# 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.

## 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.