Identification of Crack Location and Depth in Rotating Machinery ...
Identification of Crack Location and Depth in Rotating Machinery Based on Artificial Neural Network Tao Yu, Qing-Kai Han, Zhao-Ye Qin, and Bang-Chun Wen School of Mechanical Engineering & Automation, Northeastern University, 110004, Shenyang, P.R. China [email protected], [email protected], [email protected], [email protected]
Abstract. With the characteristics of ANN’s strong capability on nonlinear approximation, a new method by combining an artificial neural network with back-propagation learning algorithm and modal analysis via finite element model of cracked rotor system is proposed for fast identification of crack fault with high accuracy in rotating machinery. First, based on fracture mechanics and the energy principle of Paris, the training data are generated by a set of FEmodel-based equations in different crack cases. Then the validation of the method is verified by several selected crack cases. The results show that the trained ANN models have good performance to identify the crack location and depth with higher accuracy and efficiency, further, can be used in fast identification of crack fault in rotating machinery.
1 Introduction Rotating machinery, such as steam turbo, compressor, aeroengine and blower etc., are widely used in many industrial fields. The fault identification and diagnosis of rotating machinery, i.e. rotor system, have become a vigorous area of work during past decade[1]. Among the important rotor faults, the crack fault, which can lead to catastrophic failure and cause injuries and severe damage to machinery if undetected in its early stages, is most difficult to detect efficiently with traditional methods like waveform analysis, orbital analysis etc. Although the dynamic behaviors of rotor with transverse crack have been studied relatively enough[2], [3], [4], [5] and many methods have been introduced to detect crack(s) for non-rotating structures, they are not very suitable for fast identification of cracks in rotating machine. In the paper, based on the truth of the change of the mode shapes of the cracked structure - the most obvious effect of a structural crack, a new method, as shown in Fig.1, by combining accurate FE (Finite Element) model of rotor with transverse crack(s) and artificial neural network (ANN) is proposed to identify the location and depth of a crack in rotating machinery. Initially the accurate FE model of the object, a rotor system with a localized transverse on-edge non-propagating open crack, is built to produce the specific mode shape. Then a set of different mode shapes of a rotor system with localized crack in several different position and depth, which will be J. Wang et al. (Eds.): ISNN 2006, LNCS 3973, pp. 982 – 990, 2006. © Springer-Verlag Berlin Heidelberg 2006
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
983
treated as the input of the designed ANN model, can be obtained by repeating the above step. To insure the accuracy and quickness of the training process for the designed ANN model, a nonlinear model of a neuron is employed and a backpropagation learning algorism with Levenberg-Marquardt method is used. At last, with several selected crack cases, the errors between the results obtained by using the trained ANN model and FEM ones are compared and illustrated at last.
Fig. 1. Flow chart of identification scheme
2 Modes Calculation Based on FE Model of the Rotor System with a Transverse Crack A type of rotating machinery classified to the category with high speed and light load, can be considered as a jeffcott rotor, as shown in Fig.2.
Fig. 2. A single-span-single-disc (jeffcott) rotor system
It is assumed that the crack changes only the stiffness of the structure whereas the mass and damping coefficients remains unchanged. Cracks occurring in structures are responsible for local stiffness variations, which in consequence affect the mode shapes of the system. The finite element model of rotor system with a transverse crack and a single centrally situated disc, and also the geometry of the cracked section of the shaft are shown in Fig 3.
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T. Yu et al.
Fig. 3. The FE model of the rotor system and the geometry of the cracked section of the shaft
2.1 Calculation of Total Stiffness Matrix of the Crack Element According to the fracture mechanics and the energy principle of Paris, the additional strain energy due to a crack is given by the following equations:
U=
³ J ( A)dA .
(1)
A
where J ( A) is strain energy density function with only bending deformation taken into consideration and is expressed as:
J ( A) =
1 2 KI . E'
(2)
where E ' = E /(1 − ν ) , ν is the Poisson ratio and K I is the stress intensity factors corresponding to the bending moment M in plane crossing the axis of the beam The expression of K I is
K I = σ πz F2 ( z / h) .
