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 crosssections and different material properties. It is also possible to specify the crosssection area properties by selecting a shape from a list of predefined shapes. The following shapes are available: Circle, Hexagon, Ibeam, 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
$\left[0.5\,0.5\right]$

$m$

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 crosssection area properties are defined using a predefined shape (See section description for the available options)

useShape

section




Select a predefined section. Select Use Predefined Shape to enable this option.
Available options: Circle, Rectangle, Hexagon, and Ibeam.

sectionType




Properties for Default and Predefined Shapes


Default
Symbol

Default

Units

Description

Modelica ID

$A$

0.001
or
$\left[0.001\,0.002\right]$${}$

${m}^{2}$

Beam crosssectional 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
$\left[2\cdot {10}^{7}\,2.5\cdot {10}^{7}\right]$

${m}^{4}$

Second moment of the crosssectional 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
$\left[2\cdot {10}^{7}\,2.5\cdot {10}^{7}\right]$

${m}^{4}$

Second moment of the crosssectional 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



Ibeam
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





Advanced


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 zaxis 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 zaxis 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 bodyfixed 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





