Method for solving the nonlinear inverse problem in gas face seal diagnosis based on surrogate models

doi:10.1007/s11465-022-0689-z

Front. Mech. Eng.

2022, Vol. 17

Issue (3) : 33 https://doi.org/10.1007/s11465-022-0689-z

RESEARCH ARTICLE

Method for solving the nonlinear inverse problem in gas face seal diagnosis based on surrogate models

Yuan YIN, Weifeng HUANG(

), Decai LI, Qiang HE, Xiangfeng LIU, Ying LIU

State Key Laboratory of Tribology in Advanced Equipment, Tsinghua University, Beijing 100084, China

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Abstract

Physical models carry quantitative and explainable expert knowledge. However, they have not been introduced into gas face seal diagnosis tasks because of the unacceptable computational cost of inferring the input fault parameters for the observed output or solving the inverse problem of the physical model. The presented work develops a surrogate-model-assisted method for solving the nonlinear inverse problem in limited physical model evaluations. The method prepares a small initial database on sites generated with a Latin hypercube design and then performs an iterative routine that benefits from the rapidity of the surrogate models and the reliability of the physical model. The method is validated on simulated and experimental cases. Results demonstrate that the method can effectively identify the parameters that induce the abnormal signal output with limited physical model evaluations. The presented work provides a quantitative, explainable, and feasible approach for identifying the cause of gas face seal contact. It is also applicable to mechanical devices that face similar difficulties.

Keywords surrogate model gas face seal fault diagnosis nonlinear dynamics tribology

Corresponding Author(s): Weifeng HUANG

About author: Tongcan Cui and Yizhe Hou contributed equally to this work.

Just Accepted Date: 22 April 2022 Issue Date: 20 October 2022

Cite this article:

Yuan YIN,Weifeng HUANG,Decai LI, et al. Method for solving the nonlinear inverse problem in gas face seal diagnosis based on surrogate models[J]. Front. Mech. Eng., 2022, 17(3): 33.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0689-z
https://academic.hep.com.cn/fme/EN/Y2022/V17/I3/33

Fig.1 Typical structure of a spiral groove gas face seal.

Fig.2 Spiral grooves on the rotor face.

Fig.3 Motion of the seal rings.

Dimension index j	Parameter	Lower bound l_j	Upper bound u_j
1	Axial force exerted, $F / N$	−100	100
2	Moment exerted around X, $M x / (N ? m)$	−5	5
3	Moment exerted around Y, $M y / (N ? m)$	−5	5
4	Support stiffness and damping scale $Q s$	0	100
5	Rotor tilt around X at $t = 0$ , $γ r x 0 / mrad$	−0.5	0.5
6	Rotor tilt around Y at $t = 0$ , $γ r y 0 / mrad$	−0.5	0.5

Tab.1 Suspended causes and their respective ranges

No.	$F / N$	$M x / (N ? m)$	$M y / (N ? m)$	$Q s$	$γ r x 0 / m r a d$	$γ r y 0 / m r a d$
1	−92.0	−2.02	−3.09	82.9	0.147	−0.101
2	−60.8	−4.16	2.95	10.7	0.052	0.498
3	63.5	−3.35	−1.86	12.6	−0.075	−0.231
···	···	···	···	···	···	···
99	17.3	−1.61	3.94	17.7	−0.144	−0.433
100	82.5	4.66	0.90	18.3	−0.290	−0.378

Tab.2 Sites designed by LHD for the initial dataset

No.	$F / N$	$M x / (N ? m)$	$M y / (N ? m)$	$Q s$	$γ r x 0 / m r a d$	$γ r y 0 / m r a d$
1	−56.2	−4.53	1.79	67.9	0.435	−0.116
2	3.9	3.31	−4.65	5.3	0.030	0.171
3	−98.5	−1.17	−4.33	41.7	0.187	0.089
4	86.1	3.46	0.27	9.2	0.154	−0.084
5	40.2	4.10	2.62	26.2	−0.453	0.236
6	−34.4	1.33	2.56	99.1	−0.135	−0.253
7	96.5	2.23	2.53	65.2	−0.427	0.132
8	76.9	−2.27	−0.64	76.6	−0.022	−0.262

Tab.3 Sites randomly set for testing

Fig.4 Estimation of the surrogate model established on the initial database: (a) comparison of the true output and the estimated output; (b) density of the scaled error.

Fig.5 Schematic flowchart of the inverse problem solution procedure.