(3)
where σ = 4M R 2 − η 2 / ʌR 4 , h = 2 R 2 − η 2 is the local height of the strip, and z F2 ( ) = h
2h ʌz 0.923 + 0.199[1 − sin( ʌz / 2h)]4 tan . ʌz 2h cos( ʌz / 2h)
(4)
Substituting the Eq. (2)~(4) into Eq. (1), the local flexibility due to the crack in the ξ -axis direction can be written as
cξ =
∂ 2U 1 − ν 2 = E ∂M 2
b
z
−b
0
³ dη ³
32 z ( R 2 − η 2 ) ʌzF22 ( )dz . h ʌ2R
(5)
where b is the crack width ( b = R 2 − ( R − a) 2 )[6], z is the local depth of crack strip. The similar expression of local flexibility in the η -axis direction can be written as cη = z where F1 ( ) = h
∂ 2U 1 − ν 2 = E ∂M 2
b
z
−b
0
³ dη ³
z 32 2 η ʌzF12 ( )d z . 2 h ʌ R
ʌz 0.752 + 2.02( z / h) + 0.37[1 − sin( ʌz / 2h)]3 2h tan 2h cos( ʌz / 2h) ʌz
(6)
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
985
Now we can get the local flexibility matrix C1 of the element with a crack ªcξ C1 = « ¬0
0º . cη »¼
(7)
Here, the coupled flexibility is neglected since it is much less than the element in the main diagonal of the C1. Let C0 is the flexibility matrix of uncracked shaft element. The total flexibility matrix C is expressed as C = C 0 + C1 .
(8)
Thus, the total stiffness matrix of the crack element is written as K c = TCT T .
(9)
where T is the transformation matrix and given by ª − 1 − l 1 0º T =« ». ¬ 0 − 1 0 1¼
(10)
2.2 FE Modeling of the Rotor System
Assembling all the element mass, damping and stiffness matrices of the rotor system in stationary coordinate system, the equation of motion in stationary coordinate is Mz + Cz + Kz = F .
(11)
where M, C, K and F are total mass, damping, stiffness and external exciting force matrices of rotor system respectively. z is the displacement of the element node. 2.3 Mode Shape of the Rotor System
For the rotor system considered in the paper, each beam element has two nodes and each node has two degrees of freedom representing transverse and deflecting displacements of the corresponding cross-section. Here, only the mode shape in the ξ -axis direction is discussed by assuming the rotor system is rigid supported at the bearing position. The mode shape can be obtained by solving the homogeneous part of Eq.11 without considering the effect of the damping. Mx + Kx = 0 .
(12)
Substituting xi = A(i ) sin(ωi t + ϕ i ) into Eq.12, we get (−ωi2 M + K ) A(i ) = 0 . where ωi and A(i ) is the i-th nature frequency and eigenvector (mode shape).
(13)
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T. Yu et al.
3 Artificial Neural Network Design The architecture of a typical ANN model with back-propagation algorithm (BPA), as shown schematically in Fig.4, is found to be efficient and perform well in many practical applications [7]. During the training process, the weights moved in the direction of the negative gradient, and together with biases of the network are iteratively adjusted to minimize the network performance function - mean square error (MSE), which is the average squared error between the network outputs and the target outputs. The Levenberg-Marquardt back-propagation algorism and NguyenWidrow function [8] are used in the ANN training and initialization of weights and biases of the hidden layer for faster convergence and accuracy. The rotor system discussed in the present study, consisting of a 10 mm diameter shaft carrying a centrally situated steel disc with 50 mm diameter and 5 mm width, is divided into 12 elements as shown in Fig.3. Using the method described before, the modes of different cases are calculated by introducing a transverse crack deliberately in the middle of certain element, as shown in Fig.5. For the central symmetry of the rotor system, only the modes with crack in first 6 elements are illustrated.
