In the field of machinery diagnosis, vibration signals are often used for fault detection and state discrimination. Machinery diagnosis depends largely on the feature analysis of vibration signals measured for condition diagnosis, because the signals carry dynamic information about the machine state [1–3]. The vibration signals in different states will show different features, that is to say when plant machinery is in abnormal state, it will output signal sets which correspond to different faults. However, in most cases of condition diagnosis for rotating machinery, the values of symptom parameters calculated from vibration signals for condition monitoring and fault diagnosis are ambiguous. The main reasons for this can be explained as follows: (1) When the rotation speed and load of rotating machinery vary while vibration signals is being measured and a fault is in an early stage, the signal contains strong noise, stronger than the actual failure signal, that may lead to misrecognition of useful diagnostic information; (2) The statistical objectivity of the measured signal cannot always be satisfied because of the measurement techniques and manner of the inspectors [4]. Therefore, it is important to solve the ambiguous problem of fault diagnosis.

Roller bearings are an important part, widely used in rotating machinery. The failure of a rolling bearing may cause the breakdown of a rotating machine, and furthermore, serious consequences may arise due to the failure. Therefore, fault diagnosis of rolling bearings is extremely important for guaranteeing production efficiency and plant safety. Although fault diagnosis of rolling bearings is often artificially carried out using time or frequency analysis of vibration signals, there is a need for a reliable, fast automated diagnosis method thereof. Neural Networks (NN) have potential applications in automated detection and diagnosis of machine failure [5–9]. However, a conventional NN cannot adequately reflect the possibility of ambiguous diagnosis problems, and will never converge, when the symptom parameters, input to the 1st layer of the NN, have the same values in different states [4].

For the above reasons, this paper proposes a novel condition diagnosis method for rotating machinery developed using LSM and a fuzzy neural network realized by the PLNN. The NSPs in the time domain are defined to reflect the vibration signal features measured in each state. To raise the diagnosis sensitivity of the symptom parameters the SSPs are obtained by LSM. Using statistical theory, a detection index (DI) has also been defined to evaluate the applicability of SSPs. The DI can be used to indicate the fitness of a SSP for the PLNN. A sequential diagnosis approach is also proposed through the PLNN to sequentially identify the types of fault of rotating machinery. Diagnostic knowledge for the PLNN is acquired by possibility theory and the DST for solving the problem of ambiguous fault diagnosis. A practical example of condition diagnosis for a roller bearing verifies that the method is effective. The flowchart of the condition diagnostic procedure proposed in this paper is shown in Figure 1.

Figure 2 shows the experimental system for the roller bearing fault diagnosis test. The most commonly occurring faults in a roller element bearing are the outer-race defect, the inner-race defect, and the roller element defect. These fault bearings are shown in Figure 3 and were created artificially using a wire-cutting machine. The bearings that were utilized, and specifications of the test bearing, the size of the faults, and other necessary information is listed in Table 1.

In this work an accelerometer (PCB MA352A60) with a bandwidth from 5 Hz to 60 kHz and 10 mV/g output was used to measure the vibration signals of the vertical direction in the normal (N), the outer-race defect (O), the inner-race defect (I), and the roller element defect (R) states, respectively. The vibration signals measured by the accelerometer were transformed into a signal recorder (Scope Coder DL750) after being magnified by a sensor signal conditioner (PCB ICP Model 480C02). The original vibration signals in each state are measured at a constant speed (1,500 rpm), and a 150 kg load is also transported on the rotating shaft by the loading equipment (RCS2-RA13R) while the vibration signals are being measured. A high-pass filter with a 5 kHz cut-off frequency was used to cancel noise in the vibration signals for fault diagnosis. Examples of vibration signals measured in each state after filtering are shown in Figure 4. The sampling frequency of the signal measurement is 50 kHz, and the sampling time is 20 s.

In order to raise the diagnosis sensitivity of the symptom parameter, a method for obtaining the new synthetic symptom parameter is proposed as follows. The least squares mapping (LSM) technique aims to increase class separability and consists of the transformation of pattern vectors around arbitrary pre-selected points in the R^C space (where C is the number of states), called the decision space, in such a way that the least squares transformation error is minimized [14,15]. In this section, we propose a method used to raise the diagnosis sensitivity by projecting the SPs into discrimination space using least squares mapping. The type number of SPs (Y_k) is K, and the category number of states is M. In the coordinate space of K dimension, the endpoint of the vector Y_ij expresses statei. Y_ij is shown as follows: (15) Y ij = { y i j 1 , y i j 2 , ⋯ , y i j K | i = 1 ~ M , j = 1 ~ N } Twhere, N is the number of SPs, and the number of SPs in each state is same.

Y_ij can be projected into a new space L, and the new vector L_ij in the space L can be calculated as follows: (16) L ij = A Y ijwhere: (17) L ij = { l i j 1 , l i j 2 , ⋯ , l i j K | i = 1 ~ M , j = 1 ~ N } T

The transformation matrix A is defined by means of minimizing the least squares error (ε) between vectors L_ij and V_i for all states, where V_i is an arbitrary selected vector point in the L space. The selection of vector V_i is critical to enhance sensitiveness of the synthetic symptom parameter. In the present work, V_i is determined as a unit orthogonal vector by experience.

