# Flexible Beam from Lumped Parameters

Models a flexible beam based on lumped parameter method. Number of flexible elements, material properties as well as beam cross section and dimensions can be varied from mask parameters.

## Contents

## Model

## Diagram

The following diagram shows the generic structure of the beam represented by these blocks. A chain of flexible elements connects frames B and F which are the ports of the block. The number of elements can be varied, as can the degrees of freedom permitted by the spring-damper within each flexible element.

## Parameters, Tab `Material`

**Material**: Define the material properties of the beam. Exact values can be provided or standard values for common materials can be selected.

`Custom`- Provide exact values for relevant material properties.`Steel`- Use standard values for steel, which are shown in the dialog box.- Many other standard properties can be selected

These parameters are needed to calculate solid and deflection properties of the beam

**Material Density**: Density of the material**Modulus of Elasticity**: Young's Modulus of the material**Shear Modulus**: Shear Modulus of the material

**Damping**: Specify damping for the beam. Damping is specified by two damping factors. This parameterization enables the damping to scale with the dimensions and material used in the beam.

**Elastic Damping Factor**: Damping factor for bending and elongation**Shear Damping Factor**: Damping factor for torsion

The damping coefficient used in the flexible elements is calculated according to the following formula:

Where

= Damping coefficient for bending, elongation, and torsion

= Area moment of inertia

= Cross sectional area of beam

= Torsional constant for beam

= Modulus of Elasticity

= Shear Modulus

= Length of flexible beam element

**Print internal values to Command Window**: Prints internal values to MATLAB Command Window. Resulting values can be inspected to verify that provided parameters are correct. An example of the printed values is shown below.

Value Units __________ _______________ Area Moment of Inertia, Ixx 0.0026121 {'m^4' } Area Moment of Inertia, Iyy 0.0026121 {'m^4' } Torsional Constant, J 1.5625e-09 {'m^4' } Cross sectional area, A 7.5e-05 {'m^2' } Flexible element length 0.03 {'m' } Element Stiffness, Bending about Z 1.7414e+10 {'N*m/rad' } Element Stiffness, Bending about Y 1.7414e+10 {'N*m/rad' } Element Stiffness, Torsion about X 4020.8 {'N*m/rad' } Element Stiffness, Elongation along X 5e+08 {'N/m' } Element Damping, Bending about Z 4.494e+05 {'N*m/(rad/s)'} Element Damping, Bending about Y 4.494e+05 {'N*m/(rad/s)'} Element Damping, Torsion about X 0.012971 {'N*m/(rad/s)'} Element Damping, Elongation along X 12903 {'N/(m/s)' }

## Parameters, Tab `Geometry`

**Cross-Section Type**: Select the cross-section type for the beam. Exact values can be provided, or some standard shapes can be used.

`Hollow Rectangle`- Define cross-section as a hollow rectangle. Selecting this option exposes parameters for defining the inner and outer dimensions of the hollow rectangle. The inner dimension can be set to zero in order to define a solid rectangle. Area moments of inertia, polar moments of inertia are calculated automatically.**Note**: torsion constant calculation assumes thickly walled cross section. See code if you wish to verify formula used.`Hollow Circle`- Define cross-section as a hollow circle. Selecting this option exposes parameters for defining the inner and outer diameters of the hollow circle. The inner dimension can be set to zero in order to define a solid circle. Area moments of inertia, polar moments of inertia, and torsion constant are calculated automatically.**Note**: torsion constant calculation assumes thickly walled cross section. See code if you wish to verify formula used.`Custom`- Specify the exact properties of the cross-section. Selecting this option exposes parameters for defining the area moments of inertia, polar moments of inertia, torsion constant, and the extrusion data.

**Length**: Overall length of the beam

**Number of elements**: Number of flexible elements used to construct the beam. A higher number of elements typically results in higher accuracy but longer computation times.

**Color**: 3-vector with values between 0-1 defining color of rigid body solid [RGB]

**Opacity**: Scalar value between 0-1 defining opacity

## Parameters, Tab `Flexibility Type`

**Flexible Element Degrees of Freedom**: Select the number of degrees of freedom permitted by the spring-damper element in each flexible element.

`Rotation: Z`- Permits one rotational degree of freedom in the flexible element along the z-axis.`Rotation: X, Y, Z; Translation: X`- Permits three rotational degree of freedom and one translational degree of freedom along the x-axis.