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Engine system diagnosis using vibration data
Computers and Industrial Engineering Vol. 25, Nos 1-4, pp. 135-138, 1993
0360-8352/9356.00+0.00 Copyright © 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
ENGINE SYSTEM DIAGNOSIS USING VIBRATION DATA Steven IL-Y. I z i Mechaaical i ~ a e e r i n g Department North Carolima A&'T State Ulziversity Greensboro, North C_.aroBmt 27411
ABSTRACT This paper presents a fault diagnostic system which enables the identification of engine malfunctions using the vibration data of crank systems with experimenta. tion on one sample engine. In this system, the equations governing system characteristics are determined by a Time.domain Modal Analysis (TMA) technique. The frequencies, damping factors and mode shapes are used in a harmonic synthesis process to result in zero end torques and stresses in each crank shaft span. Two physical simulation cases were illustrated. In the first testing case, the abnormal excursion of the bearing support system was detected. The replacement of appropriate bearings is suggested. In the second testing case, the combined stresses of the fillet of the crankpin and journal units were found exceeded the allowable fatigue limit. The modification of both elements was suggested. INTRODUCTION Monitoring and diagnosis of vibration of engine [1~] have proved to be one of the most reliable means to be used as an engine health indicator. Such a technique shows good capability when predicting incipient failures in efficient time to allow repairs to be made at the most suitable time. The application of monitoring and diagnostic concept to automate the maintenance of engine crank systems is of primary importance in continuous process industries. The loss of production, due to repair stoppages, in these industries is sometimes a more economic consideration than the damage of the machine themselves. In order to reduce production losses, it is vital to have warning in advance, so that the necessary correction, purchase of spare parts or consideration for structure redesign can be made. With the increasing demand in engines with more power per unit size weight, engines were built with more cylinders and more power per cylinder. They were operated at higher rotating speeds and heavier loads than ever before. All these factors imposed more stress on the crankshaft due to cyclic bending loads, and dynamic torsional loads. Among these two loads, torsional vibration is more complex and is not predicted with any degree of accuracy by the classic Holzer Forced Vibration Method. In torsional vibration, the problem is faced usually when some speed related frequency of an
exciting force in the system is close to the torsional resonance of the system. This torsional resonance in the system causes fatigue and high stress levels in the shaft ultimately leading to its premature failure. Premature crank system failure became a major bottleneck, and it was so important to develop computer based predictive monitoring and diagnostic systems to prevent the premature failures, especially in the following areas: (1) those troublesome dynamic stress and failures at mid-span shaft and fillet junction sections, (2) those failures caused by the interaction of higher modes and higher order torsion mode shape, and (3) those complex dynamics associated with supporting structures which induced many unexpected breakdowns. The objective of this paper is to introduce a computerized diagnostic system based on recording and analyzing vibration data of crankshaft in a running engine. The time-domain model analysis technique is developed to analyze and predict possible failures of a crankshaft in torsional vibration. The code is used to calculate dynamic (time history) stress and vibration amplitudes at each shaft section. The high stress locations are identified. The proposed diagnostic system enables diagnosis of the cause and the prediction of ultimate breakdowns. TIME-DOMAIN MODAL ANALYSIS The crank system is first discretized into a series of springs, dampers and lumped inertias. Figure 1 shows such a system consisting of a front-end gear, an eight cylinder engine, and a flywheel attached to a generator rotor. The equations of motion for such a discretized system with an applied load, A, can be expressed in terms of the displacement vector, X, as . c~
+
sx - A
(l)
where M, C and S are the mass, damping, and stiffness matrices, respectively. The system is rearranged into a series of one degree of freedom equations in normal coordinates. For example, the equation for the i-th mode is given as
~., * 2,.,v&, + ,,,~ - A.,
(z)
where ¢~i and Yi are the natural h'equency and damping for the i-th mode. These parameters are identified by the time-domain transfer function using the procedure given in Fig. 2. 135
136
Proceedings of the 15th Annual Conference on Computers and Industrial Engineering
