27.3 The Structure-Based Connectionist Network

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The SBCN (Jammu et al., 1998) is developed to

take advantage of the integration capability of

neural networks, but to avoid the need for

supervised training. It defines the weights of the

network according to the structural knowledge of

the gearbox and the type of fault represented by

various vibration features. In the SBCN, the

structural influences, which represent the proximity

effect of faults on accelerometers, are

determined based on the root-mean-square

(RMS) value of the frequency response of a

simplified lumped-mass model of the gearbox.

Fault diagnosis in SBCN is performed by

propagating the abnormality-scaled vibration features

from SCBC to produce outputs representing

the fault possibility values for each gearbox

component as (Figure 27.7)

pkðtÞ ¼

Xn

i¼1

fiðtÞvik ð27:10Þ

In the above equation, pkðtÞ represents the fault possibility value for the kth component of the gearbox, fiðtÞ

denotes the abnormality-scaled value of a feature and vik represents the weighting factor determined based

on the lower and upper bounds of fuzzy influence weights ðlik; uikÞ as

vik ¼ lik þ ðuik 2 likÞfiðtÞ ð27:11Þ

Note that Equation 27.10 represents propagation of inputs through the weights of the SBCN, similar to a

regular connectionist network with vik as weights (Hertz et al., 1991). The difference is that the weights of

the SBCN vary within the range ðlik; uikÞ according to the magnitude of the corresponding input, fiðtÞ:

According to Equation 27.11, a higher abnormality-scaled feature value produces a higher weight value, vik;

emulating the reasoning by the human expert who pays more attention to the features that exhibit higher

abnormality values. In SBCN, in order to make uniform the interpretation of the fault possibility values

pkðtÞ; they are normalized to have values between zero and one as

ckðtÞ ¼

pkðtÞ Xn

i¼1

uik

ð27:12Þ

Accordingly, a ckðtÞ equal to one denotes a definite fault, whereas a value of zero represents normality. In

this system, fault diagnosis is performed hierarchically. First, the faulty subsystem within the gearbox is

identified by using the structural influences as vik: Then, the faulty components within the suspect

subsystem(s) are isolated using the product of structural and featural influences as vik:

The connection weights of the SBCN are defined based on the influences between the component

faults and vibration features. Ideally, the structural influences should represent the strength of

component vibration at the particular frequency (frequencies) represented by the feature. For this, the

attenuation property of the “travel path” between each component and accelerometer needs to be

modeled as a function of the moment of inertia, stiffness, and damping of the components in the

path (Badgley and Hartman, 1974; Smith, 1983; Lyon, 1995). In order to appreciate the difficulties

associated with vibration modeling of gearboxes, the vibration model of a simple gearbox is considered

Abnormality-Scaled

Features

Faulty

Components

uik

lik

Fuzzy

Influence

Weights

FIGURE 27.7 The structure-based connectionist network

(SBCN) with its fuzzy weights for isolating faulty

components.

27-8 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

(Choy and Qian, 1993):

½M􀀉½X€ 􀀉 þ ½ G_ v 􀀉½ X_ 􀀉 þ ½GA􀀉½X􀀉 þ ½Cb􀀉½ X_ 2 X_ c􀀉 þ ½Kb􀀉½X 2 Xc􀀉 þ ½Ks􀀉½X 2 Xr􀀉 ¼ ½FðtÞ􀀉 þ ½FGðtÞ􀀉

ð27:13Þ

where X represents the generalized displacement vectors in the lateral x, y, and z, and rotational ux ; uy ;

and uz directions, M denotes the inertia matrix, Gv represents gyroscopic forces, GA denotes the rotor

angular acceleration, Cb and Kb represent the damping and stiffness matrices of bearings, respectively,

½X 2 Xc 􀀉 denotes the casing vibration, ½X 2 Xr 􀀉 represents the shaft residual bow, and Ks denotes the

shaft bow stiffness matrix. The excitation force [F(t)] is due to mass imbalance, and [FG(t)] represents the

nonlinear gear mesh force, which in the x direction has the form (Choy and Qian, 1993)

