Fault Diagnosis and Prognosis of Bearing Based on Hidden Markov Model with Multi-Features

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Some obtained observation values as the input of HMM_

States of faults	No. of observation values
	1	2	3	4	5	6	7	8	9	10
Nor	0.71	0.03	0.1	0.1	0	0.01	0	0	0.05	0
R1	0.07	0.56	.07	0.1	0	0.17	0	0	0.03	0
R2	0.23	0.08	0.35	0.2	0.01	0.06	0.04	0	0.01	0.02
R3	0.31	0.07	0.06	0.41	0.01	0.03	0.04	0	0.07	0
I1	0	0	0.02	0.01	0.84	0.07	0.01	0	0	0.05
I2	0.04	0.3	0.07	0.06	0.13	0.37	0.02	0.01	0	0
I3	0	0	0.01	0.04	0.11	0.04	0.65	0	0	0.15
O1	0	0	0	0	0	0.02	0	0.97	0	0.01
O2	0.37	0.1	0.02	0.32	0	0	0.01	0	0.18	0
O3	0	0.01	0.03	0.01	0.15	0.03	0.07	0.03	0	0.67

Statistics of test results of various fault states for different sample lengths_

Length of observation samples	States of faults
	Nor	R1	R2	R3	I1	I2	I3	O1	O2	O3
10	64.3%	82.2%	41.1%	53.9%	93.8%	81.3%	97.5%	98.3%	58.9%	85.1%
20	86.1%	88.7%	55.0%	72.3%	93.9%	97.8%	100%	100%	69.7%	92.2%
30	96.8%	95.5%	60.2%	77.8%	99.5%	99.5%	100%	100%	71.9%	98.6%
40	100%	100%	64.0%	77.3%	100%	100%	100%	100%	67.3%	100%
50	100%	100%	63.7%	85.6%	100%	100%	100%	100%	70.6%	100%
60	100%	100%	69.1%	86.9%	100%	100%	100%	100%	74.3%	100%
70	100%	100%	76.8%	90.6%	100%	100%	100%	100%	91.7%	100%
80	100%	100%	83.6%	91.2%	100%	100%	100%	100%	100%	100%
90	100%	100%	83.2%	94.4%	100%	100%	100%	100%	100%	100%
100	100%	100%	90.7%	100%	100%	100%	100%	100%	100%	100%
110	100%	100%	95.7%	100%	100%	100%	100%	100%	100%	100%
120	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
130	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
140	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%
150	100%	100%	100%	100%	100%	100%	100%	100%	100%	100%

The statistics of the overall training results using HMM_

States of faults	Nor	R1	R2	R3	I1	I2	I3	O1	O2	O3
Accuracy	100%	99%	95%	90%	100%	99%	100%	100%	96%	99%

Time-domain statistical characteristics_

ft1=1N∑i=1Nxif{t_1} = {1 \over N}\sum\limits_{i = 1}^N {{x_i}}	ft2=1N∑i=1Nxi2f{t_2} = \sqrt {{1 \over N}\sum\limits_{i = 1}^N {x_i^2} }	ft3=[1N∑i=1N|xi|]2f{t_3} = {\left[ {{1 \over N}\sum\limits_{i = 1}^N {\sqrt {\left| {{x_i}} \right|} } } \right]^2}	ft4=1N∑i=1N|xi|f{t_4} = {1 \over N}\sum\limits_{i = 1}^N {\left| {{x_i}} \right|}
ft5=1N∑i=1Nxi3f{t_5} = {1 \over N}\sum\limits_{i = 1}^N {x_i^3}	ft6=1N∑i=1Nxi4f{t_6} = {1 \over N}\sum\limits_{i = 1}^N {x_i^4}	ft7=1N−1∑i=1N(xi−X¯)2f{t_7} = {1 \over {N - 1}}\sum\limits_{i = 1}^N {{{\left( {{x_i} - \bar X} \right)}^2}}	f t8 = max{|xi|}
f t9 = min{xi}	f t10 = max(xi) − min(xi)	ft11=ft2ft4f{t_{11}} = {{f{t_2}} \over {f{t_4}}}	ft12=ft8ft2f{t_{12}} = {{f{t_8}} \over {f{t_2}}}
ft13=ft8ft4f{t_{13}} = {{f{t_8}} \over {f{t_4}}}	ft14=ft8ft3f{t_{14}} = {{f{t_8}} \over {f{t_3}}}	ft15=ft5ft23f{t_{15}} = {{f{t_5}} \over {ft_2^3}}	ft16=ft6ft24f{t_{16}} = {{f{t_6}} \over {ft_2^4}}

