Reduced Order Flexible Solid

Create flexible body using reduced-order model data

Libraries:
Simscape / Multibody / Body Elements / Flexible Bodies

Description

The Reduced Order Flexible Solid block creates a flexible body by using reduced-order model (ROM) data.

A reduced-order model is a computationally efficient model that characterizes the properties of a flexible body that can have linear elastic deformations. The ROM data must include:

Coordinates and unit quaternions that specify the positions and orientations of all interface frames relative to a common reference frame. See Interface Frames.
A symmetric stiffness matrix that describes the elastic properties of the flexible body. See Stiffness Matrix.
A symmetric mass matrix that describes the inertial properties of the flexible body. See Mass Matrix.

To generate ROM data, you can use the Flexible Body Model Builder app or finite-element analysis tools, such as the Partial Differential Equation Toolbox™. By using the toolbox, you can start with the CAD geometry of the body, generate a finite-element mesh, apply the Craig-Bampton method, and generate the corresponding ROM data. For more information, see Model an Excavator Dipper Arm as a Flexible Body.

Common Reference Frame

The data provided to the Reduced Order Flexible Solid block must refer to a consistent common reference frame. This reference frame defines the x, y, and z directions that specify the relative position of all points in the body. The frame also defines the directions of the elastic degrees of freedom associated with each interface frame.

Requirements for the ROM Data

The ROM data for the Reduced Order Flexible Solid block must satisfy these requirements:

The ROM data must have at least one boundary node. Each boundary node determines the location of an interface frame.
Each boundary node must contribute six degrees of freedom to the reduced-order model. The degrees of freedom for node i must be retained in the order

U_i = [T_x_i, T_y_i, T_z_i, R_x_i, R_y_i, R_z_i],
where:
- T_x_i, T_y_i, and T_z_i are translational degrees of freedom along the x, y, and z directions of the common reference frame.
- R_x_i, R_y_i, and R_z_i are rotational degrees of freedom about the x, y, and z axes of the common reference frame.

The reduced-order model can also include additional degrees of freedom, D₁, D₂, ⋯, D_m, that correspond to retained normal vibration modes. For a reduced-order model that has n boundary nodes and m modal degrees of freedom, the degrees of the freedom of the model is:

U_reduced = [U₁, U₂, ⋯, U_n, D₁, D₂, ⋯, D_m].

The number of the degrees of freedom N_reduced equals 6n + m, which includes six rigid-body degrees of freedom. The number of elastic degrees of freedom equals N_reduced - 6. The number of the degrees of freedom determines the size of the stiffness and mass matrices. For the matrices, the order of the rows and columns must correspond to the order of the U_reduced.

Reduce the Degrees of Freedom for a Flexible Body

You can use the block to further reduce the degrees of freedom for a flexible body. The fewer the degrees of freedom for flexible bodies, the faster the simulation.

For example, a ROM data set generated by the Craig-Bampton method with two boundary nodes and 10 fixed-interface normal modes has a total of 22 degrees of freedom. If you use the ROM data, the block generates an internal representation of the flexible body that has six rigid-body and 16 elastic degrees of freedom. To further reduce the number of the elastic degrees of freedom, you can set Reduction to Modally Reduced. If you specify Number of Retained Modes to 8, the flexible body retains the eight lowest-frequency modal degrees of freedom in addition to the six rigid-body degrees of freedom.

Damping

To specify the damping characteristics of the flexible bodies, the block has three damping methods: proportional damping, uniform modal damping, and damping matrix methods. For more informations, see Damping.

Simulation Performance

Flexible bodies can increase the numerical stiffness of a multibody model. To avoid simulation issues, use a stiff solver such as ode15s or ode23t.

Damping can significantly influence simulation performance. For example, when modeling a body with little or no damping, undesirable high-frequency modes in the response can slow the simulation. In that case, adding a small amount of damping can improve the speed of the simulation without significantly affecting the accuracy of the model.

Examples

Model an Excavator Dipper Arm as a Flexible Body

Use the Reduced Order Flexible Solid block to model a deformable body of arbitrary geometry. Start with the CAD geometry of the body, produce a finite-element mesh, and generate reduced-order data to use with the block.

Open Live Script

Using the Reduced Order Flexible Solid Block - Flexible Dipper Arm

Model a flexible dipper arm, such as the arm for an excavator or a backhoe, by using the Reduced Order Flexible Solid block. This block captures the mechanical behavior of a deformable body through reduced-order stiffness and mass matrices that are associated with interface frame locations. In this example, the Partial Differential Equation Toolbox™ is used to create the reduced-order model.

Open Model

Ports

Frame

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F1, F2, ..., F`n` — Frames
frame

Frames that connect the flexible body to the model. Each interface frame corresponds to a boundary node in the reduced-order model. The frames allow you to connect the flexible body to other Simscape™ Multibody™ elements, such as joints, constraints, forces, and sensors. It is not required that all frame ports be connected.

