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resulting in incalculable loss. The importance of rolling bearing makes detection and
maintenance become a long-term work. Because the service life of rolling bearing
varies greatly under different working conditions, the maintenance period cannot be
completely determined according to the service life [3–5]. Therefore, the traditional
timing maintenance method is very unscientific, which not only wastes resources,
but also can not guarantee the absolute safety of rotating machinery in operation.
Based on this situation, the method of equal time difference kurtosis analysis is
proposed to diagnose the early fault of rolling bearing [6]. Through fault diagnosis
to prevent the occurrence of faults, so as to avoid mechanical accidents due to the
damage of rolling bearings, to ensure the safety of the running machinery.
12.2 Kurtosis Analysis Method
Let x(t) be a random acoustic emission signal, and have x 1 = x (t), x 2 = x (t + τ 1 ) …
x k = x (t + τ k−1 ) (τ is the delay amount), the k-order moment m k of random acoustic
emission signal x(t) is defined as:
m k (τ 1 . . . τ k−1 ) = E
x(t)x(t + τ 1 ) . . . x(t + τ k−1 )
(12.1)
The higher-order cumulants of AE signals can be expressed by higher-order
moments. For zero mean signals, the fourth-order cumulants are:
c 4 (τ 1 , τ 2 , τ 3 ) = m 4 (τ 1 , τ 2 , τ 3 ) − m 2 (τ 1 )m 2 (τ 3 − τ 2 )
− m 2 (τ 2 )m 2 (τ 3 − τ 1 ) − m 2 (τ 3 )m 2 (τ 2 − τ 1 )
(12.2)
When τ1 = τ2 = τ3 = 0, from (12.2):
c 4 (0, 0, 0) = m 4 (0, 0, 0) − 3m
2
2 (0) = E[x 4 (t)] − 3{E[x 2 (t)]}
2
(12.3)
where, c4 (0, 0, 0) is the kurtosis of the acoustic emission signal, and the kurtosis of
the zero mean random acoustic emission signal is defined as follows:
K =
E[(x)4]
{E[(x)2]}2
=
1
N
N
i=1
xi − ¯
X
σx
4
(12.4)
where formula (12.4): ¯
X =
1
N
N
i=1 xi ,σ x is the standard deviation, n is the signal
length.
According to formula (12.4), the fourth-order cumulants of AE signal is kurtosis,
because the high-order cumulants of Gaussian process is always 0, so kurtosis takes
a low value for Gaussian signal. In fault diagnosis, the acoustic emission signal
amplitude of normal rolling bearing presents normal distribution. When the rolling
bearing fails, the amplitude of AE signal will deviate from the normal distribution,
Y. Yu et al.
resulting in incalculable loss. The importance of rolling bearing makes detection and
maintenance become a long-term work. Because the service life of rolling bearing
varies greatly under different working conditions, the maintenance period cannot be
completely determined according to the service life [3–5]. Therefore, the traditional
timing maintenance method is very unscientific, which not only wastes resources,
but also can not guarantee the absolute safety of rotating machinery in operation.
Based on this situation, the method of equal time difference kurtosis analysis is
proposed to diagnose the early fault of rolling bearing [6]. Through fault diagnosis
to prevent the occurrence of faults, so as to avoid mechanical accidents due to the
damage of rolling bearings, to ensure the safety of the running machinery.
12.2 Kurtosis Analysis Method
Let x(t) be a random acoustic emission signal, and have x 1 = x (t), x 2 = x (t + τ 1 ) …
x k = x (t + τ k−1 ) (τ is the delay amount), the k-order moment m k of random acoustic
emission signal x(t) is defined as:
m k (τ 1 . . . τ k−1 ) = E
x(t)x(t + τ 1 ) . . . x(t + τ k−1 )
(12.1)
The higher-order cumulants of AE signals can be expressed by higher-order
moments. For zero mean signals, the fourth-order cumulants are:
c 4 (τ 1 , τ 2 , τ 3 ) = m 4 (τ 1 , τ 2 , τ 3 ) − m 2 (τ 1 )m 2 (τ 3 − τ 2 )
− m 2 (τ 2 )m 2 (τ 3 − τ 1 ) − m 2 (τ 3 )m 2 (τ 2 − τ 1 )
(12.2)
When τ1 = τ2 = τ3 = 0, from (12.2):
c 4 (0, 0, 0) = m 4 (0, 0, 0) − 3m
2
2 (0) = E[x 4 (t)] − 3{E[x 2 (t)]}
2
(12.3)
where, c4 (0, 0, 0) is the kurtosis of the acoustic emission signal, and the kurtosis of
the zero mean random acoustic emission signal is defined as follows:
K =
E[(x)4]
{E[(x)2]}2
=
1
N
N
i=1
xi − ¯
X
σx
4
(12.4)
where formula (12.4): ¯
X =
1
N
N
i=1 xi ,σ x is the standard deviation, n is the signal
length.
According to formula (12.4), the fourth-order cumulants of AE signal is kurtosis,
because the high-order cumulants of Gaussian process is always 0, so kurtosis takes
a low value for Gaussian signal. In fault diagnosis, the acoustic emission signal
amplitude of normal rolling bearing presents normal distribution. When the rolling
bearing fails, the amplitude of AE signal will deviate from the normal distribution,