7 crest factor to detect bearing damage breaks down.Typical examples include bearings with shallow defects which have no significant edge,bearings with advanced damage and bearings with a large number of defects or widespread damage (Norton,1989). Probability density The probability density p()(see paragraph 4.2.2)of the bearing vibration acceleration has been suggested to give an indication of the bearing condition.The probability density of a bearing in good condition has a Gaussian distribution of accelerations,whereas a damaged bearing results in a non-Gaussian distribution with a dominant tail,because of a relative increase in the number of high levels of acceleration.However some authors have reported near-Gaussian distributions in some damaged bearings also (Tandon Nakra, 1992): This ties in with an observation by Braun and Hammond(1986)that the probability density is insensitive to many variations of the signal pattern. Kurtosis A series of statistical methods can be used to define the shape of the probability density distribution.The first two statistical moments of a probability density distribution are the mean value and the mean-square value(see paragraph 4.2.1).The third statistical moment is the skewness of a distribution,which is a measure of the symmetry of the probability density function.The fourth statistical moment is widely used in machinery diagnostics, particularly for rolling element bearings.It is called kurtosis,and it is given by kurtosis 职=ne=人a (7.4) Because the fourth power is involved,the value of kurtosis is weighted towards the values in the tails of the probability density distributions-i.e.it is related to the spread in the distribution.As a general rule,odd statistical moments provide information about the disposition of the peak relative to the median value,and even statistical moments provide information about the shape of the probability distribution curve.The value of kurtosis for a Gaussian distribution is 3.A higher kurtosis value indicates that there is a larger spread of higher values than would generally be the case for a Gaussian distribution. The kurtosis of a signal is very useful for detecting the presence of an impulse within the signal.It is widely used for detecting discrete,impulsive faults in rolling element bearings.Good bearings tend to have a kurtosis value of about 3,and bearings with impulsive defects tend to have higher values (generally >4).The usage of kurtosis is however limited because,as the damage to a bearing becomes distributed,the impulsive content of the signal decreases with a subsequent decrease in the kurtosis value.This