Example 26 - Isotropic System

The rotor system shown in Figure 6.18 consists of a shaft 1.5 m long with a 50 mm diameter supported by bearings at each end. Disks are mounted on the shaft at one-third and two-third spans. Each disk is 70 mm thick and the left and right disks are 280 and 350 mm in diameter, respectively. The shaft and disks have material properties E — 211 GPa, G = 81.1 GPa, and p — 7,810 kg/m3. Determine the response of the system at the disks due to an out-of-balance on the left disk of 0.001 kgm, if each bearing has a stiffness of 1 MN/m and a damping of 100 Ns/m in both the x and y directions. The natural frequencies and mode shapes for this rotor system are calculated.

This example is based on Example 6.3.1.a page 253 from [Friswell, 2010].

import ross as rs
import plotly.graph_objects as go
import plotly.io as pio
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import ScalarFormatter

# 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=81.1e9, rho=7810)

Defining bearings, shaft and disk elements and Creating isotropic rotor

N = 6
L = 1.5 / N
idl = 0
odl = 0.050  # shaft diameter

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=100, cyy=100),
    rs.BearingElement(n=6, kxx=1e6, kyy=1e6, cxx=100, cyy=100),
]
disks = [
    rs.DiskElement.from_geometry(
        n=N / 3, material=steel, width=0.070, i_d=odl, o_d=0.280, scale_factor="mass"
    ),
    rs.DiskElement.from_geometry(
        n=2 * N / 3,
        material=steel,
        width=0.070,
        i_d=odl,
        o_d=0.350,
        scale_factor="mass",
    ),
]

rotor = rs.Rotor(shaft_elements=shaft, disk_elements=disks, bearing_elements=bearings)
rotor.plot_rotor(width=750, height=500)

Plotting the Campbell Diagram

campbell = rotor.run_campbell(
    speed_range=Q_(np.linspace(0, 6500, 65), "RPM"), frequencies=7
)

fig = campbell.plot(frequency_units="RPM", width=600, height=600)
for i in fig.data:
    try:
        i["y"] = i["y"] / 60
    except:
        pass
fig.update_yaxes(title="Natural Frequencies (Hz)", range=[0, 150])
fig.update_xaxes(title="Rotor Spin Speed (rpm)", range=[0, 6500])

Plotting the Mode Shapes and Damped Natural Frequencies

speed = Q_(3000, "RPM")
modal = rotor.run_modal(speed, num_modes=14)

for i in range(7):
    modal.plot_mode_3d(
        mode=i, frequency_units="Hz", damping_parameter="damping_ratio"
    ).show()

Creating the out-of-balancing

n1 = 2  # out-of-balancing is positioned at the left disk
m1 = 0.001  # amount of out-of-balancing expressed in kg*m
p1 = 0  # ou-of-balancing mass phase position
speed = Q_(np.linspace(0, 4500, 2451), "RPM")
results_case1 = rotor.run_unbalance_response([n1], [m1], [p1], speed)

probe1 = (2, 0)  # response in x axis (0 degrees) at the left disk of rotor (node = 2)
probe2 = (
    2,
    Q_(90, "deg"),
)  # response in y axis (90 degrees) at the left disk of rotor (node = 2)
fig = results_case1.plot(
    probe=[probe1, probe2],
    probe_units="degrees",
    frequency_units="RPM",
    amplitude_units="µm pkpk",
    phase_units="degrees",
)
# changing to log scale
fig.update_layout(
    yaxis=dict(
        type="log",
    )
)

probe1 = (4, 0)  # response in x axis (0 degrees) at the right disk of rotor (node = 4)
probe2 = (
    4,
    Q_(90, "deg"),
)  # response in y axis (90 degrees) at the right disk of rotor (node = 4)
fig = results_case1.plot(
    probe=[probe1, probe2],
    probe_units="degrees",
    frequency_units="RPM",
    amplitude_units="µm pkpk",
    phase_units="degrees",
)

# changing to log scale
fig.update_layout(
    yaxis=dict(
        # title='Amplitude (µm pkpk)',
        type="log",
        # range=[-9, -1]  # log range: 10^-9, 10^-1
    )
)

Plotting the Orbit using the Plotly library

#### Building the orbit at 496 RPM, 1346 RPM and 2596 RPM for nodes located at
#### the right and left disks (node=2 and node=4)

At 496 RPM

speed = 496 * (2 * np.pi / 60)  # Q_(496, "RPM")
time_samples = 1000001
node = 2  # out-of-balancing position
t = np.linspace(0, 43, time_samples)

# Creating the out-of-balancing force input matrix
F = np.zeros((time_samples, rotor.ndof))

# harmonic force component on x axis
F[:, rotor.number_dof * node + 0] = (
    m1 * speed**2 * np.cos(speed * t)
)  # as out-of-balancing is a harmonic force

