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Flexible Beam

A flexible beam with axial, lateral, and torsional deformations

 

Description

Connections

Parameters

See Also

Description

A Flexible Beam is shown schematically in the diagram below. Each flexible beam has three associated frames: a primary reference frame, framea (the gray frame in the diagram), an end frame, frameb, and a reference frame, base. Beam deformations are then measured with respect to the framea. When undeformed, the beam is straight and its neutral axis is aligned with the x axis of the framea.  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 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 base frame to define additional frames on the flexible beam.

 

Connections

Name

Description

framea

Base of flexible beam (inboard frame)

frameb

Tip of flexible beam (outboard frame)

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

 0.5, 0.5

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 framea.

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

-

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

 0.001, 0.002

m2

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

dly

2107 

or

 2107, 2. 5107

m4

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

dlz

2107 

or

 2107, 2. 5107

m4

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

R__o

0.01

or

0.01,0.02

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

R__i

0

or

0,0

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

0.1,0.12

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

0.2,0.25

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

b__i

0

or

0,0

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

h__i

0

or

0,0

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

0.01,0.01

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

0.1,0.12

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

0.2,0.25

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

t__1

0.001

or

0.01,0.01

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

t__2

0.001

or

0.01,0.01

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

ρ

3000

or

3000,3000

kgm3

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

71010

or

71010,71010

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

0.01,0.01

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

2.71010

or

2.71010,2.71010

Nm2

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

0.01,0.01

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

ECx

0

-

Number of elastic coordinates for axial deformation along the x axis

Axial

ECy

1

-

Number of elastic coordinates for lateral deformation along the y axis

LateralY

ECz

1

-

Number of elastic coordinates for lateral deformation along the z axis

LateralZ

ECφ

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

ICf,v

Ignore

-

Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the translational initial conditions

MechTranTree

r&conjugate0;0

000

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

Typev

Inboard

-

Indicates whether the initial velocity is expressed in the inboard or outboard frame

VelType

v&conjugate0;0

000

ms

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

ICθ,ω

Ignore

-

Indicates whether MapleSim ignores, tries to enforce, or strictly enforces the rotational initial conditions

MechRotTree

Typeθ

123

-

Indicates the sequence of body-fixed rotations used to describe the initial orientation of the center of mass frame.  For example, 1,2,3 refers to sequential rotations about the x, then y, and then z axis (123 - Euler angles).

RotType

θ&conjugate0;0

000

rad

Initial rotation of the inboard frame at the start of the simulation, based on the Typeθ parameter value

InitAng

Typeω

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

ω&conjugate0;0

000

rads

Initial velocity of the inboard frame at the start of the simulation, based on the Typeω parameter value

InitAngVel

Shape

Taylor

-

Polynomial shape function

ShapeFunc

TruncI

1

-

Truncation order of inertial terms

ITrunc

TruncEC

1

-

Truncation order of elastic coordinates

ETrunc

 

See Also

Bodies and Frames

Flexible Beam Frame

Multibody Overview