Parameters

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Unit System

Unit System — Standard or custom units for length, mass, and time
`SI (m, kg, s)` (default) | `CGS (cm, g, s)` | `English (ft, slug, s)` | `Custom`

System of units in which to express length, mass, time, and other derived units of measure used in all block parameters. Angle measure is always in radians.

Unit System	Base Units			Selected Derived Units
Unit System	Length	Mass	Time	Force	Stress and Pressure	Density
`SI`	m	kg	s	N = kg·m/s²	Pa = kg/(m·s²)	kg/m³
`CGS`	cm	g	s	dyn = g·cm/s²	Ba = g/(cm·s²)	g/cm³
`English`	ft	slug	s	lbf = slug·ft/s²	slug/(ft·s²)	slug/ft³

To specify units of length, mass, and time individually, select Custom. For example, suppose that you specify a custom unit system with mm for length, t for mass, and s for time. The derived units include N = t·mm/s² for force, MPa = t/(mm·s²) for stress and pressure, and t/mm³ for density.

Note

The specified units must be consistent with the imported data. For example, you must set the Unit System parameter to SI (m, kg, s) when using the ROM data generated by the Flexible Body Model Builder app because the generated ROM data use SI units.

Length Unit — Unit of measure for length
`m` (default) | `cm` | `mm` | `km` | `in` | `ft` | `yd` | `mi`

Unit of measure in which to express length.

Dependencies

To enable this parameter, set Unit System to Custom.

Mass Unit — Unit of measure for mass
`kg` (default) | `g` | `t` | `lbm` | `slug`

Unit of measure in which to express mass.

Dependencies

To enable this parameter, set Unit System to Custom.

Time Unit — Unit of measure for time
s (default) | `ms` | `min` | `hr`

Unit of measure in which to express time.

Dependencies

To enable this parameter, set Unit System to Custom.

Interface Frames

Number of Frames — Number of frame ports
`1` (default) | positive integer

Number of interface frame ports, specified as a positive integer. This number must match the number of boundary nodes defined in the ROM data. The parameter does not support variables or expressions.

Origins — Cartesian coordinates of interface frame origins
`[0 0 0]` (default) | n-by-3 matrix

Cartesian coordinates of all interface frame origins, specified as a n-by-3 matrix. Each row of the matrix represents one frame origin. All coordinates must be relative to the common reference frame. Each boundary node in the reduced-order model must have a corresponding interface frame on the block.

Orientations — Orientations of interface frames
`[1 0 0 0]` (default) | n-by-4 matrix of unit quaternions

Orientations of interface frames, specified as an n-by-4 matrix of unit quaternions, where n is the number of interface frames.

Reduced Order Matrices

Stiffness Matrix — Stiffness matrix from reduced-order model
`[]` (default) | r-by-r matrix

Stiffness matrix from the ROM data, specified as a r-by-r matrix. The stiffness matrix is a symmetric matrix that describes the elastic properties of the flexible body. For a ROM data set with n boundary nodes and m dynamic deformation modes, the stiffness matrix has r = 6n + m rows and columns.

Mass Matrix — Mass matrix from reduced-order model
`[]` (default) | r-by-r matrix

Mass matrix from the ROM data, specified as a r-by-r matrix. The mass matrix is a symmetric matrix that describes the inertial properties of the flexible body. For a ROM data set with n boundary nodes and m dynamic deformation modes, the mass matrix has r = 6n + m rows and columns.

Damping

Type — Type of damping method
`Proportional` (default) | `Uniform Modal` | `Damping Matrix` | `None`

Damping method to apply to the flexible body:

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]:
$[C] = α [M] + β [K]$ ,
where α and β are 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 solid. The larger the value, the faster vibrations decay.
Select Damping Matrix to use a reduced-order damping matrix that you computed with the stiffness and mass matrices. For example, you can use this option to specify a modal damping model for the flexible body.
Select None to model undamped body.

Mass Coefficient — Coefficient of mass matrix
0 `1/s` (default) | nonnegative scalar

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.

Stiffness Coefficient — Coefficient of stiffness matrix
0.001 `s` (default) | nonnegative scalar

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 — Damping ratio for uniform modal damping method
0.01 (default) | unitless nonnegative scalar

Damping ratio, ζ, applied to all vibration modes of the flexible body. The larger the value, the faster the vibrations decay.

Use ζ = 0 to model undamped solids.
Use ζ < 1 to model underdamped solids.
Use ζ = 1 to model critically damped solids.
Use ζ > 1 to model overdamped solids.

Dependencies

To enable this parameter, set Type to Uniform Modal.

Data Types: double

Damping Matrix — Reduced-order damping matrix
`[]` (default) | r-by-r matrix

Reduced-order damping matrix, specified as a r-by-r matrix. The damping matrix is a symmetric matrix that describes the damping properties of the flexible body. In a flexible body with n boundary nodes and m dynamic deformation modes, the damping matrix has r = 6n + m rows and columns.