In the transfer function identification process, the phase information preserved, both eigenvalues and response function is assumed to be a continuous one eigenvectors are faithfully transformed from discrete to with sampled input and output signals which can be continuous domain. In the single-input multiple-output described by system, if the input sequence xt is excited at reference point i and the response sequences at points j's were Y--'(B) - e~ + O, . . . . O,,B" (3) measured on q different nodal points, the corresponding x, 1+45e . . . . ~).B" transfer matrix can be written as tn
where B is a backshift operator, Yt is the system response sequence and xt the input excitation sequence. Since the parameter estimation is a nonlinear procedure, the identification requires an initial guess model and a final refinement. The initial guess model is obtained by formulating the Yule-Walker equation using a lincarized inverse structure. The parameters are estimated by a singular decomposition least squares algorithm [3]. The obtained lincarized guess model is transformed back to its rational approximant via Pade technique [4]. The final parameter refinement is accomplished by the implementation of Marquart Compromise Nonlinear Least Square search [5]. The minimum sum of squares of errors is achieved in a parameter adaptation process. The model order, n, is increased in a sequential manner until the decrease of the sum of squares of residuals is insignificant. In order to characterize physical parameters, the discrete transfer function is reconstructed in a modal superposition form: t(s) - L'(a) - ~ l x,
g~
k-, 1-XtB
+
g~ ]
1-X~B
(4)
- E E ~r,x~ + z;xZ~e Jc.t J-e
where ()-k,)-k* and (gv gk') are complex conjugate roots for underdamped dynamics and real roots for overdamped dynamics, respectively. The continuous transfer function is obtained from the discrete one by an impulse invariant transformation [6]. With both amplitude and
- ~H,~( ,)
Hj,)
(s)
.,.,.,,
1-1 .-l,~
h;l
,-I,~;
and (P'Jk,#Jk*) and (hjk, hjk°) are complex conjugate
roots
for underdamped dynamics and real roots for over. damped dynamics, respectively. The corresponding differential equation of the i-th continuous transfer function can be obtained, and the complex frequency and related physical parameters are given as
p, - 8 , . t,~ - - C ~ * iu, 1 ~ - ~
(6)
The k-th dynamic mode shape which describes the compliance elements at q different surface points is given as
{Hkl - Ihu, ha. -., h,~l'
(7)
The significant level of the k.th dynamic mode and its mode shape is obtained by calculating the contribution of the dispersion function
dk -
dklY~
w~re
dk -
g~t
,4 t-x,x,
(S)
1.-o
SIGNATURE ANALYSIS OF CRANK SYSTEM Shaft failure is generally initiated by fatigue failures and local supporting defects. To diagnose a complete crank system using the proposed approach, it is necessary to know the permissible shaft stresses and characteristic frequencies generated by defects. The dynamic stress of the vibrating system is determined by the following formula
q. _ ~ (~.,A)Ol Zl-t
Figure 1 - An equivalent crank vibratory system
lq~uure 2 - TMA modeling procedure
(9)
where Z is the modulus of shaft cross.section. 1;(Jr,)~&) is the maximum inertia torque at the node per radian at mass no. 1, and 0 t is the deflection at the free end of the system. The fillet stress of stepped shafts were determined by Jacobsen method [7]. For a complete vibration signature of rotating machinery, bearing are the best locations for measuring machinery vibration since this is where the basic dynamic loads and forces of the machine are applied and they are themselves critical components. Vibration measurements should be made near the bearing cap of each bearing in the machine with minimum possible mechanical impedance and maximum possible mechanical signature information. Bearing operations and remnant frequencies can be identified by presence
LAI: Engine System Diagnosis of peeks on the TMA spectrum at corresponding frequencies which can be evaluated from the equations given in Table 1.
137
ban .ptn trequency
rb=(DN/t20d)lt-c~Zp(tVD)2] Uz
bearing housing, as shown in Fig. 4. A charge amplifier, a low pass filter, counter and a digital oscilloscope were used for the data collection. The rotating speed of shaft was measured with a magnetic pick-up and an electronic counter. The signal was digitized with sampling frequency 500 Hz and sent to the computer for further processing. The spectrum analysis is was performed to diagnose the supporting system. The related stress analysis was conducted to diagnose possible shaft failures.
csp trequeneT
re=(N/t20)It.cadS(d/D)]
Bearing Malfunction Detection
Table 1 - Bearing operational frequencies shaft rotmflomd freqmncy
frmN/60
nz
ball pass frequency of outer race
fo=(N/60)(n/2)[l-co~(d/l))l
Hz
ban pros frequen~ or knner~
rj=(N/6O)(mv'2)[t+~(~vn)] Hz
where
Hz
d = d i a m e t e r of rolling element, D = bearing pitch dhtmeter,
= contactangle, n ~ n u m b e r o r rolling elements, N = shaft speed.
