1. Introduction
In turbo-machines like compressors, turbines or pumps, minimization of the clearance between rotor and stator is desirable to maximize energy efficiency [
1,
2]. However, vibration of the rotor may occur due to unbalance, trapped fluids, shaft cracks, thermal bends. etc. [
3]. As a consequence, rub between stator and rotor may appear. The rub can be specified as point rub, partial rub, or full annular rub [
2]. Among these kinds of rubs, full annular rub is considered most dangerous. It may lead to effects known as forward and backward whirling. Forward whirl denotes continuous rubbing in direction of the shaft rotation, whereas during backward whirling, the rubbing occurs in the opposite direction of the shaft rotation. The stator–rotor contacts may lead to heating of the shaft and occurrence of hot spots [
1,
2]. Generally, the rubbing degrades the lifetime of the machine. It may lead to damage of the contact surface, large deformation, high frequency stress, and even to complete failure [
4,
5].
Several papers have been proposed to study the dynamics induced by stator–rotor contacts. A simple but still recent modeling approach is to consider the shaft to be linear-elastic and to model the contact with the stator based on a linear dry friction model [
6]. Although the model is simple, it becomes nonlinear in the case of contact, inducing effects such as chaos, bifurcation, whirling, and wiping [
6]. Conditions of existence for forward and backward whirls depending on system parameters have been investigated [
5]. For instance, an overview about different effects resulting from stator–rotor contacts can be found in [
1,
7]. Experimental validation of the described effects has been undertaken in [
8]. A study of the rotordynamics of more complex rotating systems with multiple rotor–stator interface locations is conducted in [
9].
Besides the linear dry friction model, different methods of contact force modeling exist: approaches considering, e.g., Hertzian contact forces [
10], contact damping [
10], and surface asperities [
11].
Estimation of rotor–stator contact forces has not been considered in the literature so far. However, to predict the remaining lifetime and for the purpose of system monitoring, estimation of the contact forces could be beneficial.
A promising strategy to prevent whirl responses is the usage of active elements such as active bearings [
3]. In [
3], a state-feedback controller is applied to stabilize the rotor displacements and mitigate effects resulting from contacts. Due to the special structure of the feedback controller, the approach does not assume the system states to be measured, but it requires all interacting forces between rotor and the surroundings to be measured. To avoid additional sensors, an estimation strategy could be applied. Another active bearing control approach based on
control design is developed in [
12]. It requires several rotations and displacements to be measured, which could be reduced by applying estimation approaches.
From the perspective of filtering, the estimation of unknown contact forces is an unknown input estimation problem. Related to linear stochastic systems, two main estimation approaches exist: the minimum-variance unbiased (MVU) filtering approach and the augmented state Kalman filtering (ASKF) approach. The MVU approaches provide unbiased state estimations with minimum error variance under the influence of unknown inputs. For this purpose, the Kitanides filter is the first MVU estimation approach that has been derived [
13]. However, it does not explicitly estimate the unknown inputs itself. Later, an MVU estimator known as Gillijns-De-Moore filter was proposed that provides explicit estimation of the unknown inputs [
14]. For this filter, it is shown that, in addition to the state estimations, the estimations of the unknown inputs are unbiased and have minimum error variance. The fundamental property of the MVU approaches is that they neither know nor assume any dynamics for the unknown inputs. This is in contrast to the ASKF approaches. In augmented state Kalman filtering, the system model is augmented by the dynamics of the unknown inputs. This was first described in [
15] for the treatment of a constant bias. However, in the case of time-varying inputs, the related dynamics of the unknown inputs are typically not known and have to be approximated by, e.g., a piece-wise constant model [
16]. The variance of the process noise related to the augmented state influences the estimation performance of the filter. It is a design parameter that is typically tuned by trial and error. In [
17], it was recently shown that a direct connection between the MVU and ASKF approaches exists. If the considered noise variance related to the augmented state is selected as infinity, then the ASKF algorithm equals the Gillijns–De-Moore filter. Consequently, the MVU approaches appear as a special case of augmented-state Kalman filtering.
In this paper, an estimation strategy to estimate states and contact forces in rotating machines is proposed. First, a linear-elastic model based on the finite-element method is derived to describe the elastic movement of the shaft. The contact with the stator is described based on a dry friction model. As a consequence, a nonlinear augmented state description is obtained that exactly describes the system dynamics in case of contact and no contact. No assumptions about the dynamics of the unknown input are required to be made. By applying a nonlinear filtering approach to the augmented state description, the displacements and bendings, as well as the contact forces can be estimated. As the augmented state description is an exact description, an optimal estimation approach like the particle filter would lead to optimal estimation results of the unknown contact force. However, due to the large number of states and the resulting computational costs, particle filtering is considered to be unapplicable. Instead, the extended Kalman filter is applied, and the related Jacobians are derived. In the conducted simulation example, contact force estimation under the hazardous backward whirl is explicitly considered.