Fig. 4. The architecture of a BP ANN model
(a) modes of rotor system with crack in No.1 element Fig. 5. The 1st, 2nd and 3rd modes of rotor system when crack in different element
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
(b) modes of rotor system with crack in No.2 element
(c) modes of rotor system with crack in No.3 element
(d) modes of rotor system with crack in No.4 elementr
(e) modes of rotor system with crack in No.5 element
(f) modes of rotor system with crack in No.6 element Fig. 5. (continued)
987
988
T. Yu et al.
In each case, the modes corresponding to a/R ratios from 0.01 to 1 at a interval of 0.01 are the vectors consisting of 13 elements, which represent the amplitude of each node in the x -direction. The network architecture is built in terms of the numerals as NI, NH1, , NHn and NO, which denotes the number of neurons in the input layer, Hnth hidden layer and output layer respectively. The input and target output for training are expressed respectively as P=[P1 P2 Pn] and T=[T1 T2 Tn], where Pi=[p1 p2 pm]13hm (i=1, 2, n), denotes the modes matrix corresponding to crack at a certain position with m kinds crack depth; Ti=[t1 t2 tm]12hm (i=1, 2, n), denotes the crack depth vectors corresponding with different position. That is to say, the target output vector is filled with values equal to 0 or a number between 0 and 1, and a value not equal to 0 is symbolic of the localization and depth of a crack. For example, the T column vector [0 0 0.5 0]12 ×1 indicates the presence of a crack with a a/R ratio equal to 0.5 in the middle of No. 3 element. Since the dimensions of input and output vectors have been decided, the ANN needs to have a 13-(NH1…NHn)-12 architecture. But how many hidden layers and neurons in each hidden layers should be adopted has not a routine way to decide. Here, a single hidden layer with 20 neurons is designed for the following training process and proved to perform well with proper training data.
Ă
Ă
Ă
Ă
Ă
Ă
Ă
4 Training and Validation of the ANN The training process can be started with the prepared training data and ANN model. To evaluate the effects of training data on the success of crack identification, different training scheme is designed. That is to say, with total 12 shaft elements and 100 kinds of depth in each element corresponding to any type of 3 modes, the training data are arranged at intervals of 0, 1, 2, … and 14 respectively. Thus, 15 ANN models corresponding to each mode are trained. Obviously, ANN models trained with an interval of 0 have the best performance of almost zero error for the full use of the training data. To compare the performances of the other trained ANN models, several crack cases excluded in the training data of different schemes are adopted as test cases, and the errors between predicted crack parameters (location and depth) and actual ones under different trained ANN models and test cases are illustrated in Fig.6. It can be seen from Fig.6 that under condition of a given testing case, i.e. with a constant crack depth ratio (a/R), the increase of the system modes leads to performances of trained ANN models dropping down, and the differences between them are more and more remarkable with the decrease of the testing crack depth (a/R); under all training schemes, the performances of ANN models trained by 1st modes training data vary little when the crack depth ranges from 0.4~1. For the little effect of a tiny crack on the stiffness of the rotor system, the performances of trained ANN models may relatively lower when the training scheme with a large interval of training data is adopted. And for the effects of the discs on the modes of the rotor system, the errors are magnified with the decrease of the quantity of training data.
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
989
(a) testing case with a depth ratio (a/R) 0.80
(b) testing case with a depth ratio (a/R) 0.62
(c) testing case with a depth ratio (a/R) 0.48
(d) testing case with a depth ratio (a/R) 0.32 Fig. 6. Performances of different trained ANN models with several testing cases (figures from left to right in each case correspond to the performances of ANN models trained with 1st modes, 2nd modes and 3rd modes of the cracked rotor system respectively)
With the method and measured 1st mode shapes of cracked rotor system as shown in Fig.2, the crack location and depth can be identified quantificationally to a good degree of accuracy with the trained ANN models.