No.	Parameter type	$F / N$	$M / (N ? m)$	$Q s$	$γ r / m r a d$	$Q s γ r / m r a d$	Iteration count	Discrepancy
1	Truth	−56.2	4.87	67.9	0.450	30.6	34	0.019
	Identified	−71.9	4.36	69.0	0.462	31.9
	Error	−15.7	−0.51	1.1	0.012	1.4
2	Truth	3.9	5.71	5.3	0.174	0.9	8	0.009
	Identified	−92.8	4.20	39.2	0.034	1.3
	Error	−96.7	−1.51	33.8	−0.139	0.4
3	Truth	−98.5	4.49	41.7	0.207	8.6	9	0.006
	Identified	−92.9	4.68	85.5	0.105	9.0
	Error	5.5	0.20	43.7	−0.102	0.3
4	Truth	86.1	3.47	9.2	0.175	1.6	43/100	0.045
	Identified	−3.8	2.72	15.1	0.158	2.4
	Error	−89.9	−0.75	5.9	−0.017	0.8
5	Truth	40.2	4.87	26.2	0.510	13.4	7	0.017
	Identified	60.4	4.91	80.0	0.160	12.8
	Error	20.2	0.04	53.8	−0.350	−0.6
6	Truth	−34.4	2.89	99.1	0.287	28.4	35	0.019
	Identified	−31.0	2.58	68.5	0.451	35.9
	Error	3.3	−0.31	−30.6	0.165	2.5
7	Truth	96.5	3.37	65.2	0.447	29.1	14/100	0.040
	Identified	96.9	2.82	81.8	0.439	35.9
	Error	0.4	−0.55	16.6	−0.008	6.7
8	Truth	76.9	2.36	76.6	0.263	20.2	56/100	0.048
	Identified	87.6	2.49	68.1	0.278	19.0
	Error	10.7	0.13	−8.5	0.015	−1.2

Tab.4 Validation of the simulated cases by comparing the true and identified parameters

Fig.6 Actual values of the queried parameters and their respective identified values. M and Q_sγ_r are identified well. Q_s and γ_r are difficult to be discriminated because of physical reasons.

Fig.7 Iteration process toward the targeted y′ for the simulated cases. The dashed lines denote the best in the initial database and the improving steps. The black and red solid lines denote the searched target and the result found, respectively. (a)–(h) correspond to cases 1–8.

Fig.8 (a) Test rig for exerting abnormities and online monitoring and (b) filtered acoustic emission RMS during startup.

No.	Parameter type	$F / N$	$M / (N ? m)$	$Q s γ r / mrad$	Iteration count	Discrepancy
1	Exerted	−29.7	1.51	?	88/100	0.195
	Identified	−37.5	2.16	6.0
	Error	−7.8	0.65	?
2	Exerted	−54.6	2.78	?	7/100	0.088
	Identified	−85.7	3.72	12.1
	Error	−31.1	0.94	?
3	Exerted	−79.5	4.05	?	65/100	0.047
	Identified	−87.5	5.01	5.3
	Error	−8.0	0.96	?

Tab.5 Validation of the experimental cases by comparing the exerted and identified parameters and checking the consistency of parameters not changed through the experiments

Fig.9 Experimental data and results found by the method for each case. The dashed lines denote the initial and improving steps; the black and red solid lines denote the searched target and the result found, respectively. (a)–(c) correspond to cases 1–3.

Fig.10 Reconstructing the detailed relative motion and the separation and pressure distributions by taking experimental case 2 as an example.

Classification	Parameter	Value
Geometric parameters of the seal face	Outer radius, $r o / mm$	61.6
	Inner radius, $r i / mm$	51.6
	Spiral groove inner radius, $r spr / mm$	55.5
	Spiral groove angle, $β spr /$ (° )	15
	Groove depth, $h spr / μm$	6
	Groove count, $N spr$	12
Mechanical parameters	Stator mass, $m / kg$	0.2
	Stator moment of inertia, $J / (kg ? m 2)$	3.5 × 10⁻⁴
	Baseline axial stiffness, $k s z 0 / (N ? m − 1)$	1.64 × 10⁴
	Baseline axial damping, $c s z 0 / (N ? s ? m − 1)$	1 × 10³
	Baseline angular stiffness, $k s γ 0 / (N ? m ? ra d − 1)$	31
	Baseline angular damping, $c s γ 0 / (N ? m ? s ? ra d − 1)$	1.4
Equivalent contact pair parameters	Standard deviation of surface height, $σ a / μm$	0.177
	Standard deviation of asperity height, $σ s / μm$	0.170
	Mean of asperity height, $w ¯ s / μm$	0.114
	Asperity density, $η s / μ m − 2$	0.133
	Average radius of asperity tips, $R s / μm$	3.02
	Elastic modulus, $E / GPa$	151
Working medium	Gas dynamic viscosity, $μ / (Pa ? s)$	1.8 × 10⁻⁵
Working medium	Gas density under 0.1 MPa pressure, $ρ / (kg ? m − 3)$	1.2
Operating condition parameters	Gas pressure at inner radius, $p i / M P a$	0.1
	Gas pressure at outer radius, $p o / M P a$	0.4
	Closing force, $P b / N$	1320.1
	Rotation speed, $n ω / (r ? mi n − 1)$	600