# harmonic force component on y axis
F[:, rotor.number_dof * node + 1] = (
    m1 * speed**2 * np.sin(speed * t)
)  # as out-of-balancing is a harmonic force

# Using the ROSS’ method to calculate displacements due a force in time domain: run_time_response().

response3 = rotor.run_time_response(speed, F, t)

# Editing the ross plots in order to explicit the orbit whirl in node 2

orb2 = response3.plot_2d(node=2, width=500, height=500)
cutoff = int(1000 * 2.7)
x_new2 = orb2.data[0].x[-cutoff:]
y_new2 = orb2.data[0].y[-cutoff:]

# Inserting the orbit starting point
starting_point2 = go.Scatter(
    x=[x_new2[0]],
    y=[y_new2[0]],
    marker={"size": 10, "color": "orange"},
    showlegend=False,
    name="Starting Point2",
)

# Inserting the orbit of node 2
orb2_curve = go.Scatter(
    x=x_new2,
    y=y_new2,
    mode="lines",
    name="orb2",
    showlegend=False,
    line=dict(color="orange"),
)

# Editing the ross plots in order to explicit the orbit whirl in node 4

orb4 = response3.plot_2d(node=4, width=500, height=500)

x_new4 = orb4.data[0].x[-cutoff:]
y_new4 = orb4.data[0].y[-cutoff:]

# Inserting the orbit starting point
starting_point4 = go.Scatter(
    x=[x_new4[0]],
    y=[y_new4[0]],
    marker={"size": 10, "color": "#636EFA"},
    showlegend=False,
    name="Starting Point4",
)

# Proper scaling x and y axis
max_amp = max(np.max(x_new2), np.max(y_new2), np.max(x_new4), np.max(y_new4))

# Merging orbit at node 2 and node 4
orb4.update_xaxes(range=[-1.2 * max_amp, 1.2 * max_amp])
orb4.update_yaxes(range=[-1.2 * max_amp, 1.2 * max_amp])
orb4.update_traces(x=x_new4, y=y_new4, name="orb4")
orb4.update_layout(title="Response at node 2 and 4 at 496 RPM")
orb4.add_trace(starting_point4)
orb4.add_trace(starting_point2)
orb4.add_trace(orb2_curve)
orb4

At 1346 RPM

speed = 1346 * (2 * np.pi / 60)  # Q_(496, "RPM")
time_samples = 1000001
node = 2  # out-of-balancing position
t = np.linspace(0, 43, time_samples)

# Creating the out-of-balancing force input matrix
F = np.zeros((time_samples, rotor.ndof))

# harmonic force component on x axis
F[:, rotor.number_dof * node + 0] = (
    m1 * speed**2 * np.cos(speed * t)
)  # as out-of-balancing is a harmonic force

# harmonic force component on y axis
F[:, rotor.number_dof * node + 1] = (
    m1 * speed**2 * np.sin(speed * t)
)  # as out-of-balancing is a harmonic force

# Using the ROSS’ method to calculate displacements due a force in time domain: run_time_response().

response3 = rotor.run_time_response(speed, F, t)

# Editing the ross plots in order to explicit the orbit whirl in node 2

orb2 = response3.plot_2d(node=2, width=500, height=500)
cutoff = int(1000 * 1)
x_new2 = orb2.data[0].x[-cutoff:]
y_new2 = orb2.data[0].y[-cutoff:]

# Inserting the orbit starting point
starting_point2 = go.Scatter(
    x=[x_new2[0]],
    y=[y_new2[0]],
    marker={"size": 10, "color": "orange"},
    showlegend=False,
    name="Starting Point2",
)

# Inserting the orbit of node 2
orb2_curve = go.Scatter(
    x=x_new2,
    y=y_new2,
    mode="lines",
    name="orb2",
    showlegend=False,
    line=dict(color="orange"),
)

# Editing the ross plots in order to explicit the orbit whirl in node 4

orb4 = response3.plot_2d(node=4, width=500, height=500)

x_new4 = orb4.data[0].x[-cutoff:]
y_new4 = orb4.data[0].y[-cutoff:]

# Inserting the orbit starting point
starting_point4 = go.Scatter(
    x=[x_new4[0]],
    y=[y_new4[0]],
    marker={"size": 10, "color": "#636EFA"},
    showlegend=False,
    name="Starting Point4",
)

# Proper scaling x and y axis
max_amp = max(np.max(x_new2), np.max(y_new2), np.max(x_new4), np.max(y_new4))

# Merging orbit at node 2 and node 4
orb4.update_xaxes(range=[-1.2 * max_amp, 1.2 * max_amp])
orb4.update_yaxes(range=[-1.2 * max_amp, 1.2 * max_amp])
orb4.update_traces(x=x_new4, y=y_new4, name="orb4")
orb4.update_layout(title="Response at node 2 and 4 at 1346 RPM")
orb4.add_trace(starting_point4)
orb4.add_trace(starting_point2)
orb4.add_trace(orb2_curve)
orb4