990
T. Yu et al.
5 Conclusion By combining FE model of the rotor system and ANN models with architecture of 3layer and 20 neurons in hidden layer, a novel method to non-destructively identify a crack location and depth in rotating machinery is presented in the paper. With the ANN models trained by modes, especially the 1st modes of cracked rotor system, the given testing crack cases are identified to a good degree of accuracy. It is proved that the effectiveness of identification process depends to a good extent on the number of training data when the crack is tiny or crack depth is shallow. But it has also been found that there are few effects of the quantity of training data on the performances of trained ANN models when the training modes are reasonably selected. However, for the relative simplicity of ANN models designed here and the complexity of 2nd and 3rd modes, the performances of ANN models trained by them are limited. Research to improve the crack identification method is continuing as the follows: First, more accurate FE model of the rotor system and efficient modes measuring method should be developed according to the practical rotor machinery. Second, the attempt to apply the method on detecting multiple cracks should be conducted, though the single crack case has been studied in the paper.
Acknowledgements The paper is supported by the Science & Technology Foundation of Liaoning Province (No. 20044003) and the Key Project of National Natural Science Foundation of China (No. 50535010). The authors would like to thank for their financial support.
References 1. Wauer, J.: On the Dynamics of the Cracked Rotors: A Literature Survey. Applied Mechanics Review 43 (1) (1990) 13–17 2. Dimarogonas, A.D.: Vibration of Cracked Structures: A State of the Art Review. Engineering Fracture Mechanics 55 (5) (1996) 831–857 3. Bachschmid, N., Pennacchi, P., Tanzi, E., Audebert, S.: Identification of Transverse Cracks in Rotors Systems. Proc. ISROMAC-8 Conf. Honolulu Hawaii (2000) 1065–1072 4. Wen, B.C., Wu, X.H., Ding, Q., Han, Q.K.: Dynamics and Experiments of Fault Rotors (in Chinese). 1st edn. Science Press, Beijing (2004) 5. Wen, B.C., Gu, J.L., Xia, S.B., Wang, Z.: Advanced Rotor Dynamics (in Chinese). 1st edn. China Machine Press, Beijing (2000) 6. Zou, J., Chen, J., Niu, J.C., Geng, Z.M.: Discussion on the Local Flexibility Due To the Crack in A Cracked Rotor System. Journal of Sound and Vibration 262 (2003) 365–369 7. Vyas, N.S., Satishkumar, D.: Artificial Neural Network Design for Fault Identification in a Rotor-bearing System. Mechanism and Machine Theory 36 (2001) 157–175 8. Hagan, M.T., Demuth, H.B., Beale, M.H.: Neural Network Design. PWS Publishing Company, Boston (1996)
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Abstract. With the characteristics of ANN’s strong capability on nonlinear approximation, a new method by combining an artificial neural network with back-propagation learning algorithm and modal analysis via finite element model of cracked rotor system is proposed for fast identification of crack fault with high accuracy in rotating machinery. First, based on fracture mechanics and the energy principle of Paris, the training data are generated by a set of FEmodel-based equations in different crack cases. Then the validation of the method is verified by several selected crack cases. The results show that the trained ANN models have good performance to identify the crack location and depth with higher accuracy and efficiency, further, can be used in fast identification of crack fault in rotating machinery.
1 Introduction Rotating machinery, such as steam turbo, compressor, aeroengine and blower etc., are widely used in many industrial fields. The fault identification and diagnosis of rotating machinery, i.e. rotor system, have become a vigorous area of work during past decade[1]. Among the important rotor faults, the crack fault, which can lead to catastrophic failure and cause injuries and severe damage to machinery if undetected in its early stages, is most difficult to detect efficiently with traditional methods like waveform analysis, orbital analysis etc. Although the dynamic behaviors of rotor with transverse crack have been studied relatively enough[2], [3], [4], [5] and many methods have been introduced to detect crack(s) for non-rotating structures, they are not very suitable for fast identification of cracks in rotating machine. In the paper, based on the truth of the change of the mode shapes of the cracked structure - the most obvious effect of a structural crack, a new method, as shown in Fig.1, by combining accurate FE (Finite Element) model of rotor with transverse crack(s) and artificial neural network (ANN) is proposed to identify the location and depth of a crack in rotating machinery. Initially the accurate FE model of the object, a rotor system with a localized transverse on-edge non-propagating open crack, is built to produce the specific mode shape. Then a set of different mode shapes of a rotor system with localized crack in several different position and depth, which will be J. Wang et al. (Eds.): ISNN 2006, LNCS 3973, pp. 982 – 990, 2006. © Springer-Verlag Berlin Heidelberg 2006
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
983
treated as the input of the designed ANN model, can be obtained by repeating the above step. To insure the accuracy and quickness of the training process for the designed ANN model, a nonlinear model of a neuron is employed and a backpropagation learning algorism with Levenberg-Marquardt method is used. At last, with several selected crack cases, the errors between the results obtained by using the trained ANN model and FEM ones are compared and illustrated at last.