Table A1 Fixed parameters of the gas face seal

$c s z,$ $c s γ$	Axial and angular damping, respectively
$c s z 0$ , $c s γ 0$	Baseline axial and angular damping, respectively
$C s$	Damping matrix of the stator
$d$	Discrepancy measurement
$d T$	Terminating threshold for discrepancy measurement
$D, d i j$	Random permutation matrix for Latin hypercube design ( $i = 1, 2, . . ., M 0; j = 1, 2, . . ., P$ )
$E$	Elastic modulus of the equivalent contact pair
$F$	Exerted axial force modeling seal faults
$F b$	Closing force at $U s − U r = 0$
$f q (x)$	Different forms of products of the components of x (of order not higher than q)
g	Probability density of consistent output
$h$	Film thickness field
$h 0$	Film thickness on the land at $U s − U r = 0$
$h g r v$	Groove contribution to film thickness field
$h s p r$	Depth of the spiral grooves
$J$	Moment of inertia of the stator
$k s z,$ $k s γ$	Axial and angular stiffness, respectively
$k s z 0$ , $k s γ 0$	Baseline axial and angular stiffness, respectively
$k T$	Maximum iteration count
$K$	Coefficient of the acoustic emission transmission
$K s$	Stiffness matrix of the stator
$l j,$ $u j$	Lower bound and upper bound of design variables ( $j = 1, 2, . . ., P$ ), respectively
$L (θ, p)$	Object function for training the parameters ( $θ, p$ ) of the correlation function
$m$	Mass of the stator
$M$	Moment exerted modeling seal faults
$M 0$	Target count of sites for Latin hypercube design
$M x,$ $M y$	Moment around X and Y axes exerted modeling seal faults, respectively
M_s	Inertia matrix of the stator
$N$	Nugget effect matrix
$p$	Film pressure field (absolute pressure)
$p c$	Solid contact pressure field
$p i,$ $p o$	Boundary pressure on inner and outer radii, respectively
$p, p i$	Exponential parameters as vector and components ( $i = 1, 2, . . ., P$ ) of the correlation function, respectively
$P$	Count of design variables
$P$	Generalized force exerted modeling seal faults
$P b$	Generalized closing force at $U s − U r = 0$
$P c$	Generalized force of solid contact
$P f$	Generalized force of the fluid film
$Q$	Count of time steps
$Q s$	Scaling coefficient of the support stiffness and damping
$r,$ $θ$	Polar coordinates
$r i,$ $r o$	Inner and outer radii of the tribol-pair, respectively
$R$	Correlation function
R_s	Average radius of asperity tips of the equivalent contact pair
$R$	Correlation matrix of the training samples
$s^j$	Estimated standard variance of the dimension $j$ of the Kriging model output ( $j = 1, 2, . . ., Q$ )
$t$	Time
$U r$	Generalized displacement of the rotor
U_s	Generalized displacement of the stator
$v i j$	Design results for the $P$ variables on $M 0$ sites by Latin hypercube design
$V$	Root-mean-square value of the monitored acoustic emission signal
$V 0$	Root-mean-square value of the generated acoustic emission signal
$w$	Surface height of the equivalent contact pair
$w ¯ s$	Mean of asperity height of the equivalent contact pair
$W$	Dynamically updated database for surrogate models
x, y	Orthogonal coordinates
$x,$ $x i$	Input vector and components ( $i = 1, 2, . . ., P$ ) to the surrogate models, respectively
$X$ , $x (k)$	Dataset of input to the surrogate models as matrix and samples ( $k = 1, 2, . . ., M$ ), respectively
$y,$ $y j$	Output vector and components ( $j = 1, 2, . . ., Q$ ) to the surrogate models, respectively
$y^,$ $y^j$	Estimated output vector and components ( $j = 1, 2, . . ., Q$ ) to the surrogate models, respectively
$Y,$ $y (k)$	Dataset of output to the surrogate models as matrix and samples ( $k = 1, 2, . . ., M$ ), respectively
$z r$	Axial displacement of the rotor
$z s$	Axial displacement of the stator
$z j (x)$	Random variables submitting to the Gaussian process of $x$ ( $j = 1, 2, . . ., Q$ )
$β j$	Coefficient toward dimension $j$ of the output vector
$γ r$	Tilt angle of the rotor
$γ r x,$ $γ r y$	Tilt angle around X and Y axes of the rotor, respectively
$γ r x 0,$ $γ r y 0$	Tilt angle around X and Y axes of the rotor at $t = 0$ , respectively
$γ s x,$ $γ s y$	Tilt angle around X and Y axes of the stator, respectively
δ_ij	Random variables for Latin hypercube design ( $i = 1, 2, . . ., M 0;$ $j = 1, 2, . . ., P$ )
η_s	Asperity density of equivalent contact pair
$ϕ$	Probability density function of the standard normal distribution
$φ$	Flow factor for the average flow Reynolds equation
$μ$	Dynamic viscosity
$σ a$	Standard deviation of the surface height of the equivalent contact pair
$σ j$	Standard variance of the Gaussian process ( $j = 1, 2, . . ., Q$ )
$σ s$	Standard deviation of the asperity height of the equivalent contact pair
$Ω$	Area of the tribol-pair
$ω$	Rotation speed
$θ,$ $θ i$	Scaling parameters as vector/components ( $i = 1, 2, . . ., P$ ) of the correlation function

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