Example 17 - Features of Eigenvalues and Eigenvectors - Isotropic Bearings

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Example 17 - Features of Eigenvalues and Eigenvectors - Isotropic Bearings#

This example is based on Example 5.9.1 from [Friswell, 2010].

Isotropic Bearings. A 1.5-m-long shaft, shown in Figure 5.27, has a diameter of 0.05 m. The disks are keyed to the shaft at 0.5 and 1 m from one end. The left disk is 0.07 m thick with a diameter of 0.28 m; the right disk is 0.07 m thick with a diameter of 0.35 m. For the shaft, E = 211 GN/m2 and G = 81.2 GN/m2. There is no internal shaft damping. For both the shaft and the disks, p = 7,810 kg/m3. The shaft is supported by identical bearings at its ends.Constant

These bearings are isotropic and have a stiffness of 1 MN/m in both the x and y directions. The bearings contribute no additional stiffness to the rotational degrees of freedom and there is no damping or cross-coupling in the bearings. Create an FE model of the shaft using six Timoshenko beam elements and investigate the dynamics of the machine at 0 and 4,000 rev/min.

import ross as rs
import numpy as np
import plotly.graph_objects as go
from IPython.display import display

# Make sure the default renderer is set to 'notebook' for inline plots in Jupyter
import plotly.io as pio

pio.renderers.default = "notebook"

Q_ = rs.Q_

steel = rs.Material("steel", E=211e9, G_s=81.2e9, rho=7810)

L = 0.25
N = 6
idl = 0
odl = 0.05

shaft = [rs.ShaftElement(L=L, idl=idl, odl=odl, material=steel) for i in range(N)]
bearings = [
    rs.BearingElement(n=0, kxx=1e6, cxx=0, scale_factor=2),
    rs.BearingElement(n=len(shaft), kxx=1e6, cxx=0, scale_factor=2),
]
disks = [
    rs.DiskElement.from_geometry(
        n=2, material=steel, width=0.07, i_d=odl, o_d=0.28, scale_factor="mass"
    ),
    rs.DiskElement.from_geometry(
        n=4, material=steel, width=0.07, i_d=odl, o_d=0.35, scale_factor="mass"
    ),
]

rotor = rs.Rotor(shaft_elements=shaft, disk_elements=disks, bearing_elements=bearings)
rotor.plot_rotor()

campbell = rotor.run_campbell(
    speed_range=Q_(list(range(0, 4500, 50)), "RPM"), frequencies=7
)

campbell.plot(frequency_units="RPM")

modal = rotor.run_modal(speed=Q_(4000, "RPM"), num_modes=14)

for mode in range(7):
    display(modal.plot_mode_3d(mode, frequency_units="Hz"))

for mode in range(7):
    display(modal.plot_orbit(mode, nodes=[2, 4]))

References#

[Fri10]

Michael I Friswell. Dynamics of rotating machines. Cambridge University Press, 2010.

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