In addition, the bearing elements themselves also exhibit their own resonant frequencies which may be excited by the periodic impacting of defective elements. Resonant frequencies are functions of the bearing configuration and bearing materials. From the analysis by Stokey [8], the resonant frequency of race is given as k(~-l)
1.['~
(10)
where k is the order of the resonance, r, is the radius to the neutral axis, I is the moment of inertia, and m is the mass of race per linear foot. Bearing malfunctions like spalls, pits and cracks in races or balls can be identified from the calculated values. When the bearing rotates at constant speed each defect generates a regular set of impacts with a characteristic rotational frequency. There are four basic modes of bearing defects: (1) defects on the outer race, (2) defects on the inner race, (3) rolling element defects, (4) cage defects. FAULT DIAGNOSIS OF A V-12 ENGINE The experimental set-up is shown in Fig. 3. It consists of a shaft supported by seven bail bearings. The bearing dimensions are: Pitch diameter D=18", inner diameter d=16", contact angle/3=15 °, number of balls n=12. Several identical bearings were used for the off-line test. These bearings could be dismantled easily. Various types of defects were simulated by electric. discharged machining. The rotating speed was kept approximately constant. The bearing vibration signal was measured with a piezo-electric accelerometer mounted on the rear
It is sometimes difficult to identify resonant frequencies related to bearing defects since high frequency amplitude may be obscured by the broadband noise levels associated with many factors like bearing friction, external vibration and instrumentation noise. An overall range of frequency monitoring seems to be highly difficult. Therefore, we isolated the particular mode (characteristic frequency) of vibration for detailed monitoring. Four different type of defective spectra and one normal operating spectrum were generated. As compared with the spectrum of the I2V engine shaft vibration data, the bearing inner ball defect at f~quency around 138 Hz was identified. The results are presented in Fig. 5. The replacement of the test engine bearing is suggested. Shaft Stress Analysis Fig. 6 shows the diagram of shaft vibration amplitude plotted against engine speed at 2128 bpm. The corresponding torsional stresses of the largest 4 orders along the shaft section are given in Fig. 7. These high order stresses contribute to the high sum of orders stress level. Section 4.5 gives the highest stress. The stress and mode shape at that particular section are plotted in Figures 8 and 9. They are found within the permissible range. However, a detailed calculation of the fillet stress at the crankpin and journal exceed the
!
A
A
A
s m
488
4) I p*-
[*
h
t7 ~6 ~s ~4zs
'I IiIiIl
~a
tm
Figure 6 - Stress check of shaft dynamic modes
£ |11
588
t t.o~ I t.t.c¢
.
[
11~a H,
,,
I )
I
4)
| .E-0~ t .8-ol I ,l-o~
t.l-os
rrlw,ucT
3 - The PCV-I2 crank system ~
4 - The location of signal pick-up Irtgure$ - PSD check of bearing malfunctions
138
Proceedings of the 15th Annual Conference on Computers and Industrial Engineering
maximum allowable fatigue limit. The results are tabulated in Table 2. The immediate suggestion from the analysis, is a modification in both crank-pin and journal configurations.
