General Flexible Beam

Slender extrusion with elastic properties for deformation

  • Library:
  • Simscape / Multibody / Body Elements / Flexible Bodies / Beams

Description

The General Flexible Beam block models a slender beam of constant, general cross-section that can have small and linear deformations. These deformations include extension, bending, and torsion. The block calculates the beam cross-sectional properties, such as the axial, flexural, and torsional rigidities, based on the geometry and material properties that you specify.

The geometry of the flexible beam is an extrusion of its cross-section. The beam cross-section, defined in the xy-plane, is extruded along the z-axis. You can use this block to create flexible beams with simply or multiply connected cross-sections. For example, you can create the beam shown in the figure by entering these values for the Cross-section in the block's dialog box: {[-0.25,-0.50;0.25,-0.50;0.25,0.50;-0.25,0.50],[-0.15,-0.40;0.15,-0.40;0.15,-0.05;-0.15,-0.05],[-0.15,0.05;0.15,0.05;0.15,0.40;-0.15,0.40]}.

Flexible beams are assumed to be made of a homogeneous, isotropic, and linearly elastic material. You can specify the beam's density, Young’s modulus, and Poisson’s ratio or shear modulus in the Stiffness and Inertia section of the block dialog box. Additionally, this block supports two damping methods and a discretization option to increase the accuracy of the modeling. For more information, see Overview of Flexible Beams.

Ports

Frame

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Frame by which to connect the beam in a model. In the undeformed configuration, this frame is at half the beam length in the -z direction relative to the origin of the local reference frame.

Frame by which to connect the beam in a model. In the undeformed configuration, this frame is at half the beam length in the +z direction relative to the origin of the local reference frame.

Parameters

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Geometry

Coordinates used to specify the boundaries of a beam cross-section. Specify the beam cross-section using one of the following methods:

  • Use an N-by-2 matrix of xy coordinates to specify a simply connected section. Each row gives the [x,y] coordinates of a point on the cross-section boundary. The points connect in the order given to form a closed polyline. To ensure that the polyline is closed, a line segment is always inserted between the last and first points.

  • Use an M-by-1 or 1-by-M cell array of N-by-2 matrices of xy coordinates to specify a multiply connected section. The first entry in the cell represents the outer boundary and subsequent entries specify the hole boundaries.

Note

To properly define the cross-section of beams, any two boundaries should not intersect, overlap, or touch.

Additionally, each individual boundary should have:

  • No repeated vertices.

  • No self-intersections.

  • At least three non-collinear points.

The beam's length. The beam is modeled by extruding the specified cross-section along the z-axis of the local reference frame. The extrusion is symmetric about the xy-plane, with half of the beam being extruded in the negative direction of the z-axis and half in the positive direction.

Stiffness and Inertia

Mass per unit volume of material—assumed here to be distributed uniformly throughout the beam. The default value corresponds to aluminum.

Elastic properties in terms of which to parameterize the beam. These properties are commonly available from materials databases.

Young's modulus of elasticity of the beam. The greater its value, the stronger the resistance to bending and axial deformation. The default value corresponds to aluminum.

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

Shear modulus (or modulus of rigidity) of the beam. The greater its value, the stronger the resistance to torsional deformation. The default value corresponds to aluminum.

Calculated values of the mass and stiffness sectional properties of the beam. Click Update to calculate and display those values.

The properties given include Centroid and Shear Center. The centroid is the point at which an axial force extends (or contracts) the beam without bending. The shear center is that through which a transverse force must pass to bend the beam without twisting.

The stiffness sectional properties are computed as follows:

  • Axial Rigidity: EA

  • Flexural Rigidity: [EIx, EIy]

  • Cross Flexural Rigidity: EIxy

  • Torsional Rigidity: GJ

The mass sectional properties are computed as follows:

  • Mass per Unit Length: ρA

  • Mass Moment of Inertia Density: [ρIx, ρIy]

  • Mass Product of Inertia Density: ρIxy

  • Polar Mass Moment of Inertia Density: ρIp

The equation parameters include:

  • A — Cross-sectional area

  • ρ — Density

  • E — Young's modulus

  • G — Shear modulus

  • J — Torsional constant (obtained from the solution of Saint-Venant's warping partial differential equation)

The remaining parameters are the relevant moments of area of the beam. These are calculated about the axes of a centroidal frame—one aligned with the local reference frame but located with its origin at the centroid. The moments of area are:

  • Ix, Iy — Centroidal second moments of area:

    [Ix,Iy]=[A(yyc)2dA,A(xxc)2dA],

  • Ixy — Centroidal product moment of area:

    Ixy=A(xxc)(yyc)dA,

  • Ip — Centroidal polar moment of area:

    IP=Ix+Iy,

where xc and yc are the coordinates of the centroid.

Damping

Damping method to apply to the beam:

  • Select None to model undamped beams.

  • 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 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 beam. 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 beam vibration modes in the uniform modal damping model. The larger the value, the faster beam vibrations decay.

  • Use ζ = 0 to model undamped beams.

  • Use ζ < 1 to model underdamped beams.

  • Use ζ = 1 to model critically damped beams.

  • Use ζ > 1 to model overdamped beams.

Dependencies

To enable this parameter, set Type to Uniform Modal.

Data Types: double

Discretization

Number of finite elements in the beam model. Increasing the number of elements always improves accuracy of the simulation. But practically, at some point, the increase in accuracy is negligible when there are many elements. Additionally, a higher number of elements increases the computational cost and slows down the speed of the simulation.

Graphic

Choice of graphic used in the visualization of the beam. The graphic is by default the geometry specified for the beam. Change this parameter to None to eliminate this beam altogether from the model visualization.

Parameterization for specifying visual properties. Select Simple to specify color and opacity. Select Advanced to add specular highlights, ambient shadows, and self-illumination effects.

RGB color vector with red (R), green (G), and blue (B) color amounts specified on a 0–1 scale. A color picker provides an alternative interactive means of specifying a color.

Dependencies

To enable this parameter, set :

  1. Type to From Geometry.

  2. Visual Properties to Simple.

Graphic opacity specified on a scale of 0–1. An opacity of 0 corresponds to a completely transparent graphic and an opacity of 1 to a completely opaque graphic.

Dependencies

To enable this parameter, set :

  1. Type to From Geometry.

  2. Visual Properties to Simple.

True color 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 specifies the color opacity also 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 specular highlights 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. 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 shadow areas in diffuse ambient 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. 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.

Surface color due to self illumination, 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. 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.

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Introduced in R2018b