Overview of Flexible Beams
Flexible Beam Blocks
You can use flexible beam blocks in the Simscape™ Multibody™ to model slender bodies with constant cross-sections that can have small and linear elastic deformations. These deformations include extension, bending, and torsion. To use these blocks, in the Library Browser, click Simscape > Multibody > Body Elements > Flexible Bodies > Beams.
The following figure shows a flexible channel beam model. In this example, the beam
undergoes both bending and torsion under an applied transverse point load.
The degree to which the beam bends and twists varies with the point of
application of the force in the plane of the cross-section. Enter
smdoc_flexible_cantilever_channel at the
MATLAB® command prompt to open the model.
The geometry of a beam is an extrusion of its cross-section. The general cross-sections, with or without holes, are supported by the General Flexible Beam block. Additionally, the beam cross-section can take many standard shapes, such as channel, angle, and hollow cylindrical. For beams with standard cross-sectional shapes, use the following flexible beam blocks:
Each beam has two connection frames labeled A and B. Each connection frame has a frame port on the block that can connect to another block. The connection frames are located at the ends of the beam and fall on the z-axis of the local reference frame labeled R. The reference frame serves merely as an internal reference for the beam and has no frame port.
In Simscape Multibody, all the flexible beams can have elastic bending, axial, and torsional deformations. The beams are assumed to be slender bodies whose length must far exceed its overall cross-sectional dimensions, and all the deformations should be linear and small.
The bending and axial deformations of a beam follow classical (Euler-Bernoulli) beam theory. The bending can be about any axis in the cross-sectional plane (xy-plane) of the beam. Cross-sectional slices are assumed to be rigid in-plane, to stay planar during deformation, and to always be perpendicular to the deformed neutral axis of the beam. The twisting of a beam derives from Saint-Venant torsion theory, and the cross-sectional slices are rigid in-plane but free to warp out-of-plane.
When one or more of these assumptions are not met, the result may be inaccurate. For example, in the figure, a cantilevered beam that is subjected to a transverse point load will get an inaccurate result when the bending deformation, δ, is large. During the bending, the free end of the beam moves downward perpendicularly instead of following the true physical path, which is indicated by the dotted trajectory. The discrepancy, ε, increases as the δ increases.
Flexible beams in Simscape Multibody are assumed to be made of a homogeneous, isotropic, and linearly elastic material. You can specify the material properties, such as density and Young's modulus in the Stiffness and Inertia section of the block dialog box. The beam cross-sectional properties, such as the axial, flexural, and torsional rigidities, are automatically calculated by the block using the material and geometry properties that you specify. To see the computed values, in the beam block dialog box, open Stiffness and Inertia > Derived Values and click the Update button.
The beam blocks support two damping methods: uniform modal damping and proportional damping. The uniform modal damping method applies identical damping ratios to all the vibration modes of the beam. In the proportional damping method, the damping matrix [C] is a linear combination of the mass matrix [M] and the stiffness matrix [K]:
where α and β are scalar coefficients.
The Number of Elements parameter in the Discretization section of the beam block dialog box specifies the number of finite elements used to discretize the beam. You can select its value to obtain a good compromise between simulation accuracy, which may require more elements, and simulation speed, which requires fewer elements. Use the fewest elements needed to satisfy your accuracy requirements.
For bending deformations, the beam blocks use the cubic Hermite interpolation method to compute the displacement distributions throughout each element. The distributions of axial displacement and torsional rotation are obtained by linear interpolation method.
When using beam blocks in a model, several factors impact the accuracy and speed of the simulation performance. This section discusses the impact of the three most important factors: flexible beam usage, solver selection, and damping settings.
Even though using flexible beams can increase the accuracy of a multibody simulation, the flexible beams tend to slow it down by increasing the numerical stiffness and the number of degrees of freedom of the system. To speed up the simulation, you should use a rigid body whenever the deformation of the body is negligible. Moreover, the Number of Elements parameter in the Discretization section heavily impacts the performance of the simulation. For more information, see the Discretization section.
The solver is critical to the performance of a multibody simulation. The stiff solvers, such as ode15s, ode23t, or daessc, tend to work better for systems with flexible beams due to the stiff nature of these systems. Additionally, solver tolerances and maximum order also impact the accuracy and speed of the simulation. For more information, see Choose a Solver.
All the solvers, except ode23t, provide some level of numerical dissipation, which can be helpful for modeling flexible multibody systems.
When modeling a flexible beam with little or no damping, undesirable high-frequency modes in the response can slow down the simulation if the solver does not already provide adequate numerical dissipation. In that case, adding a small amount of damping can improve the speed of the simulation without significantly affecting the accuracy of the model.
Deformation Under Gravity
Flexible beams in Simscape Multibody respond to gravity, but only that specified in the Mechanism Configuration block. The force due to a Gravitational Field block is ignored. If the frame network of which the flexible beam block is a part contains a Gravitational Field block, the body behaves as though in zero gravity. Using flexible body and Gravitational Field blocks in the same frame network causes Diagnostic Viewer to issue a compilation warning.
Modeling gravity with both the Mechanism Configuration and Gravitational Field blocks results in a compilation error.
The dialog box of each flexible beam block contains a collapsible visualization pane. This pane provides instant visual feedback on the beam you are modeling. Use it to find and fix any issues with the cross-section, length, and color of the beam. You can examine the beam from different views by selecting a standard view or by rotating, panning, and zooming.
In the toolstrip of the visualization pane, click the Update Visualization button to view the latest changes to the beam. Click Apply or OK to commit any changes to the model.
You can point to any button to see its function.
Additionally, you can right-click the visualization pane for a context-sensitive menu. This menu provides additional options to change the background color, modify the view convention setting, and split the visualization pane into multiple windows that display different views of the beam.