Table 2 - Diagnostic check of engine crank stresses
388O W
• . - - ee4~ 6--- er4er
Z480
$ tO Input
Figure7. Single ordertorsionalstresses
Output ¢a z88
46o
~48
?~
F]gare 8 - The mode shape of shaft section 4-5
stress
i
RLI - bearing center to shaft center (hi) RL2 - bearing center to crankpin center RL3 - distance between hearings (in) DB • diameter ofjourlml (in) DP - diameter of cranlq)in (in) 2"2 - crank throw thiclmess (in) B . crank throw width (In) DC - diameter of cylinder (in) R . radius of crank (in) W R E . reciprocating weight (lb) WRO - rotating weight (lb) VA - hank angle of V-engine (degree) RPM - rotation speed (rev/mln) RLP - length of ¢A-ankpln (in) PMAX. r m ~ pressure in cylinder (psi) DBH - diameter of bore hi crankpin (in) DBG - diameter of hare In journal (in) RH - fillet radius of crankpin (In) RG - fillet radius of Journal (in) TH - recess of crankpln TG - recess of journal N S T . number of stroke RK - factor for different type of forge THBA- mln. tensile strength of shaft (psi) THQ • shearing stress (psi) THTH- torsional stress at crankpin (psi) THTG- torsional stress at Journal (psi) THADD- additional stress (psi) THDEF- bedpiote deformation stress
piston Area (in 2) Crank Section Module (In 2) Gas pressure (psi) Reciprocating Weight pressure (psi) Rotation Weight pressure (psi) Max. pressure hi Cylinder (psi) Nominal Alternating Bending Press (psi) Nominal Alternating Bending Stress (psi) Nominal Alternating Shearing Stress Nominal Alternating Torsional Stress Bending Concentration Factor Shearing Concentration Factor Torsional Concentration Factor Bending Stress on the Fillet (psi) Torsional Stress on the Fillet (psi) Comblnafion Stress on the Fillet (psi) Allowable Fatigue Limit (psi) Factor of Safety
Computed CIMAC Data
6,100 L2.000 24,000 13.000 12.000 4.5O0 21.000 1%000 10.$00 8L6.40 401.00 0.000 450.00 7.125 1650.0 0.000 0.000 0.750 0.876 0.688 0.000 4 1.050 100770 1500.0 6594.O 5186.0 290~0 0.000 Crankpin 227.0 84.9 1650, 217.2 106.7 1650. 825. 6817.4 1500. 6594, 2.758
Journal 227.0 84.9 1650. 217.2 106.7 1650. 825. 6817.4 1500. 6594. 2.077 3.571 1.848 1.782 18804.3 24970.7 1218&7 9241.6 30278.2 32140.8 34731.9 34014.8 1.150 1.150 1.147 1.147 Not Safe Not Safe
REFERENCES ¢a 2m
46O
548
m
I.
Flpu~ ~ - The stress history of shaft section 4-5 2. CONCLUSIONS The typical defects of engine crank system are diagnosed by using the proposed Time-domain Modal Analysis (TMA) technique. The technique is capable of detecting both single and multiple crank system defects by producing an accurate and smooth stress spectra as compared with the spectra of analytical approaches. The transfer function model provides more information than the classic Holzer method. Dispersion of each frequency component and damping ratio of corresponding mode give the quantitative status evaluation to the engine crank system. Based on the fact that only relatively short period of signal is needed for modelin~ the TMA method is suitable for on.liue monitoring. Since the TMA method is based on vibration data, both amplitude and stress are obtained directly from the experimental data which reflect the operating status of the underlying system.
3.
4. 5. 6. 7. 8.
Malay Ghosh and A. Rajamani, " Modified Torsional Vibration Analysis of Multi-mass Shaft Systems," Proceedings of the 4th IMAC, pp. 101-106. Collacott, R.A., "Mechanical Fault Diagnosis and Condition Monitoring," John Wiley & Sons, New York, 1977, pp. 218. Goulb, G.H. and Reinsch, C., "Singular Value Decomposition and Least Square Solution," in Wilkinson, J.H. and Reinsch, C. (editors), Handbook of Automatic Computation, Vol. II, Linear Algebra, Heidelberg, Spring. Baker, G.A.,"Essentials of Pade Approximant," Academic Press Inc., N.Y., 1975. Marquart, D.W.,"AnAlgorithm for Least Squares Estimation of Nonlinear Parameters," J. of SOc. lndust. Appl. Math., 11, 431, 1963. Bozic, S.M.,'Digital Kalman Filtering," Edward Arnold Ltd., England, 1982. Nestorides, E. J. "A Handbook on Torsional Vibrations," Cambridge University Press, 1958. Braun, S.G. and Datner, B., "Analysis of Roller/Ball Bearing Vibration," Transaction of the ASME, Jan. 1979.
