Flexible Beam

A flexible beam with axial, lateral, and torsional deformations

 Description A Flexible Beam is shown schematically in the diagram below. Each flexible beam has three associated frames: a primary reference frame, ${\mathrm{frame}}_{a}$ (the gray frame in the diagram), an end frame, ${\mathrm{frame}}_{b}$, and a reference frame, $\mathrm{base}$. Beam deformations are then measured with respect to the ${\mathrm{frame}}_{a}$. When undeformed, the beam is straight and its neutral axis is aligned with the x axis of the ${\mathrm{frame}}_{a}$.  The Rayleigh beam model includes axial deformation along the neutral axis, bending about both the y and z axes, and torsional deformations about the x axis.    The beam material is assumed to be linearly elastic, that is, the stress tensor is a linear function of the strain tensor. The Flexible Beam component can represent beams with multiple sections having different cross-sections and different material properties. It is also possible to specify the cross-section area properties by selecting a shape from a list of predefined shapes. The following shapes are available: Circle, Hexagon, I-beam, and Rectangle. The $\mathrm{base}$ frame is used only with the Flexible Beam Frame component as shown in the following figure. Any number of Flexible Beam Frame components can be connected to the $\mathrm{base}$ frame to define additional frames on the flexible beam.

Connections

 Name Description ${\mathrm{frame}}_{a}$ Base of flexible beam (inboard frame) ${\mathrm{frame}}_{b}$ Tip of flexible beam (outboard frame) $\mathrm{base}$ Special port for connecting Flexible Beam Frames

Parameters

Geometry

 Symbol Default Units Description Modelica ID Use multiple sections $-$ When selected, the parameter $L$ becomes a vector (list) of shaft section lengths from frame_a to frame_b useMultiSection $L$ 1 or $m$ Beam length. This is either a single value or a vector of values, with each vector element representing the length of a beam section. The vector option is available when you select the Use multiple sections option. Note: When undeformed, the beam is straight and its neutral axis is aligned with the x axis of the ${\mathrm{frame}}_{a}$. L_s or L_v Use Predefined Shape $-$ When selected the cross-section area properties are defined using a predefined shape (See section description for the available options) useShape section CircleHexagonI-beamRectangle - Select a predefined section. Select Use Predefined Shape to enable this option. Available options: Circle, Rectangle, Hexagon, and I-beam. sectionType

Properties for Default and Predefined Shapes

Default

 Symbol Default Units Description Modelica ID $A$ 0.001  or   ${m}^{2}$ Beam cross-sectional area. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. A_g_s or A_g_v ${\mathrm{dl}}_{y}$ $2\cdot {10}^{-7}$  or ${m}^{4}$ Second moment of the cross-sectional area about the y axis. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. dIy_g_s or dIy_g_v ${\mathrm{dl}}_{z}$ $2\cdot {10}^{-7}$  or ${m}^{4}$ Second moment of the cross-sectional area about the z axis. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. dIz_g_s or dIz_g_v

Circle

 Symbol Default Units Description Modelica ID $\mathrm{R__o}$ $0.01$ or $\left[0.01,0.02\right]$ $m$ Outer radius. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. Ro_s or Ro_v $\mathrm{R__i}$ $0$ or $\left[0,0\right]$ $m$ Inner radius. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. Ri_s or Ri_v

Rectangle

 Symbol Default Units Description Modelica ID $b$ $0.1$ or $\left[0.1,0.12\right]$ $m$ Width. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. b_s or b_v $h$ $0.2$ or $\left[0.2,0.25\right]$ $m$ Height. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. h_s or h_v $\mathrm{b__i}$ $0$ or $\left[0,0\right]$ $m$ Inner width. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. bi_s or bi_v $\mathrm{h__i}$ $0$ or $\left[0,0\right]$ $m$ Inner height. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. hi_s or hi_v

Hexagon

 Symbol Default Units Description Modelica ID $a$ $0.01$ or $\left[0.01,0.01\right]$ $m$ Side length. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. a_s or a_v

I-beam

 Symbol Default Units Description Modelica ID $b$ $0.1$ or $\left[0.1,0.12\right]$ $m$ Width. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. b_s or b_v $h$ $0.2$ or $\left[0.2,0.25\right]$ $m$ Height. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. h_s or h_v $\mathrm{t__1}$ $0.001$ or $\left[0.01,0.01\right]$ $m$ Flange thickness. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. t1_s or t1_v $\mathrm{t__2}$ $0.001$ or $\left[0.01,0.01\right]$ $m$ Web thickness. This is either a single value or a vector of values for each section of the beam (that is, Use multiple sections is selected). The vector must have the same number of elements as $L$. t2_s or t2_v

