Example 31 - Isotropic Bearings with Damping

Example 31 - Isotropic Bearings with Damping#

This example is based on Example 5.9.5 from {cite}friswell2010dynamics

The isotropic bearing Example 5.9.1 (Example 17) is repeated but with damping in the bearings. The, x and y directions are uncoupled, with a translational stiffness of 1 MN/m and a damping of 3 kNs/m in each direction.

import ross as rs
import numpy as np
import pandas as pd

from ross.units import 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, kyy=1e6, cxx=3e3, cyy=3e3, scale_factor=2),
    rs.BearingElement(n=len(shaft), kxx=1e6, kyy=1e6, cxx=3e3, cyy=3e3, 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()

Plotting the Campbell Diagram#

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

campbell.plot(frequency_units="RPM", frequencies=8)

Plotting the Mode Shapes and Damped Natural Frequencies#

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

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

Creating table with eigenvalues, natural frequencies and damping ratios#

def modal_table(modal_results, rpm):
    eig = modal_results.evalues
    omega_n = np.abs(eig)
    omega_d = np.imag(eig)
    zeta = -np.real(eig) / omega_n
    fn = omega_n / (2 * np.pi)
    fd = omega_d / (2 * np.pi)

    return pd.DataFrame(
        {
            "Speed (RPM)": [rpm] * 8,
            "Root s (rad/s)": eig[:8],
            "Wn (Hz)": fn[:8],
            "Wd (Hz)": fd[:8],
            "Damping Ratio": zeta[:8],
        }
    )


modal_0 = rotor.run_modal(speed=Q_(0, "RPM"))
modal_4000 = rotor.run_modal(speed=Q_(4000, "RPM"))

df_modal_0 = modal_table(modal_0, rpm=0)
df_modal_4000 = modal_table(modal_4000, rpm=4000)
df_combined = pd.concat([df_modal_0, df_modal_4000], ignore_index=True)

display(df_combined)
Speed (RPM) Root s (rad/s) Wn (Hz) Wd (Hz) Damping Ratio (%)
0 0 -4.423835+ 87.257936j 13.905368 13.887532 5.063333e+00
1 0 -4.423835+ 87.257936j 13.905368 13.887532 5.063333e+00
2 0 -78.240826+292.422766j 48.177624 46.540529 2.584688e+01
3 0 -78.240826+292.422766j 48.177624 46.540529 2.584688e+01
4 0 -566.540378+648.581404j 137.060578 103.224936 6.578675e+01
5 0 -566.540378+648.581404j 137.060578 103.224936 6.578675e+01
6 0 0.000000+650.407866j 103.515627 103.515627 -5.070025e-13
7 0 -657.308076+834.757536j 169.099715 132.855788 6.186517e+01
8 4000 -4.083081+ 85.965810j 13.697307 13.681884 4.744309e+00
9 4000 -4.741740+ 88.414706j 14.091860 14.071637 5.355371e+00
10 4000 -74.101769+263.786351j 43.607964 41.982902 2.704475e+01
11 4000 -78.810422+318.235036j 52.178705 50.648679 2.403867e+01
12 4000 -0.000000+650.407866j 103.515627 103.515627 1.019396e-11
13 4000 -402.590859+655.012043j 122.365228 104.248404 5.236318e+01
14 4000 -667.237536+663.899669j 149.806039 105.662914 7.088776e+01
15 4000 -694.666683+818.186345j 170.822321 130.218401 6.472201e+01
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