0360-8352/9356.00+0.00 Copyright © 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
ENGINE SYSTEM DIAGNOSIS USING VIBRATION DATA Steven IL-Y. I z i Mechaaical i ~ a e e r i n g Department North Carolima A&'T State Ulziversity Greensboro, North C_.aroBmt 27411
ABSTRACT This paper presents a fault diagnostic system which enables the identification of engine malfunctions using the vibration data of crank systems with experimenta. tion on one sample engine. In this system, the equations governing system characteristics are determined by a Time.domain Modal Analysis (TMA) technique. The frequencies, damping factors and mode shapes are used in a harmonic synthesis process to result in zero end torques and stresses in each crank shaft span. Two physical simulation cases were illustrated. In the first testing case, the abnormal excursion of the bearing support system was detected. The replacement of appropriate bearings is suggested. In the second testing case, the combined stresses of the fillet of the crankpin and journal units were found exceeded the allowable fatigue limit. The modification of both elements was suggested. INTRODUCTION Monitoring and diagnosis of vibration of engine [1~] have proved to be one of the most reliable means to be used as an engine health indicator. Such a technique shows good capability when predicting incipient failures in efficient time to allow repairs to be made at the most suitable time. The application of monitoring and diagnostic concept to automate the maintenance of engine crank systems is of primary importance in continuous process industries. The loss of production, due to repair stoppages, in these industries is sometimes a more economic consideration than the damage of the machine themselves. In order to reduce production losses, it is vital to have warning in advance, so that the necessary correction, purchase of spare parts or consideration for structure redesign can be made. With the increasing demand in engines with more power per unit size weight, engines were built with more cylinders and more power per cylinder. They were operated at higher rotating speeds and heavier loads than ever before. All these factors imposed more stress on the crankshaft due to cyclic bending loads, and dynamic torsional loads. Among these two loads, torsional vibration is more complex and is not predicted with any degree of accuracy by the classic Holzer Forced Vibration Method. In torsional vibration, the problem is faced usually when some speed related frequency of an
exciting force in the system is close to the torsional resonance of the system. This torsional resonance in the system causes fatigue and high stress levels in the shaft ultimately leading to its premature failure. Premature crank system failure became a major bottleneck, and it was so important to develop computer based predictive monitoring and diagnostic systems to prevent the premature failures, especially in the following areas: (1) those troublesome dynamic stress and failures at mid-span shaft and fillet junction sections, (2) those failures caused by the interaction of higher modes and higher order torsion mode shape, and (3) those complex dynamics associated with supporting structures which induced many unexpected breakdowns. The objective of this paper is to introduce a computerized diagnostic system based on recording and analyzing vibration data of crankshaft in a running engine. The time-domain model analysis technique is developed to analyze and predict possible failures of a crankshaft in torsional vibration. The code is used to calculate dynamic (time history) stress and vibration amplitudes at each shaft section. The high stress locations are identified. The proposed diagnostic system enables diagnosis of the cause and the prediction of ultimate breakdowns. TIME-DOMAIN MODAL ANALYSIS The crank system is first discretized into a series of springs, dampers and lumped inertias. Figure 1 shows such a system consisting of a front-end gear, an eight cylinder engine, and a flywheel attached to a generator rotor. The equations of motion for such a discretized system with an applied load, A, can be expressed in terms of the displacement vector, X, as . c~
+
sx - A
(l)
where M, C and S are the mass, damping, and stiffness matrices, respectively. The system is rearranged into a series of one degree of freedom equations in normal coordinates. For example, the equation for the i-th mode is given as
~., * 2,.,v&, + ,,,~ - A.,
(z)
where ¢~i and Yi are the natural h'equency and damping for the i-th mode. These parameters are identified by the time-domain transfer function using the procedure given in Fig. 2. 135
136
Proceedings of the 15th Annual Conference on Computers and Industrial Engineering