Dependencies

To enable this parameter, set Type to Damping Matrix.

Fidelity

Type — Method to model flexible bodies
`None` (default) | `Modally Reduced`

Method to use to model flexible bodies, specified as None or Modally Reduced.

If you select None, the block models a flexible body with all the degrees of freedom.
If you select Modally Reduced, the block performs an eigenvalue decomposition to the mass and stiffness matrices of the flexible body. You can then use the Number of Retained Modes parameter to specify the desired normal modes of the flexible body for the simulation.

Number of Retained Modes — Retained normal modes
1 (default) | nonnegative integer

Retained normal modes, specified as an integer in range [0, n], where n equals the number of elastic degrees of freedom of the flexible body.

The block uses this parameter to specify how many low-frequency modes to retain for the simulation. If you set this parameter to 0, the flexible body is treated as a rigid body. The larger the number, the slower the simulation.

Dependencies

To enable this parameter, set Type to Modally Reduced.

Graphic

Type — Visual representation of flexible body
`None` (default) | `Partitioned Geometry` | `Deformed Geometry`

Visual representation of the flexible body, specified as None, Partitioned Geometry, or Deformed Geometry.

To hide the body, use None.
To show the deformed body by using an approximate representation that consists of several rigid pieces, use Partitioned Geometry. In the visualization, the block divides the body into several pieces according to the interface frames and rigidly attaches each piece to its corresponding interface frame.
To show the reconstructed deformed shape of the body, use Deformed Geometry. The block uses the provided triangulation of the body and the recovery data to reconstruct the elastic deformations of the vertices on the triangulation.

File Name — Path of CAD geometry file
custom character vector

Path of the CAD file that defines the undeformed solid geometry, specified as a custom character vector. The file location can be specified as an absolute path starting from the root directory of the file system or a relative path starting from a folder on the MATLAB^® path.

The CAD file must define the geometry of the flexible body by using the same reference frame as the reduced-order model. See Supported Software and File Formats for details of supported file formats.

Example: 'C:/Users/JDoe/Documents/myShape.STEP' or 'Documents/myShape.STEP'

Dependencies

To enable this parameter, set Type to Partitioned Geometry.

Unit Type — Source of length unit for geometry
`From File` (default) | `From Unit System`

Source of the solid geometry unit, specified as From File or From Unit System. Select From File to use the units specified in the imported file. Select From Unit System to use the units specified by the Unit System parameter.

Dependencies

To enable this parameter, set Type to Partitioned Geometry.

Vertex Coordinates — Coordinates of vertices on triangulation representation
`[]` (default) | n_v-by-3 matrix

Coordinates of the vertices on the triangulation representation for the solid, specified as a n_v-by-3 matrix. n_v is the number of vertices, and each row of the matrix contains the x, y, and z coordinates for a vertex. The coordinates refer to the common reference frame.

Dependencies

To enable this parameter, set Type to Deformed Geometry.

Facet Vertex Indices — Indices of vertices that define facets of triangulation representation
`[]` (default) | n_f-by-3 matrix

Indices of vertices that define the facets of the triangulation representation for the solid, specified as a n_f-by-3 matrix. n_f is the number of facets, and each row of the matrix contains the indices of three vertices that define their corresponding facet. The index of a vertex is the row number of the vertex in the matrix for the Vertex Coordinates parameter.

Dependencies

To enable this parameter, set Type to Deformed Geometry.

Reconstruction Matrix — Recovery data to visualize deformed body
`[]` (default) | 3n_v-by-r matrix

Recovery data to visualize the deformed body, specified as a 3n_v-by-r matrix. n_v is the number of vertices and r is the number of columns of the mass or stiffness matrix in the ROM data.

The Reduced Order Flexible Solid block uses the reconstruction matrix to calculate the displacements of the vertices on the triangulation representation to display the deformed shape of the body. Each set of three rows of the matrix corresponds to the x, y, and z displacements of a vertex.

Dependencies

To enable this parameter, set Type to Deformed Geometry.

Visual Properties — Parameterizations for color and opacity
`From File` (default) | `Advanced` | `Simple`

Parameterization for specifying visual properties.

Select From File to use color data from the imported CAD geometry file. Note that not all file formats allow color data. In formats that allow color data, that data is often optional. If your file does not specify color, the solid is rendered in gray.
Select Simple to specify Color and Opacity.
Select Advanced to specify more visual properties, such as Diffuse Color, Specular Color, Ambient Color, Emissive Color, and Shininess.

Dependencies

To enable this parameter, set Type to Partitioned Geometry.

Color — Color of light due to diffuse reflection
[0.5 0.5 0.5] (default) | 3-by-1 or 1-by-3 vector with values in the range of 0 to 1 | 4-by-1 or 1-by-4 vector with values in the range of 0 to 1

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.