At 2596 RPM

speed = 2596 * (2 * np.pi / 60)  # Q_(496, "RPM")
time_samples = 1000001
node = 2  # out-of-balancing position
t = np.linspace(0, 43, time_samples)

# Creating the out-of-balancing force input matrix
F = np.zeros((time_samples, rotor.ndof))

# harmonic force component on x axis
F[:, rotor.number_dof * node + 0] = (
    m1 * speed**2 * np.cos(speed * t)
)  # as out-of-balancing is a harmonic force

# harmonic force component on y axis
F[:, rotor.number_dof * node + 1] = (
    m1 * speed**2 * np.sin(speed * t)
)  # as out-of-balancing is a harmonic force

# Using the ROSS’ method to calculate displacements due a force in time domain: run_time_response().

response3 = rotor.run_time_response(speed, F, t)

# Editing the ross plots in order to explicit the orbit whirl in node 2

orb2 = response3.plot_2d(node=2, width=500, height=500)
cutoff = int(1000 * 0.52)
x_new2 = orb2.data[0].x[-cutoff:]
y_new2 = orb2.data[0].y[-cutoff:]

# Inserting the orbit starting point
starting_point2 = go.Scatter(
    x=[x_new2[0]],
    y=[y_new2[0]],
    marker={"size": 10, "color": "orange"},
    showlegend=False,
    name="Starting Point2",
)

# Inserting the orbit of node 2
orb2_curve = go.Scatter(
    x=x_new2,
    y=y_new2,
    mode="lines",
    name="orb2",
    showlegend=False,
    line=dict(color="orange"),
)

# Editing the ross plots in order to explicit the orbit whirl in node 4

orb4 = response3.plot_2d(node=4, width=500, height=500)

x_new4 = orb4.data[0].x[-cutoff:]
y_new4 = orb4.data[0].y[-cutoff:]

# Inserting the orbit starting point
starting_point4 = go.Scatter(
    x=[x_new4[0]],
    y=[y_new4[0]],
    marker={"size": 10, "color": "#636EFA"},
    showlegend=False,
    name="Starting Point4",
)

# Proper scaling x and y axis
max_amp = max(np.max(x_new2), np.max(y_new2), np.max(x_new4), np.max(y_new4))

# Merging orbit at node 2 and node 4
orb4.update_xaxes(range=[-1.2 * max_amp, 1.2 * max_amp])
orb4.update_yaxes(range=[-1.2 * max_amp, 1.2 * max_amp])
orb4.update_traces(x=x_new4, y=y_new4, name="orb4")
orb4.update_layout(title="Response at node 2 and 4 at 2596 RPM")
orb4.add_trace(starting_point4)
orb4.add_trace(starting_point2)
orb4.add_trace(orb2_curve)
orb4

Plotting the Orbit using the Matplotlib library

#### Building the orbit at 496 RPM, 1346 RPM and 2596 RPM for nodes located at
#### the right and left disks (node=2 and node=4)

####  ORBIT AT 496 RPM

speed = 496 * (2 * np.pi / 60)  # Q_(496, "RPM")
time_samples = 1000001
node = 2  # out-of-balancing position
t = np.linspace(0, 43, time_samples)

F = np.zeros((time_samples, rotor.ndof))

# Creating the out-of-balancing force input matrix

# harmonic force component on x axis
F[:, rotor.number_dof * node + 0] = (
    m1 * speed**2 * np.cos(speed * t)
)  # as out-of-balancing is a harmonic force

# harmonic force component on y axis
F[:, rotor.number_dof * node + 1] = (
    m1 * speed**2 * np.sin(speed * t)
)  # as out-of-balancing is a harmonic force

# Using the ROSS’ method to calculate displacements due a force in time domain: run_time_response().

response3 = rotor.run_time_response(speed, F, t)

# Extracting the response vector throught attribute (.yout)  in x and y axis in both nodes 2 and 4

node_response = 2
x1_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 0]
y1_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 1]
t_vector = response3.t

node_response = 4
x2_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 0]
y2_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 1]

####  ORBIT AT 1346 RPM

speed = 1346 * (2 * np.pi / 60)  # Q_(496, "RPM")
time_samples = 1000001
node = 2  # out-of-balancing position
t = np.linspace(0, 43, time_samples)

F = np.zeros((time_samples, rotor.ndof))