In the transfer function identification process, the phase information preserved, both eigenvalues and response function is assumed to be a continuous one eigenvectors are faithfully transformed from discrete to with sampled input and output signals which can be continuous domain. In the single-input multiple-output described by system, if the input sequence xt is excited at reference point i and the response sequences at points j's were Y--'(B) - e~ + O, . . . . O,,B" (3) measured on q different nodal points, the corresponding x, 1+45e . . . . ~).B" transfer matrix can be written as tn
where B is a backshift operator, Yt is the system response sequence and xt the input excitation sequence. Since the parameter estimation is a nonlinear procedure, the identification requires an initial guess model and a final refinement. The initial guess model is obtained by formulating the Yule-Walker equation using a lincarized inverse structure. The parameters are estimated by a singular decomposition least squares algorithm [3]. The obtained lincarized guess model is transformed back to its rational approximant via Pade technique [4]. The final parameter refinement is accomplished by the implementation of Marquart Compromise Nonlinear Least Square search [5]. The minimum sum of squares of errors is achieved in a parameter adaptation process. The model order, n, is increased in a sequential manner until the decrease of the sum of squares of residuals is insignificant. In order to characterize physical parameters, the discrete transfer function is reconstructed in a modal superposition form: t(s) - L'(a) - ~ l x,
g~
k-, 1-XtB
+
g~ ]
1-X~B
(4)
- E E ~r,x~ + z;xZ~e Jc.t J-e
where ()-k,)-k* and (gv gk') are complex conjugate roots for underdamped dynamics and real roots for overdamped dynamics, respectively. The continuous transfer function is obtained from the discrete one by an impulse invariant transformation [6]. With both amplitude and
- ~H,~( ,)
Hj,)
(s)
.,.,.,,
1-1 .-l,~
h;l
,-I,~;
and (P'Jk,#Jk*) and (hjk, hjk°) are complex conjugate
roots
for underdamped dynamics and real roots for over. damped dynamics, respectively. The corresponding differential equation of the i-th continuous transfer function can be obtained, and the complex frequency and related physical parameters are given as
p, - 8 , . t,~ - - C ~ * iu, 1 ~ - ~
(6)
The k-th dynamic mode shape which describes the compliance elements at q different surface points is given as
{Hkl - Ihu, ha. -., h,~l'
(7)
The significant level of the k.th dynamic mode and its mode shape is obtained by calculating the contribution of the dispersion function
dk -
dklY~
w~re
dk -
g~t
,4 t-x,x,
(S)
1.-o
SIGNATURE ANALYSIS OF CRANK SYSTEM Shaft failure is generally initiated by fatigue failures and local supporting defects. To diagnose a complete crank system using the proposed approach, it is necessary to know the permissible shaft stresses and characteristic frequencies generated by defects. The dynamic stress of the vibrating system is determined by the following formula
q. _ ~ (~.,A)Ol Zl-t
Figure 1 - An equivalent crank vibratory system
lq~uure 2 - TMA modeling procedure
(9)
where Z is the modulus of shaft cross.section. 1;(Jr,)~&) is the maximum inertia torque at the node per radian at mass no. 1, and 0 t is the deflection at the free end of the system. The fillet stress of stepped shafts were determined by Jacobsen method [7]. For a complete vibration signature of rotating machinery, bearing are the best locations for measuring machinery vibration since this is where the basic dynamic loads and forces of the machine are applied and they are themselves critical components. Vibration measurements should be made near the bearing cap of each bearing in the machine with minimum possible mechanical impedance and maximum possible mechanical signature information. Bearing operations and remnant frequencies can be identified by presence
LAI: Engine System Diagnosis of peeks on the TMA spectrum at corresponding frequencies which can be evaluated from the equations given in Table 1.
137
ban .ptn trequency
rb=(DN/t20d)lt-c~Zp(tVD)2] Uz
bearing housing, as shown in Fig. 4. A charge amplifier, a low pass filter, counter and a digital oscilloscope were used for the data collection. The rotating speed of shaft was measured with a magnetic pick-up and an electronic counter. The signal was digitized with sampling frequency 500 Hz and sent to the computer for further processing. The spectrum analysis is was performed to diagnose the supporting system. The related stress analysis was conducted to diagnose possible shaft failures.
csp trequeneT
re=(N/t20)It.cadS(d/D)]
Bearing Malfunction Detection
Table 1 - Bearing operational frequencies shaft rotmflomd freqmncy
frmN/60
nz
ball pass frequency of outer race
fo=(N/60)(n/2)[l-co~(d/l))l
Hz
ban pros frequen~ or knner~
rj=(N/6O)(mv'2)[t+~(~vn)] Hz
where
Hz
d = d i a m e t e r of rolling element, D = bearing pitch dhtmeter,
= contactangle, n ~ n u m b e r o r rolling elements, N = shaft speed.
