16 Modeling and Parameter Identification for Active Lubricated …
201
M ¨
d(t) + c ˙
d(t) + kd(t) = bu(t).
(16.2)
This grey-box modelling is eased by reformulation of dynamic equation to state
space form, where u(t) is the input of the system and d(t) is the output of the system.
The model inevitably gives rise to modelling errors, which are include as process
noise v(t), measurement noise is modelled as an additive signal w(t). This combination of process noise and measurement noise can be reduced to the equivalent noise
term e(t) entering both the state and the output equation. Therefore, model errors and
measurement noise are included as a signal e(t) entering both through an input gain
E and the measurements directly as given in Eq. (16.3):
˙
x 0 (t) = Ax 0 (t) + Bu(t) + Ee(t)
d(t) = C x 0 (t) + e(t).
(16.3)
where the system matrix, input gain matrix, output matrix and state vector are defined
in Eq. (16.4):
A =
0 I
K D
, B =
0
B
, E =
0
E
, C =
I 0
, x 0 =
d x , d y , ˙
d x , ˙
d y
T
(16.4)
and K = −M
−1 k, D = −M
−1 c, B = M
−1 b. The parameters in Eqs. (16.3)
and (16.4) are identified by recasting the problem to a model parameterized in
θ = {K, D, B, E, x 0 } as (θ). Each matrix K, D, B, E has four elements. e.g.
K =
k xx k xy
k yx k yy
.
16.3 Parameters Identification of Active Lubricated
Bearing-Rotor System
16.3.1 Description of Experiments
The experimental equipment setup is shown in Fig. 16.3. Physical dimensions of the
equipment are shown in Table 16.1. In the experiment, the oil supply pressure p =
1.1 MPa and the rotational speed = 0 rad/s, all measurements are sampled with
period t 0 = 0.2 ms.
In the experiment, the input signal is PRBS. Because of the incompressibility of
the oil, the stiffness and damping of the system are not constant. These parameters
will change with the position of the shaft. Therefore, in this experiment, the initial
position of the shaft can be changed by changing the valley value of the PRBS, and
201
M ¨
d(t) + c ˙
d(t) + kd(t) = bu(t).
(16.2)
This grey-box modelling is eased by reformulation of dynamic equation to state
space form, where u(t) is the input of the system and d(t) is the output of the system.
The model inevitably gives rise to modelling errors, which are include as process
noise v(t), measurement noise is modelled as an additive signal w(t). This combination of process noise and measurement noise can be reduced to the equivalent noise
term e(t) entering both the state and the output equation. Therefore, model errors and
measurement noise are included as a signal e(t) entering both through an input gain
E and the measurements directly as given in Eq. (16.3):
˙
x 0 (t) = Ax 0 (t) + Bu(t) + Ee(t)
d(t) = C x 0 (t) + e(t).
(16.3)
where the system matrix, input gain matrix, output matrix and state vector are defined
in Eq. (16.4):
A =
0 I
K D
, B =
0
B
, E =
0
E
, C =
I 0
, x 0 =
d x , d y , ˙
d x , ˙
d y
T
(16.4)
and K = −M
−1 k, D = −M
−1 c, B = M
−1 b. The parameters in Eqs. (16.3)
and (16.4) are identified by recasting the problem to a model parameterized in
θ = {K, D, B, E, x 0 } as (θ). Each matrix K, D, B, E has four elements. e.g.
K =
k xx k xy
k yx k yy
.
16.3 Parameters Identification of Active Lubricated
Bearing-Rotor System
16.3.1 Description of Experiments
The experimental equipment setup is shown in Fig. 16.3. Physical dimensions of the
equipment are shown in Table 16.1. In the experiment, the oil supply pressure p =
1.1 MPa and the rotational speed = 0 rad/s, all measurements are sampled with
period t 0 = 0.2 ms.
In the experiment, the input signal is PRBS. Because of the incompressibility of
the oil, the stiffness and damping of the system are not constant. These parameters
will change with the position of the shaft. Therefore, in this experiment, the initial
position of the shaft can be changed by changing the valley value of the PRBS, and