Tutorial - Faults#
In this tutorial, we present examples of fault modeling in rotor systems using ROSS. Different fault conditions — such as misalignment, rotor–stator rubbing, and transverse cracks — are introduced into the model to illustrate their influence on the rotor’s dynamic response.
Section 1: Rotor model#
The examples in this tutorial are based on a finite element (FE) model of the rotor test rig available at LMEst - UFU. The physical system comprises a flexible steel shaft (\(0.914\) m long and \(0.019\) m in diameter, modeled with Timoshenko beam elements), two rigid steel discs (\(0.150\) m in diameter and \(0.020\) m thick), and two self-aligning ball bearings.
To ensure the numerical simulations accurately reflect the physical system, a model updating procedure was performed. Specifically, a heuristic optimization technique (Differential Evolution) was used to identify the model’s unknown parameters: the stiffness and damping coefficients of the bearings, as well as the proportional damping.
import numpy as np
import ross as rs
from ross.faults import *
from ross.units import Q_
from ross.probe import Probe
# Make sure the default renderer is set to 'notebook' for inline plots in Jupyter
import plotly.io as pio
pio.renderers.default = "notebook"
"""Create example rotor with given number of elements."""
steel2 = rs.Material(name="Steel", rho=7850, E=2.17e11, G_s=81.2e9)
# Rotor with 6 DoFs, with internal damping, with 10 shaft elements, 2 disks and 2 bearings.
i_d = 0
o_d = 0.019
n = 33
# fmt: off
L = np.array(
[0 , 25, 64, 104, 124, 143, 175, 207, 239, 271,
303, 335, 345, 355, 380, 408, 436, 466, 496, 526,
556, 586, 614, 647, 657, 667, 702, 737, 772, 807,
842, 862, 881, 914]
)/ 1000
# fmt: on
L = [L[i] - L[i - 1] for i in range(1, len(L))]
shaft_elem = [
rs.ShaftElement(
material=steel2,
L=l,
idl=i_d,
odl=o_d,
idr=i_d,
odr=o_d,
alpha=8.0501,
beta=1.0e-5,
rotary_inertia=True,
shear_effects=True,
)
for l in L
]
Id = 0.003844540885417
Ip = 0.007513248437500
disk0 = rs.DiskElement(n=12, m=2.6375, Id=Id, Ip=Ip)
disk1 = rs.DiskElement(n=24, m=2.6375, Id=Id, Ip=Ip)
kxx1 = 4.40e5
kyy1 = 4.6114e5
kzz = 0
cxx1 = 27.4
cyy1 = 2.505
czz = 0
kxx2 = 9.50e5
kyy2 = 1.09e8
cxx2 = 50.4
cyy2 = 100.4553
bearing0 = rs.BearingElement(
n=4, kxx=kxx1, kyy=kyy1, cxx=cxx1, cyy=cyy1, kzz=kzz, czz=czz
)
bearing1 = rs.BearingElement(
n=31, kxx=kxx2, kyy=kyy2, cxx=cxx2, cyy=cyy2, kzz=kzz, czz=czz
)
rotor = rs.Rotor(shaft_elem, [disk0, disk1], [bearing0, bearing1])
"""Inserting a mass and phase unbalance and defining the local response."""
mass_unb = [3e-4, 0]
phase_unb = [-np.pi / 2, 0]
node_unb = [12, 24]
probe1 = Probe(14, 0)
probe2 = Probe(22, 0)
Section 2: Misalignment#
In practical terms within mechanics and rotordynamics, misalignment is a geometric condition that occurs when the centerlines of two coupled shafts do not perfectly coincide during the machine’s normal operation.
You can assign three different string values to mis_type, each representing a specific mechanical condition as show in Figure 1 ([Desouki et al., 2020]):
a) "parallel": Simulates a pure radial offset. The centerlines of the two connected shafts remain strictly parallel but are displaced by a certain distance (defined by mis_distance_x and mis_distance_y).
b) "angular": Simulates a pure angular intersection. The centerlines intersect at the coupling center at a specific angle (defined by mis_angle), with no radial offset.
c) "combined": Applies both parallel and angular misalignments simultaneously, requiring you to input both the translation distances and the intersection angle.
Figure 1: Types of misalignment. a) Parallel Misalignment b) Angular Misalignment c) Combined Misalignment.
Example 2.1: Rotor connected by a flexible mechanical coupling with parallel, angular or combined misalignment.#
This example uses the rotor described in Section 1 to demonstrate the effects of a flexible coupling misalignment fault within the system.