In addition, the bearing elements themselves also exhibit their own resonant frequencies which may be excited by the periodic impacting of defective elements. Resonant frequencies are functions of the bearing configuration and bearing materials. From the analysis by Stokey [8], the resonant frequency of race is given as k(~-l)
1.['~
(10)
where k is the order of the resonance, r, is the radius to the neutral axis, I is the moment of inertia, and m is the mass of race per linear foot. Bearing malfunctions like spalls, pits and cracks in races or balls can be identified from the calculated values. When the bearing rotates at constant speed each defect generates a regular set of impacts with a characteristic rotational frequency. There are four basic modes of bearing defects: (1) defects on the outer race, (2) defects on the inner race, (3) rolling element defects, (4) cage defects. FAULT DIAGNOSIS OF A V-12 ENGINE The experimental set-up is shown in Fig. 3. It consists of a shaft supported by seven bail bearings. The bearing dimensions are: Pitch diameter D=18", inner diameter d=16", contact angle/3=15 °, number of balls n=12. Several identical bearings were used for the off-line test. These bearings could be dismantled easily. Various types of defects were simulated by electric. discharged machining. The rotating speed was kept approximately constant. The bearing vibration signal was measured with a piezo-electric accelerometer mounted on the rear
It is sometimes difficult to identify resonant frequencies related to bearing defects since high frequency amplitude may be obscured by the broadband noise levels associated with many factors like bearing friction, external vibration and instrumentation noise. An overall range of frequency monitoring seems to be highly difficult. Therefore, we isolated the particular mode (characteristic frequency) of vibration for detailed monitoring. Four different type of defective spectra and one normal operating spectrum were generated. As compared with the spectrum of the I2V engine shaft vibration data, the bearing inner ball defect at f~quency around 138 Hz was identified. The results are presented in Fig. 5. The replacement of the test engine bearing is suggested. Shaft Stress Analysis Fig. 6 shows the diagram of shaft vibration amplitude plotted against engine speed at 2128 bpm. The corresponding torsional stresses of the largest 4 orders along the shaft section are given in Fig. 7. These high order stresses contribute to the high sum of orders stress level. Section 4.5 gives the highest stress. The stress and mode shape at that particular section are plotted in Figures 8 and 9. They are found within the permissible range. However, a detailed calculation of the fillet stress at the crankpin and journal exceed the
!
A
A
A
s m
488
4) I p*-
[*
h
t7 ~6 ~s ~4zs
'I IiIiIl
~a
tm
Figure 6 - Stress check of shaft dynamic modes
£ |11
588
t t.o~ I t.t.c¢
.
[
11~a H,
,,
I )
I
4)
| .E-0~ t .8-ol I ,l-o~
t.l-os
rrlw,ucT
3 - The PCV-I2 crank system ~
4 - The location of signal pick-up Irtgure$ - PSD check of bearing malfunctions
138
Proceedings of the 15th Annual Conference on Computers and Industrial Engineering
maximum allowable fatigue limit. The results are tabulated in Table 2. The immediate suggestion from the analysis, is a modification in both crank-pin and journal configurations.