Fig. 1. Flow chart of identification scheme
2 Modes Calculation Based on FE Model of the Rotor System with a Transverse Crack A type of rotating machinery classified to the category with high speed and light load, can be considered as a jeffcott rotor, as shown in Fig.2.
Fig. 2. A single-span-single-disc (jeffcott) rotor system
It is assumed that the crack changes only the stiffness of the structure whereas the mass and damping coefficients remains unchanged. Cracks occurring in structures are responsible for local stiffness variations, which in consequence affect the mode shapes of the system. The finite element model of rotor system with a transverse crack and a single centrally situated disc, and also the geometry of the cracked section of the shaft are shown in Fig 3.
984
T. Yu et al.
Fig. 3. The FE model of the rotor system and the geometry of the cracked section of the shaft
2.1 Calculation of Total Stiffness Matrix of the Crack Element According to the fracture mechanics and the energy principle of Paris, the additional strain energy due to a crack is given by the following equations:
U=
³ J ( A)dA .
(1)
A
where J ( A) is strain energy density function with only bending deformation taken into consideration and is expressed as:
J ( A) =
1 2 KI . E'
(2)
where E ' = E /(1 − ν ) , ν is the Poisson ratio and K I is the stress intensity factors corresponding to the bending moment M in plane crossing the axis of the beam The expression of K I is
K I = σ πz F2 ( z / h) .
(3)
where σ = 4M R 2 − η 2 / ʌR 4 , h = 2 R 2 − η 2 is the local height of the strip, and z F2 ( ) = h
2h ʌz 0.923 + 0.199[1 − sin( ʌz / 2h)]4 tan . ʌz 2h cos( ʌz / 2h)
(4)
Substituting the Eq. (2)~(4) into Eq. (1), the local flexibility due to the crack in the ξ -axis direction can be written as
cξ =
∂ 2U 1 − ν 2 = E ∂M 2
b
z
−b
0
³ dη ³
32 z ( R 2 − η 2 ) ʌzF22 ( )dz . h ʌ2R
(5)
where b is the crack width ( b = R 2 − ( R − a) 2 )[6], z is the local depth of crack strip. The similar expression of local flexibility in the η -axis direction can be written as cη = z where F1 ( ) = h
∂ 2U 1 − ν 2 = E ∂M 2
b
z
−b
0
³ dη ³
z 32 2 η ʌzF12 ( )d z . 2 h ʌ R
ʌz 0.752 + 2.02( z / h) + 0.37[1 − sin( ʌz / 2h)]3 2h tan 2h cos( ʌz / 2h) ʌz
(6)
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
985
Now we can get the local flexibility matrix C1 of the element with a crack ªcξ C1 = « ¬0
0º . cη »¼
(7)
Here, the coupled flexibility is neglected since it is much less than the element in the main diagonal of the C1. Let C0 is the flexibility matrix of uncracked shaft element. The total flexibility matrix C is expressed as C = C 0 + C1 .
(8)
Thus, the total stiffness matrix of the crack element is written as K c = TCT T .
(9)
where T is the transformation matrix and given by ª − 1 − l 1 0º T =« ». ¬ 0 − 1 0 1¼
(10)
2.2 FE Modeling of the Rotor System
Assembling all the element mass, damping and stiffness matrices of the rotor system in stationary coordinate system, the equation of motion in stationary coordinate is Mz + Cz + Kz = F .