"""Calculate the response of the rotor connected by a flexible mechanical coupling with parallel,
angular or combined misalignment."""
results_misalignment_flexible = rotor.run_misalignment(
coupling="flex",
mis_type="parallel", # change here the type: mis_type="angular" or mis_type="combined"
radial_stiffness=40e3,
bending_stiffness=38e3,
mis_distance_x=2e-4,
mis_distance_y=2e-4,
mis_angle=5 * np.pi / 180,
input_torque=0,
load_torque=0,
n=0,
speed=Q_(1200, "RPM"),
node=node_unb,
unbalance_magnitude=mass_unb,
unbalance_phase=phase_unb,
t=np.arange(0, 5, 0.0001),
model_reduction={"num_modes": 12}, # Pseudo-modal method, optional
)
"""Plots the time response for the two given probes."""
results_misalignment_flexible.plot_1d([probe1, probe2]).show()
"""Plots the frequency response for the two given probes."""
results_misalignment_flexible.plot_dfft(
[probe1, probe2], frequency_range=Q_((0, 100), "Hz"), yaxis_type="log"
).show()
Example 2.2: Rotor connected by a rigid mechanical coupling with parallel misalignment.#
In this example, we use the rotor introduced in Section 1 to demonstrate the results obtained when a rigid coupling misalignment fault is present in the system.
"""Calculate the response of the rotor connected by a rigid mechanical coupling with parallel misalignment."""
results_misalignment_rigid = rotor.run_misalignment(
coupling="rigid", # ADICIONAR VARIAVEL AQUI EXPLICANDO PA MIAU
mis_distance=2e-4,
input_torque=0,
load_torque=0,
n=0,
speed=Q_(1200, "RPM"),
node=node_unb,
unbalance_magnitude=mass_unb,
unbalance_phase=phase_unb,
t=np.arange(0, 5, 0.0001),
model_reduction={"num_modes": 12}, # Pseudo-modal method, optional
)
Running with model reduction: pseudomodal
"""Plots the time response for the two given probes."""
results_misalignment_rigid.plot_1d([probe1, probe2]).show()
"""Plots the frequency response for the two given probes."""
results_misalignment_rigid.plot_dfft(
[probe1, probe2], frequency_range=Q_((0, 100), "Hz"), yaxis_type="log"
).show()
Section 3: Rubbing#
Rubbing is a severe, non-linear dynamic fault in rotating machinery. It happens when the rotor’s vibration amplitude exceeds the physical radial clearance, causing the rotating shaft or blades to impact the stationary casing (stator).
Here, the run_rubbing method simulates this fault in the time domain. At each integration step, it checks if the nodal displacement is greater than the clearance. If true, it applies the stator stiffness (k_stator) and dynamic friction (mu).
Example 3.1 Impact phenomena in rotor-casing dynamical systems#
This example uses the rotor described in Section 1 to demonstrate the effects of vibrations induced by contact between the rotor and a stationary component, such as a casing or stator.
"""Calculate the response when a rotor makes contact with a casing or stator."""
results_rubbing = rotor.run_rubbing(
distance=7.95e-5,
contact_stiffness=1.1e6,
contact_damping=40,
friction_coeff=0.3,
n=12,
speed=Q_(1200, "RPM"),
node=node_unb,
unbalance_magnitude=mass_unb,
unbalance_phase=phase_unb,
t=np.arange(0, 5, 0.0001),
model_reduction={"num_modes": 12}, # Pseudo-modal method, optional
)
Running with model reduction: pseudomodal
"""Plots the time response for the two given probes."""
results_rubbing.plot_1d([probe1, probe2]).show()
"""Plots the frequency response for the two given probes."""
results_rubbing.plot_dfft(
[probe1, probe2], frequency_range=Q_((0, 100), "Hz"), yaxis_type="log"
).show()
Section 4: Cracks#
In rotating machinery, such as shafts and rotors, cracks are particularly critical because they can propagate under cyclic loading due to fatigue. The non-linear “breathing” behavior of the crack introduces a time-varying stiffness. This causes the rotor to vibrate not just at the synchronous speed (1X) due to unbalance, but also at higher harmonics (especially 2X and 3X). The run_crack method simulates this time-domain response using models like “Mayes” or “Gasch” to represent the stiffness transition.
Example 4.1 Evaluation of the influence of transverse cracks in a rotating shaft.#
This example uses the rotor described in Section 1 to demonstrate the effects of a transverse crack within the system. The crack’s breathing behavior is modeled following the approaches proposed by [Mayes and Davies, 1984] and [Gasch, 1993].
results_crack = rotor.run_crack(
depth_ratio=0.2,
n=18,
speed=Q_(1200, "RPM"),
crack_model="Gasch", # or crack_model="Mayes"
node=node_unb,
unbalance_magnitude=mass_unb,
unbalance_phase=phase_unb,
t=np.arange(0, 5, 0.0001),
model_reduction={"num_modes": 12}, # Pseudo-modal method, optional
)
Running with model reduction: pseudomodal
"""Plots the time response for the two given probes."""
results_crack.plot_1d([probe1, probe2]).show()
"""Plots the frequency response for the two given probes."""
results_crack.plot_dfft(
[probe1, probe2], frequency_range=Q_((0, 100), "Hz"), yaxis_type="log"
).show()
References#
Mohamed Desouki, Sadok Sassi, Jamil Renno, and Samer Gowid. Dynamic response of a rotating assembly under the coupled effects of misalignment and imbalance. Shock and Vibration, 2020:1–26, 10 2020. doi:10.1155/2020/8819676.
R Gasch. A survey of the dynamic behaviour of a simple rotating shaft with a transverse crack. Journal of sound and vibration, 160(2):313–332, 1993.
IW Mayes and WGR Davies. Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 106(1):139–145, 1984.