Table 2 - Diagnostic check of engine crank stresses
388O W
• . - - ee4~ 6--- er4er
Z480
$ tO Input
Figure7. Single ordertorsionalstresses
Output ¢a z88
46o
~48
?~
F]gare 8 - The mode shape of shaft section 4-5
stress
i
RLI - bearing center to shaft center (hi) RL2 - bearing center to crankpin center RL3 - distance between hearings (in) DB • diameter ofjourlml (in) DP - diameter of cranlq)in (in) 2"2 - crank throw thiclmess (in) B . crank throw width (In) DC - diameter of cylinder (in) R . radius of crank (in) W R E . reciprocating weight (lb) WRO - rotating weight (lb) VA - hank angle of V-engine (degree) RPM - rotation speed (rev/mln) RLP - length of ¢A-ankpln (in) PMAX. r m ~ pressure in cylinder (psi) DBH - diameter of bore hi crankpin (in) DBG - diameter of hare In journal (in) RH - fillet radius of crankpin (In) RG - fillet radius of Journal (in) TH - recess of crankpln TG - recess of journal N S T . number of stroke RK - factor for different type of forge THBA- mln. tensile strength of shaft (psi) THQ • shearing stress (psi) THTH- torsional stress at crankpin (psi) THTG- torsional stress at Journal (psi) THADD- additional stress (psi) THDEF- bedpiote deformation stress
piston Area (in 2) Crank Section Module (In 2) Gas pressure (psi) Reciprocating Weight pressure (psi) Rotation Weight pressure (psi) Max. pressure hi Cylinder (psi) Nominal Alternating Bending Press (psi) Nominal Alternating Bending Stress (psi) Nominal Alternating Shearing Stress Nominal Alternating Torsional Stress Bending Concentration Factor Shearing Concentration Factor Torsional Concentration Factor Bending Stress on the Fillet (psi) Torsional Stress on the Fillet (psi) Comblnafion Stress on the Fillet (psi) Allowable Fatigue Limit (psi) Factor of Safety
Computed CIMAC Data
6,100 L2.000 24,000 13.000 12.000 4.5O0 21.000 1%000 10.$00 8L6.40 401.00 0.000 450.00 7.125 1650.0 0.000 0.000 0.750 0.876 0.688 0.000 4 1.050 100770 1500.0 6594.O 5186.0 290~0 0.000 Crankpin 227.0 84.9 1650, 217.2 106.7 1650. 825. 6817.4 1500. 6594, 2.758
Journal 227.0 84.9 1650. 217.2 106.7 1650. 825. 6817.4 1500. 6594. 2.077 3.571 1.848 1.782 18804.3 24970.7 1218&7 9241.6 30278.2 32140.8 34731.9 34014.8 1.150 1.150 1.147 1.147 Not Safe Not Safe
REFERENCES ¢a 2m
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Flpu~ ~ - The stress history of shaft section 4-5 2. CONCLUSIONS The typical defects of engine crank system are diagnosed by using the proposed Time-domain Modal Analysis (TMA) technique. The technique is capable of detecting both single and multiple crank system defects by producing an accurate and smooth stress spectra as compared with the spectra of analytical approaches. The transfer function model provides more information than the classic Holzer method. Dispersion of each frequency component and damping ratio of corresponding mode give the quantitative status evaluation to the engine crank system. Based on the fact that only relatively short period of signal is needed for modelin~ the TMA method is suitable for on.liue monitoring. Since the TMA method is based on vibration data, both amplitude and stress are obtained directly from the experimental data which reflect the operating status of the underlying system.
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4. 5. 6. 7. 8.
Malay Ghosh and A. Rajamani, " Modified Torsional Vibration Analysis of Multi-mass Shaft Systems," Proceedings of the 4th IMAC, pp. 101-106. Collacott, R.A., "Mechanical Fault Diagnosis and Condition Monitoring," John Wiley & Sons, New York, 1977, pp. 218. Goulb, G.H. and Reinsch, C., "Singular Value Decomposition and Least Square Solution," in Wilkinson, J.H. and Reinsch, C. (editors), Handbook of Automatic Computation, Vol. II, Linear Algebra, Heidelberg, Spring. Baker, G.A.,"Essentials of Pade Approximant," Academic Press Inc., N.Y., 1975. Marquart, D.W.,"AnAlgorithm for Least Squares Estimation of Nonlinear Parameters," J. of SOc. lndust. Appl. Math., 11, 431, 1963. Bozic, S.M.,'Digital Kalman Filtering," Edward Arnold Ltd., England, 1982. Nestorides, E. J. "A Handbook on Torsional Vibrations," Cambridge University Press, 1958. Braun, S.G. and Datner, B., "Analysis of Roller/Ball Bearing Vibration," Transaction of the ASME, Jan. 1979.