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.




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.

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

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.

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

$b_{b} = factor_{elastic} \cdot E \cdot Ia_{beam} / l_{element}$

$b_{e} = factor_{elastic} \cdot E \cdot A_{beam} / l_{element}$

$b_{t} = factor_{shear} \cdot G \cdot J_{beam} / l_{element}$


$b_{b}, b_{e}, b_{t}$ = Damping coefficient for bending, elongation, and torsion

$Ia_{beam}$ = Area moment of inertia

$A_{beam}$ = Cross sectional area of beam

$J_{beam}$ = Torsional constant for beam

$E$ = Modulus of Elasticity

$G$ = Shear Modulus

$l_{element}$ = 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.

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.