(11)
where M, C, K and F are total mass, damping, stiffness and external exciting force matrices of rotor system respectively. z is the displacement of the element node. 2.3 Mode Shape of the Rotor System
For the rotor system considered in the paper, each beam element has two nodes and each node has two degrees of freedom representing transverse and deflecting displacements of the corresponding cross-section. Here, only the mode shape in the ξ -axis direction is discussed by assuming the rotor system is rigid supported at the bearing position. The mode shape can be obtained by solving the homogeneous part of Eq.11 without considering the effect of the damping. Mx + Kx = 0 .
(12)
Substituting xi = A(i ) sin(ωi t + ϕ i ) into Eq.12, we get (−ωi2 M + K ) A(i ) = 0 . where ωi and A(i ) is the i-th nature frequency and eigenvector (mode shape).
(13)
986
T. Yu et al.
3 Artificial Neural Network Design The architecture of a typical ANN model with back-propagation algorithm (BPA), as shown schematically in Fig.4, is found to be efficient and perform well in many practical applications [7]. During the training process, the weights moved in the direction of the negative gradient, and together with biases of the network are iteratively adjusted to minimize the network performance function - mean square error (MSE), which is the average squared error between the network outputs and the target outputs. The Levenberg-Marquardt back-propagation algorism and NguyenWidrow function [8] are used in the ANN training and initialization of weights and biases of the hidden layer for faster convergence and accuracy. The rotor system discussed in the present study, consisting of a 10 mm diameter shaft carrying a centrally situated steel disc with 50 mm diameter and 5 mm width, is divided into 12 elements as shown in Fig.3. Using the method described before, the modes of different cases are calculated by introducing a transverse crack deliberately in the middle of certain element, as shown in Fig.5. For the central symmetry of the rotor system, only the modes with crack in first 6 elements are illustrated.
Fig. 4. The architecture of a BP ANN model
(a) modes of rotor system with crack in No.1 element Fig. 5. The 1st, 2nd and 3rd modes of rotor system when crack in different element
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
(b) modes of rotor system with crack in No.2 element
(c) modes of rotor system with crack in No.3 element
(d) modes of rotor system with crack in No.4 elementr
(e) modes of rotor system with crack in No.5 element
(f) modes of rotor system with crack in No.6 element Fig. 5. (continued)
987
988
T. Yu et al.
In each case, the modes corresponding to a/R ratios from 0.01 to 1 at a interval of 0.01 are the vectors consisting of 13 elements, which represent the amplitude of each node in the x -direction. The network architecture is built in terms of the numerals as NI, NH1, , NHn and NO, which denotes the number of neurons in the input layer, Hnth hidden layer and output layer respectively. The input and target output for training are expressed respectively as P=[P1 P2 Pn] and T=[T1 T2 Tn], where Pi=[p1 p2 pm]13hm (i=1, 2, n), denotes the modes matrix corresponding to crack at a certain position with m kinds crack depth; Ti=[t1 t2 tm]12hm (i=1, 2, n), denotes the crack depth vectors corresponding with different position. That is to say, the target output vector is filled with values equal to 0 or a number between 0 and 1, and a value not equal to 0 is symbolic of the localization and depth of a crack. For example, the T column vector [0 0 0.5 0]12 ×1 indicates the presence of a crack with a a/R ratio equal to 0.5 in the middle of No. 3 element. Since the dimensions of input and output vectors have been decided, the ANN needs to have a 13-(NH1…NHn)-12 architecture. But how many hidden layers and neurons in each hidden layers should be adopted has not a routine way to decide. Here, a single hidden layer with 20 neurons is designed for the following training process and proved to perform well with proper training data.