Figure 5 shows an illustration of the projection by the LSM, where K = 2 and M = 2. Namely, the two states (state 1 and state 2) should be classified using two SP series.

The error vector is: (18) ɛ = 1 N ∑ j = 1 N ‖ L ij − V i ‖ 2

ε minimization is performed by solving the following equation over A: (19) ∇ A ɛ = 0which, in conjunction with (18), leads to: (20) A = [ ∑ j = 1 N { V i Y ij ′ } ]   [ ∑ j = 1 N { Y ij Y ij ′ } ] − 1

When M ≥ 2, A is decided as follows: (21) A = [ ∑ i = 1 M ∑ j = 1 N { V i Y ij ′ } ]   [ ∑ i = 1 M ∑ j = 1 N { Y ij Y ij ′ } ] − 1

For diagnosis, the new synthetic symptom parameter can be obtained as follows: (22) SSP = A ⋅ SPwhere SP indicates symptom parameter (here P₁∼P₈).

According to the projected results shown in Figure 5(b), the points in state 1 and state 2 are congregated to vector V₁ and V₂, respectively. The two states in the space L can be distinguished more easily than in the space Y.

To explain the efficiency of the LSM method, some examples are given. In the present example, we used two symptom parameters (P₁ and P₂) to distinguish the inner race defect (I) and roller element defect (R) states of the bearing. SSP₁ and SSP₂ are the new synthetic parameter obtained by the LSM. Tables 3 and 4 show the parameters and the values of the DI and the DR before projection and after projection by the LSM, respectively. According to those examples, the states can be clearly distinguished by the SSPs. It is obvious that the sensitivity of the SSPs obtained by the LSM is higher than the original SPs. In Tables 3 and 4, μ_p1, μ_p2, μ_ssp1 and μ_ssp2 are the mean values of P₁, P₂, SSP₁ and SSP₂, respectively. σ_p1, σ_p2, σ_ssp1 andσ_ssp2 are the standard deviations of P₁, P₂, SSP₁ and SSP₂, respectively.

The main mathematic symbols used in Section 6 are:

N_m: the neuron number of the m-th layer of an NN, m = 1 to M.

X ( 1 ) = { X i ( i , j ) }: the pattern input to the 1st layer. Here, X i ( 1 , j ) is the value input to the j-th neuron in the input (1st) layer, i = 1 to P, j =1 to N₁.

X ( M ) = { X i ( M , k ) }: the training (teaching) data for the last layer (M-th layer). Here, X i ( M , k ) is the output value of the k-th neuron in the output (M-th) layer; k = 1 to N_M.

X ( 1 ) ∗ = { X i ( 1 , j ) ∗ } and X ( M ) ∗ = { X ( M , k ) ∗ 1 }: new data that has not yet been learnt by the NN.

X i ( m , t ): the value of the t-th neuron in the hidden (m-th) layer; t =1 to N_M.

W u v ( m ): the weight between the u-th neuron in the m-th layer and the v-th neuron in the (m+1)-th layer, m =1 to M − 1;u = 1 to N_m; v =1 to N_m+1.

The fuzzy neural network is applied to diagnose the fault types of a rolling bearing by the sequential diagnosis algorithm, and realized with a developed back propagation neural network called as “the partially-linearized neural network” (PLNN). A back propagation neural network is only used for training the data, and the PLNN is used for testing the learned NN. Here, the basic principle of the PLNN for the fault diagnosis is described as follows.

The neuron number of the m-th layer of an NN is N_m. The set X ( 1 ) = { X i ( 1 , j ) } represents the pattern input to the 1st layer and the set X ( M ) = { X i ( M , k ) } is the training data for the last layer (M-th layer). Here, i = 1 to P, j = 1 to N₁, k = 1 to N_M and, X i ( 1 , j ): the value input to the j-th neuron in the input (1st) layer; X i ( M , k ) the output value of the k-th neuron in the output (M-th) layer, k =1 to N_M.

Even if the NN converges by learning X⁽¹⁾ and X^(M), it cannot adequately deal with the ambiguous relationship between the new X⁽¹⁾* and X^(M)*, which has not been learnt. In order to predict X^(M)* according to the probability distribution of X⁽¹⁾*, partial linear interpolation of the NN is introduced as shown in Figure 9.

In the NN that has converged with the data X⁽¹⁾ and X^(M), the following symbols are used:

X i ( m , t ) the value of the t-th neuron in the hidden (m-th) layer; t =1 to N_m.

W u v ( m ): the weight between the u-th neuron in the m-th layer and the v-th neuron in the (m+1)-th layer, m =1 to M;u = 1 to N_m: v = 1 to N_m+1.

