Euler–Bernoulli Beam

This model is an extended version of an Euler–Bernoulli Beam element. In addition to the Euler–Bernoulli Beam element, axial and twisting deformation is also taken into account. Background theory can be found in [1].

The axial direction of the beam is by default in the z direction, i.e. [3]-direction. See the following figure.

Figure 1: Euler–Bernouilli beam element.

The forces to deformation relation can be written as:

where the mass matrix M can be written as:

  

the damping matrix C can be written as:

the stiffness matrix K can be written as:

The load vector f, which describes loads and moments in the two directions that are not the axial for the beam (i.e. x and y axes) can be written as:

The deformation vector q can be written as:

where f is forces, t is the bending moment, r is displacement, and θ is the rotation angle. The index a refers to frame_a, and b refers to frame_b. The x, y and z directions correspond to the Modelica notation [1], [2] and [3], respectively. For instance, f_ax is the force in direction [1] on flange a and t_by is the bending moment around axis [2] on flange b.

The velocities and accelerations become:

In addition to the Euler–Bernoulli Beam element axial, and twisting deformation is also taken into account. The axial deformation becomes:

Parameters:

• length = Length of beam
• diameter = Outer diameter of beam
• innerDiameter =  Inner diameter of beam
• density = Density of beam material
• Emod = Young’s modulus
• G = Element material modulus of rigidity (Shear modulus)
• alpha = Rayleigh constant, i.e. material damping proportional to the stiffness matrix
• enforceStates_a = true, if absolute variables of frame_a shall be used as states (StateSelect.always)
• enforceStates_b = true, if absolute variables of frame_b shall be used as states (StateSelect.always)
• resolveInFrame = RotatingMachinery.Components.Shafts.Components.Types.ResolveInFrame.frame_a, if variables of the beam shall be resolved in frame_a

A vibration due to material damping is frequency dependent, i.e. alpha needs to be adjusted depending which mode is studied.

As a rule of thumb, enforceStates_a is false for all elements and  enforceStates_b is true for all elements until it is rooted, i.e. normally the last element. Several examples describe this; see RotatingMachinery.Examples.Shafts. 

Normally, the default value for resolveInFrame is sufficient. If the model is built from "left to right", i.e. from frame_a to frame_b, this value should be the default. If a beam is connected only in frame_b, this value might have to be changed to frame_b instead.

References

[1] Wikipedia. "Euler-Bernoulli Beam." https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory.

[2] Adams, M. L . Rotating Machinery Vibration: From Analysis to Troubleshooting (2nd ed.). CRC Press, 2010.

animation	Value: true Type: Boolean Description: =false, if the animation is disabled
shapeType	Value: "pipecylinder" Type: ShapeType Description: Shape of beam
length	Value: Type: Length (m) Description: Length vector from frame_a to frame_b, resolved resolveInFrame
diameter	Value: Type: Diameter (m) Description: Outer radius of beam
innerDiameter	Value: 0 Type: Diameter (m) Description: Inner radius of beam
density	Value: 7850 Type: Density (kg/m³) Description: Density of beam material
Emod	Value: 210000000000.0 Type: ModulusOfElasticity (Pa) Description: Young's Modulus
G	Value: 80000000000.0 Type: ShearModulus (Pa) Description: Element material modulus of rigidity (Shear modulus)
alpha	Value: 1 / 1000 Type: Real Description: Rayleigh constant
enforceStates_a	Value: false Type: Boolean Description: = true, if absolute variables of frame_a shall be used as states (StateSelect.always)
enforceStates_b	Value: true Type: Boolean Description: = true, if absolute variables of frame_b shall be used as states (StateSelect.always)
resolveInFrame	Value: Shafts.Types.ResolveInFrame.frame_a Type: ResolveInFrame Description: Enumeration to define the frame in which the beam equations are resolved (frame_a, frame_b)
theta_start	Value: if theta_b_Fixed then theta_b_start else theta_a_start Type: Angle[3] (rad)
r_a_Fixed	Value: false Type: Boolean Description: = true, if theta_a_start are used as initial values, else as guess values
r_a_start	Value: {0, 0, 0} Type: Position[3] (m) Description: Initial values of position
r_b_Fixed	Value: false Type: Boolean Description: = true, if theta_b_start are used as initial values, else as guess values
r_b_start	Value: {0, 0, 0} Type: Position[3] (m) Description: Initial values of position
v_a_Fixed	Value: false Type: Boolean Description: = true, if v_a_start are used as initial values, else as guess values
v_a_start	Value: {0, 0, 0} Type: Velocity[3] (m/s) Description: Initial values of velocity
v_b_Fixed	Value: false Type: Boolean Description: = true, if v_b_start are used as initial values, else as guess values
v_b_start	Value: {0, 0, 0} Type: Velocity[3] (m/s) Description: Initial values of velocity
theta_a_Fixed	Value: false Type: Boolean Description: = true, if theta_a_start are used as initial values, else as guess values
theta_a_start	Value: {0, 0, 0} Type: Angle[3] (rad) Description: Initial values of angles
theta_b_Fixed	Value: false Type: Boolean Description: = true, if theta_b_start are used as initial values, else as guess values
theta_b_start	Value: {0, 0, 0} Type: Angle[3] (rad) Description: Initial values of angles
thetad_a_Fixed	Value: false Type: Boolean Description: = true, if der(theta_a_start) are used as initial values, else as guess values
thetad_a_start	Value: {0, 0, 0} Type: AngularVelocity[3] (rad/s) Description: Initial values of angles
thetad_b_Fixed	Value: false Type: Boolean Description: = true, if der(theta_b_start) are used as initial values, else as guess values
thetad_b_start	Value: {0, 0, 0} Type: AngularVelocity[3] (rad/s) Description: Initial values of angles