Frequency-domain statistical characteristics_

ff1=∑k=1Ks(k)kf{f_1} = {{\sum\limits_{k = 1}^K {s(k)} } \over k}	ff2=∑k=1K(s(k)−ff1)2k−1f{f_2} = {{\sum\limits_{k = 1}^K {{{\left( {s(k) - f{f_1}} \right)}^2}} } \over {k - 1}}	ff3=∑k=1K(s(k)−ff1)3k(p2)3f{f_3} = {{\sum\limits_{k = 1}^K {{{\left( {s(k) - f{f_1}} \right)}^3}} } \over {k{{\left( {\sqrt {{p_2}} } \right)}^3}}}	ff4=∑k=1K(s(k)−ff1)4k⋅ff22f{f_4} = {{\sum\limits_{k = 1}^K {{{\left( {s(k) - f{f_1}} \right)}^4}} } \over {k \cdot ff_2^2}}
ff5=∑k=1Kfks(k)∑k=1Ks(k)f{f_5} = {{\sum\limits_{k = 1}^K {{f_k}s(k)} } \over {\sum\limits_{k = 1}^K {s(k)} }}	ff6=∑k=1K(fk−ff5)2s(k)kf{f_6} = \sqrt {{{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^2}s(k)} } \over k}}	ff7=∑k=1Kfk2s(k)∑k=1Ks(k)f{f_7} = \sqrt {{{\sum\limits_{k = 1}^K {f_k^2s(k)} } \over {\sum\limits_{k = 1}^K {s(k)} }}}	ff8=∑k=1Kfk4s(k)∑k=1Kfk2s(k)f{f_8} = \sqrt {{{\sum\limits_{k = 1}^K {f_k^4s(k)} } \over {\sum\limits_{k = 1}^K {f_k^2s(k)} }}}
ff9=∑k=1Kfk2s(k)∑k=1Ks(k)∑k=1Kfk4s(k)f{f_9} = {{\sum\limits_{k = 1}^K {f_k^2s(k)} } \over {\sqrt {\sum\limits_{k = 1}^K {s(k)} \sum\limits_{k = 1}^K {f_k^4s(k)} } }}	ff10=ff6ff5f{f_{10}} = {{f{f_6}} \over {f{f_5}}}	ff11=∑k=1K(fk−ff5)3s(k)kp63f{f_{11}} = {{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^3}s(k)} } \over {kp_6^3}}	ff12=∑k=1K(fk−ff5)4s(k)kp64f{f_{12}} = {{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^4}s(k)} } \over {kp_6^4}}
ff13=∑k=1K(|fk−p5|)12s(k)kp6f{f_{13}} = {{\sum\limits_{k = 1}^K {{{\left( {\left| {{f_k} - {p_5}} \right|} \right)}^{{1 \over 2}}}s(k)} } \over {k\sqrt {{p_6}} }}	ff14=∑k=1K(fk−ff5)2s(k)∑k=1Ks(k)f{f_{14}} = {{\sum\limits_{k = 1}^K {{{\left( {{f_k} - f{f_5}} \right)}^2}s(k)} } \over {\sum\limits_{k = 1}^K {s(k)} }}