# Creating the out-of-balancing force input matrix

# harmonic force component on x axis
F[:, rotor.number_dof * node + 0] = (
    m1 * speed**2 * np.cos(speed * t)
)  # as out-of-balancing is a harmonic force

# harmonic force component on y axis
F[:, rotor.number_dof * node + 1] = (
    m1 * speed**2 * np.sin(speed * t)
)  # as out-of-balancing is a harmonic force

# Using the ROSS’ method to calculate displacements due a force in time domain: run_time_response().

response3 = rotor.run_time_response(speed, F, t)

# Extracting the response vector throught attribute (.yout)  in x and y axis in both nodes 2 and 4

node_response = 2
x3_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 0]
y3_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 1]


node_response = 4
x4_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 0]
y4_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 1]

###### ORBIT AT 2596 RPM

speed = 2596 * (2 * np.pi / 60)  # Q_(496, "RPM")
time_samples = 1000001
node = 2  # out-of-balancing position
t = np.linspace(0, 43, time_samples)

F = np.zeros((time_samples, rotor.ndof))

# Creating the out-of-balancing force input matrix

# harmonic force component on x axis
F[:, rotor.number_dof * node + 0] = (
    m1 * speed**2 * np.cos(speed * t)
)  # as out-of-balancing is a harmonic force

# harmonic force component on y axis
F[:, rotor.number_dof * node + 1] = (
    m1 * speed**2 * np.sin(speed * t)
)  # as out-of-balancing is a harmonic force

# Using the ROSS’ method to calculate displacements due a force in time domain: run_time_response().

response3 = rotor.run_time_response(speed, F, t)

# Extracting the response vector throught attribute (.yout)  in x and y axis in both nodes 2 and 4

node_response = 2
x5_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 0]
y5_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 1]


node_response = 4
x6_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 0]
y6_axis_displacement = response3.yout[:, rotor.number_dof * node_response + 1]

##### PLOTTING THE ORBITS

fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(13, 4))

# Scaling the graphs in multiples of 1e-6

formatter = ScalarFormatter(useMathText=True)
formatter.set_powerlimits((-6, -6))

# Choosing a proper cutoff the avoiding transient data and build an almost 360 degrees of the orbit
cutoff = -int(time_samples * 0.0027)

# Plot orbit data at 496 RPM
ax1.plot(x1_axis_displacement[cutoff:], y1_axis_displacement[cutoff:], label="Orbit")
ax1.plot(
    x1_axis_displacement[cutoff:][0],
    y1_axis_displacement[cutoff:][0],
    "o",
    markersize=6,
    color="#636EFA",
)
ax1.plot(x2_axis_displacement[cutoff:], y2_axis_displacement[cutoff:], label="Orbit")
ax1.plot(
    x2_axis_displacement[cutoff:][0],
    y2_axis_displacement[cutoff:][0],
    "o",
    markersize=6,
    color="#636EFA",
)
ax1.set_title("496 RPM")
ax1.xaxis.set_major_formatter(formatter)
ax1.yaxis.set_major_formatter(formatter)

cutoff = -int(time_samples * 0.001)

# Plot orbit data at 1346 RPM
ax2.plot(x3_axis_displacement[cutoff:], y3_axis_displacement[cutoff:], label="Orbit")
ax2.plot(
    x3_axis_displacement[cutoff:][0],
    y3_axis_displacement[cutoff:][0],
    "o",
    markersize=10,
    color="#636EFA",
)
ax2.plot(x4_axis_displacement[cutoff:], y4_axis_displacement[cutoff:], label="Orbit")
ax2.plot(
    x4_axis_displacement[cutoff:][0],
    y4_axis_displacement[cutoff:][0],
    "o",
    markersize=10,
    color="#636EFA",
)
ax2.set_title("1346 RPM")
ax2.xaxis.set_major_formatter(formatter)
ax2.yaxis.set_major_formatter(formatter)

cutoff = -int(time_samples * 0.00052)

# Plot orbit data at 2596 RPM
ax3.plot(x5_axis_displacement[cutoff:], y5_axis_displacement[cutoff:], label="Orbit")
ax3.plot(
    x5_axis_displacement[cutoff:][0],
    y5_axis_displacement[cutoff:][0],
    "o",
    markersize=10,
    color="#636EFA",
)
ax3.plot(x6_axis_displacement[cutoff:], y6_axis_displacement[cutoff:], label="Orbit")
ax3.plot(
    x6_axis_displacement[cutoff:][0],
    y6_axis_displacement[cutoff:][0],
    "o",
    markersize=10,
    color="#636EFA",
)
ax3.set_title("2596 RPM")
ax3.xaxis.set_major_formatter(formatter)
ax3.yaxis.set_major_formatter(formatter)

REFERENCES

[1] M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees, Dynamics of Rotating Machines. Cambridge: Cambridge University Press, 2010.