If all these values are memorized by the computer, when new values X j ( 1 , u ) ∗ ( X j ( 1 , u ) < X j ( 1 , u ) ∗ < X j + 1 ( 1 , u )) are input into the first layer, the predicted value of the v-th neuron (v=1 to N_m) in the (m+1)-th layer (m = 1 to M - 1) can be estimated by: (41) X j ( m + 1 , ν ) = X i + 1 ( m + 1 , ν ) − { ∑ u = 1 N m W u v ( m ) ( X i + 1 ( m , u ) − X j ( m , u ) ) } ( X i + 1 ( m + 1 , v ) − X i ( m + 1 , v ) ) ∑ u = 1 N m W u v ( m ) ( X i + 1 ( m , u ) − X i ( m , u ) )

Using the operation above, the sigmoid function is partially linearized, as shown in Figure 9. If a function must be learned, the PLNN will learn the points indicated by the ● symbols shown in Figure 8. When new data (s₁′, s₂′) are input into the converged PLNN, the values depicted by the ■ symbols corresponding to the data (s₁′, s₂′) will quickly be identified as P_e. Thus, the PLNN can be used to deal with ambiguous diagnosis problems.

As shown in Figure 10, the new data (s₁′, s₂′) input into the converged PLNN, and which are not learnt by the PLNN for recognizing, must satisfy the following condition: (42) S 1 ( min ) < S 1 ′ < S 1 ( max ) and S 2 ( min ) < S 2 ′ < S 2 ( max )where S_1(min), S_2(min) and S_1(max), S_2(max) are the minimum values and the maximum values of S₁ and S₂, respectively, which have been learned by the PLNN. Therefore, in this work, the values (P_i* and P_j*) of symptom parameters input to the PLNN for fault diagnosis must satisfy the following condition: (43) P i ( min ) < P 1 ∗ < P i ( max ) and P j ( min ) < P 1 ∗ < P j ( max )where P_i(min), P_j(min) and P_i(max), P_j(max) are the minimum values and the maximum values of P_i and P_j, respectively.

Figure 11 shows the PLNNs constructed for the condition diagnosis, which consists of the first layer, the hidden layer and the last layer. The SSPs selected by DI are input into the neurons in the first layer. The number of neurons in hidden layer is eighty. The outputs in the last layer are W(N)′, W(B)′, W(O)′, W(IR)′, W(I)′, W(R)′and W(U)′, which mean the possibility grades of normal state, bearing fault state, outer race defect state, other bearing defect, inner race defect, rolling element defect and unknown states, respectively.

In this study, the diagnosis knowledge for training of the PLNN is acquired by the possibility theory and the Dempster & Shafer theory (DST). The possibility functions of the SSPs used for each diagnostic step, as examples, are shown in Figures 12–14, respectively.

In Figure 12 P(N), P(B) and P(U) are the possibility functions of the normal, bearing defect and the unknown states, respectively. Using the matching method explained in Section 5.2, W₁(N), W₁(B) and W₁(U) that the possibilities of SSP₁ in the normal, the bearing defect and the unknown states can be obtained, respectively; W₅(N), W₅(B) and W₅(U) that the possibilities of SSP₅ in the normal, the bearing defect and the unknown states can also be obtained, respectively.

In Figure 13 P(O), P(IR) and P(U) are the possibility functions of the outer-race defect, other bearing faults (the rolling element defect and the inner-race defect), and the unknown states, respectively. Using the matching method explained in Section 5.2, W₁(O), W₁(IR) and W₁(U) that the possibilities of SSP₁ in the outer-race defect, other bearing faults and the unknown states can be obtained, respectively; W₅(O), W₅(IR) and W₅(U) that the possibilities of SSP₅ in the outer-race defect, other bearing faults and the unknown states can also be obtained, respectively.

In Figure 14 P(I), P(R) and P(U) are the possibility functions of the inner-race defect, the rolling element defect and the unknown states, respectively. Using the matching method explained in Section 5.2, W₁(I), W₁(R) and W₁(U) that the possibilities of SSP₁ in the inner-race defect, the rolling element defect and the unknown states can be obtained, respectively; W₂(I), W₂(R) and W₂(U) that the possibilities of SSP₂ in the inner-race defect, the rolling element defect and the unknown states can also be obtained, respectively.

After obtaining the possibilities of the SSPs for each diagnostic step, the combination possibility function of each state W(N)′, W(B)′, W(O)′, W(IR)′, W(I)′, W(R)′ and W(U)′ can be obtained by the Dempster & Shafer theory. As an example, parts of training data for each diagnosis step are shown in Tables 6–8.

In order to verify the diagnostic capability of the PLNN, we used the data measured in each state had not been learned by the PLNN. When inputting the test data into the learnt PLNNs, they can correctly and quickly diagnose those faults with the possibility grades of the corresponding states. The diagnosis results are shown in Tables 9–11.

According to the diagnosis results above, the normal (N), the outer-race defect (O), the inner-race defect (I), and the roller element defect (R) states of roller bearing can be automatically and correctly identified using the diagnosis methods proposed in this paper.

In order to solve the problem of ambiguity between the symptom parameters and fault types, effectively diagnose faults and automatically identify the condition of a rotating machine, an intelligent diagnosis method was proposed on the basis of the least squares mapping (LSM) and a fuzzy neural network. The main conclusions can be summarized as follows:

A sequential diagnosis method was proposed through which the fuzzy neural network realized by the partially-linearized neural network (PLNN) could sequentially distinguish fault types.