
The R port must not be rigidly connected to any blocks that exert forces and torques or possess inertia.
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:
Type to
From GeometryVisual 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:
Type to
From GeometryVisual 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:
Type to
From GeometryVisual 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:
Type to
From GeometryVisual 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:
Type to
From GeometryVisual 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 an emissive color, the beam can be seen even if there is no external light source.
Dependencies
To enable this parameter, set:
Type to
From GeometryVisual 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:
Type to
From GeometryVisual 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 Multibody 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 Use Selected Feature 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 Use Selected Feature 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.
Extended Capabilities
Version History
Introduced in R2021b