The paper is organized as follows. The modeling of the elastic shaft and the contact force is described in
Section 2. In
Section 3 the estimation strategy to estimate the states and contact forces is proposed. The contact force estimation under forward and backward whirl is considered in the simulation study of
Section 4.
Throughout the paper denotes an identity matrix of dimension and denotes a zero matrix of dimension .
3. State and Contact Force Estimation
It is assumed that the parameters
and
of the introduced model (
17) are unknown. The contact force can be estimated by estimating the state
and the parameters
and
of (
17). By applying the contact force model (
7), the estimations of the contact force are achieved. The parameters
and
are estimated based on augmented state Kalman filtering. The discrete-time model (
17) is augmented by the dynamics of
and
leading to
where
is the augmented state vector. The parameters
and
are indeed constants such that (
18) describes the true dynamics. The vector of process noise
is introduced to account for discretization and modeling errors. The noise
is assumed to be zero-mean, white, and to have covariance
with standard deviations
and
. The augmented state model (
18) is nonlinear. Consequently, the extended Kalman filter is applied. Therefore, the Jacobian
has to be calculated, which is
with
as
and
specified as
where
and
. The switching functions
,
are assumed to be independent of
, i.e.,
.
As illustrated in
Figure 1a, the displacements of nodes two and eight are measured. The corresponding measurement model
with
describes four measurements
, i.e., the displacements of nodes two and eight in
x and
y directions. The measurement noise
is assumed to be white, zero-mean, with covariance
and standard deviation
.
The Kalman filter is applied to estimate the states of the augmented system. Quantities
,
,
,
,
denote the estimation of
,
,
,
,
. First, the estimated displacement
of node five is determined. It is checked if the known clearance
is exceeded or not, i.e.,
The a priori state estimation
is obtained from the nonlinear augmented model
f. The input vector
is known and equals
u from (
12) at time step
k. The a priori error covariance is determined as
where
denotes the Jacobian
known from (
20). The a posteriori state estimation is calculated as
where
are the measured node displacements and
denotes Kalman filter gain according to
The a posteriori error covariance is obtained from
According to (
7), the estimation of the contact forces at time step
is given by
4. Results
In the following, the lateral behavior of a rotating shaft under the influence of rotor–stator contact forces is simulated. The system parameters used in the simulation are summarized in
Table 1.
The behavior of the shaft during forward and backward whirling is considered with the aim to estimate the appearing contact forces. During forward whirling, the angular velocities
and
of
Figure 2a have the same algebraic signs, whereas during backward whirling,
and
have opposite signs. The lowest eigenfrequency of the shaft is 23.71 [1/s]. The rotational speed
of the shaft is increased according to
A simulation duration of 5 [s] is considered. During that time, the shaft runs through its first eigenmode. The resulting resonance peak leads to stator–rotor contacts. Dependent on the values , of the dry friction model, forward or backward whirling occurs. The simulated measurement noise has standard deviation [mm] and the sampling time of the filter is chosen as [s]. The initial states and error covariance are , . The design parameters , are chosen as , .
Selecting
,
as
,
establishes a forward whirl as shown in
Figure 3. The unknown radial displacement of node five is estimated well by the filter. The exceedance of the clearance can be correctly detected from the estimated displacements. The estimated contact forces are shown in
Figure 4. The forward whirl remains stable until the end of the simulation. A strong vibrational effect can be detected. For both x− and y−components of the contact force, shape as well as amplitude of the true and estimated signal closely fit to each other.
Selecting
,
as
,
establishes a backward whirl as visualized by
Figure 5. It can be seen that the displacement of node five is correctly estimated so that the exceedance of the clearance can be detected. The backward whirl remains stable for the remaining simulation time. In comparison to the forward whirl, more than four times higher amplitude values can be observed, which also vary in time. The estimations of the contact forces are visualized in
Figure 6. Again, shape and amplitude of true and estimated signals closely fit to each other.
In the following, the contact force estimation performance is quantified based on the measures
For both scenarios, the performance measures are stated in
Table 2. The performance measures are sensitive to relatively high estimation errors occuring when
or
are small. However, from
Figure 4 and
Figure 6, it is known that the amplitudes of the contact forces are large most of the time. For that reason, the performance values (
44), (
45) are evaluated for all
and
additionally.