Material

 Symbol Default Units Description Modelica ID Use single material $-$ Only available when you select Use multiple sections. When selected, material parameters are single values (that is, the beam is uniform). When cleared, material parameters are vectors (lists), and each element in the vector is the material property value for that section of the beam. useSameMaterial $\mathrm{ρ}$ $3000$ or $\left[3000,3000\right]$ $\frac{\mathrm{kg}}{{m}^{3}}$ Beam density. This is either a single value or a vector of values. The vector is available when you select Use multiple sections and clear Use single material.  The vector must have the same number of elements as $L$. rho_s or rho_v $E$ $7\cdot {10}^{10}$ or $\left[7\cdot {10}^{10},7\cdot {10}^{10}\right]$ $\mathrm{Pa}$ Young's modulus of elasticity. This is either a single value or a vector of values. The vector is available when you select Use multiple sections and clear Use single material.  The vector must have the same number of elements as $L$. E_s or E_v ${E}^{*}$/$E$ $0.01$ or $\left[0.01,0.01\right]$ $s$ Bending/Elongation damping as a proportion of Young's modulus of elasticity. This is either a single value or a vector of values. The vector is available when you select Use multiple sections and clear Use single material.  The vector must have the same number of elements as $L$. cE_s or cE_v $G$ $2.7\cdot {10}^{10}$ or $\left[2.7\cdot {10}^{10},2.7\cdot {10}^{10}\right]$ $\frac{N}{{m}^{2}}$ Shear modulus of elasticity. This is either a single value or a vector of values. The vector is available when you select Use multiple sections and clear Use single material.  The vector must have the same number of elements as $L$. G_s or G_v ${G}^{*}$/$G$ $0.01$ or $\left[0.01,0.01\right]$ $s$ Torsional damping as a proportion of shear modulus of elasticity. This is either a single value or a vector of values. The vector is available when you select Use multiple sections and clear Use single material.  The vector must have the same number of elements as $L$. cG_s or cG_v

Formulation

 Symbol Default Units Description Modelica ID ${\mathrm{EC}}_{x}$ $0$ - Number of elastic coordinates for axial deformation along the x axis Axial ${\mathrm{EC}}_{y}$ $1$ - Number of elastic coordinates for lateral deformation along the y axis LateralY ${\mathrm{EC}}_{z}$ 1 - Number of elastic coordinates for lateral deformation along the z axis LateralZ ${\mathrm{EC}}_{\mathrm{φ}}$ $0$ - Number of elastic coordinates for torsional deformation about the x axis Torsional

 Symbol Default Units Description Modelica ID Straight Strictly Enforce - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces a straight beam at the start of the simulation ElasticICs ${\mathrm{IC}}_{f,v}$ Ignore - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the translational initial conditions MechTranTree ${\stackrel{&conjugate0;}{r}}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $m$ Initial displacement of the inboard frame at the start of the simulation, expressed along the x-, y-, and z-axis of the inboard frame respectively InitPos ${\mathrm{Type}}_{v}$ $\mathrm{Inboard}$ - Indicates whether the initial velocity is expressed in the inboard or outboard frame VelType ${\stackrel{&conjugate0;}{v}}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $\frac{m}{s}$ Initial velocity of the inboard frame frame at the start of the simulation, expressed along the x-, y-, and z-axis of the inboard frame respectively InitVel ${\mathrm{IC}}_{\mathrm{θ},\mathrm{ω}}$ Ignore - Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the rotational initial conditions MechRotTree ${\mathrm{Type}}_{\mathrm{θ}}$ $\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$ - Indicates the sequence of body-fixed rotations used to describe the initial orientation of the center of mass frame.  For example, $\left[1,2,3\right]$ refers to sequential rotations about the x, then y, and then z axis (123 - Euler angles). RotType ${\stackrel{&conjugate0;}{\theta }}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $\mathrm{rad}$ Initial rotation of the inboard frame at the start of the simulation, based on the ${\mathrm{Type}}_{\mathrm{\theta }}$ parameter value InitAng ${\mathrm{Type}}_{\mathrm{ω}}$ $\mathrm{Euler}$ - Indicates whether the initial angular velocity is expressed in the inboard or outboard frame.  If Euler is selected, the initial angular velocities are assumed to be the direct derivatives of the Euler angles. AngVelType ${\stackrel{&conjugate0;}{\omega }}_{0}$ $\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ $\frac{\mathrm{rad}}{s}$ Initial velocity of the inboard frame at the start of the simulation, based on the ${\mathrm{Type}}_{\mathrm{ω}}$ parameter value InitAngVel $\mathrm{Shape}$ $\mathrm{Taylor}$ - Polynomial shape function ShapeFunc ${\mathrm{Trunc}}_{I}$ 1 - Truncation order of inertial terms ITrunc ${\mathrm{Trunc}}_{\mathrm{EC}}$ 1 - Truncation order of elastic coordinates ETrunc