Ă
Ă
Ă
Ă
Ă
Ă
Ă
4 Training and Validation of the ANN The training process can be started with the prepared training data and ANN model. To evaluate the effects of training data on the success of crack identification, different training scheme is designed. That is to say, with total 12 shaft elements and 100 kinds of depth in each element corresponding to any type of 3 modes, the training data are arranged at intervals of 0, 1, 2, … and 14 respectively. Thus, 15 ANN models corresponding to each mode are trained. Obviously, ANN models trained with an interval of 0 have the best performance of almost zero error for the full use of the training data. To compare the performances of the other trained ANN models, several crack cases excluded in the training data of different schemes are adopted as test cases, and the errors between predicted crack parameters (location and depth) and actual ones under different trained ANN models and test cases are illustrated in Fig.6. It can be seen from Fig.6 that under condition of a given testing case, i.e. with a constant crack depth ratio (a/R), the increase of the system modes leads to performances of trained ANN models dropping down, and the differences between them are more and more remarkable with the decrease of the testing crack depth (a/R); under all training schemes, the performances of ANN models trained by 1st modes training data vary little when the crack depth ranges from 0.4~1. For the little effect of a tiny crack on the stiffness of the rotor system, the performances of trained ANN models may relatively lower when the training scheme with a large interval of training data is adopted. And for the effects of the discs on the modes of the rotor system, the errors are magnified with the decrease of the quantity of training data.
Identification of Crack Location and Depth in Rotating Machinery Based on ANN
989
(a) testing case with a depth ratio (a/R) 0.80
(b) testing case with a depth ratio (a/R) 0.62
(c) testing case with a depth ratio (a/R) 0.48
(d) testing case with a depth ratio (a/R) 0.32 Fig. 6. Performances of different trained ANN models with several testing cases (figures from left to right in each case correspond to the performances of ANN models trained with 1st modes, 2nd modes and 3rd modes of the cracked rotor system respectively)
With the method and measured 1st mode shapes of cracked rotor system as shown in Fig.2, the crack location and depth can be identified quantificationally to a good degree of accuracy with the trained ANN models.
990
T. Yu et al.
5 Conclusion By combining FE model of the rotor system and ANN models with architecture of 3layer and 20 neurons in hidden layer, a novel method to non-destructively identify a crack location and depth in rotating machinery is presented in the paper. With the ANN models trained by modes, especially the 1st modes of cracked rotor system, the given testing crack cases are identified to a good degree of accuracy. It is proved that the effectiveness of identification process depends to a good extent on the number of training data when the crack is tiny or crack depth is shallow. But it has also been found that there are few effects of the quantity of training data on the performances of trained ANN models when the training modes are reasonably selected. However, for the relative simplicity of ANN models designed here and the complexity of 2nd and 3rd modes, the performances of ANN models trained by them are limited. Research to improve the crack identification method is continuing as the follows: First, more accurate FE model of the rotor system and efficient modes measuring method should be developed according to the practical rotor machinery. Second, the attempt to apply the method on detecting multiple cracks should be conducted, though the single crack case has been studied in the paper.
Acknowledgements The paper is supported by the Science & Technology Foundation of Liaoning Province (No. 20044003) and the Key Project of National Natural Science Foundation of China (No. 50535010). The authors would like to thank for their financial support.
References 1. Wauer, J.: On the Dynamics of the Cracked Rotors: A Literature Survey. Applied Mechanics Review 43 (1) (1990) 13–17 2. Dimarogonas, A.D.: Vibration of Cracked Structures: A State of the Art Review. Engineering Fracture Mechanics 55 (5) (1996) 831–857 3. Bachschmid, N., Pennacchi, P., Tanzi, E., Audebert, S.: Identification of Transverse Cracks in Rotors Systems. Proc. ISROMAC-8 Conf. Honolulu Hawaii (2000) 1065–1072 4. Wen, B.C., Wu, X.H., Ding, Q., Han, Q.K.: Dynamics and Experiments of Fault Rotors (in Chinese). 1st edn. Science Press, Beijing (2004) 5. Wen, B.C., Gu, J.L., Xia, S.B., Wang, Z.: Advanced Rotor Dynamics (in Chinese). 1st edn. China Machine Press, Beijing (2000) 6. Zou, J., Chen, J., Niu, J.C., Geng, Z.M.: Discussion on the Local Flexibility Due To the Crack in A Cracked Rotor System. Journal of Sound and Vibration 262 (2003) 365–369 7. Vyas, N.S., Satishkumar, D.: Artificial Neural Network Design for Fault Identification in a Rotor-bearing System. Mechanism and Machine Theory 36 (2001) 157–175 8. Hagan, M.T., Demuth, H.B., Beale, M.H.: Neural Network Design. PWS Publishing Company, Boston (1996)
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