FGxk ¼

Xn

i¼1;i–k

Ktki½2Rciuci 2 Rckuck þ ðXci 2 XckÞ cosðakiÞ

þ ðYci 2 YckÞ sinðakiÞ􀀉½cosðakiÞ þ SIGNðmÞ sinðakiÞ􀀉 ð27:14Þ

where FGxk denotes the gear mesh force in the x direction on the (n 2 1)th gear due to its mesh with

(n 2 1) other gears, Ktki represents the nonlinear gear mesh stiffness between the kth gear and ith gear, Rci

denotes the radius of the ith gear, aki represents the orientation angle between the kth and ith gears, and

m denotes the coefficient of friction. Mathematical relationships similar to Equation 27.14 can be defined

to represent the force in the y direction as well as torsional gear mesh forces. Furthermore, the vibration

of the casing due to the vibration of the gear– shaft system needs to be represented by a separate set of

coupled equations of motion similar to Equation 27.13. The above equations need to be integrated

numerically in order to estimate the vibration signal recorded on the housing, but the following

difficulties exist. (1) The values of stiffness and damping coefficients for the components are not readily

available. (2) The cross-coupling terms in the stiffness matrices in the x, y, and z directions cannot be

easily defined (Mitchell and Davis, 1985; Choy and Qian, 1993). (3) It is difficult to take into account the

multitude of travel paths and the associated models of attenuation for the many component –

accelerometer pairs in the gearbox. For example, Ktki, the gear mesh stiffness, which is obtained by

considering the gear tooth as a nonuniform cantilever beam (Lin et al., 1988; Boyd and Pike, 1989), is a

function of the cross section of the tooth at the point of loading as well as load variation due to changes in

the direction of load application (Mark, 1987; Lin et al., 1988; Choy and Qian, 1993), friction between the

meshing teeth (Rebbechi et al., 1991), contact ratio (Cornell and Westervelt, 1978), the type of gears

(spur, helical, etc.) (Mark, 1987; Lin et al., 1988; Boyd and Pike, 1989), and gear errors such as profile,

transmission, and manufacturing errors (Smith, 1983; Mark, 1987). Similarly, the stiffness of bearings is a

time-varying, nonlinear function of bearing displacement and the number of rolling elements in the load

zone, as well as the bearing type (roller, ball, etc.), axial preload, clearance, and race waviness (Harris,

1966; While, 1979; Walford and Stone, 1983). All these factors make it very difficult to obtain an accurate

and computationally inexpensive vibration attenuation model for gearboxes.

In order to avoid the difficulties associated with accurate modeling of vibration transfer, a simplified

method is described here that accounts separately for the two main aspects of vibration change by faulty

components: (1) the proximity effect of the faulty component on the accelerometer generating the feature

(structural influence), and (2) the frequency related information represented by the feature (featural

influence). The main simplification in this method is in the definition of structural influences as

the average strength of the vibration signal across all frequencies measured by an accelerometer due

to a component fault. To compute this average vibration, several simplifications have been adopted:

1. A lumped-mass model of the gearbox is used to model vibration.

2. Only the average static values for the stiffness coefficients are used, to cope with the absence of

accurate values for stiffness coefficients.

3. Damping ratios of bearings and shafts are neglected (Lin et al., 1988).

4. A damping ratio of 0.1 is used for all the gears as an approximation to actual gear ratios estimated

between 0.03 and 0.17 (Kasuba and Evans, 1981).

Fault Diagnosis of Helicopter Gearboxes 27-9

© 2005 by Taylor & Francis Group, LLC

5. The cross-coupling terms in the stiffness matrix are neglected.

6. Only the shortest vibration travel path between each component – accelerometer pair is

considered.

Using the above simplifications, the average vibration registered by an accelerometer due to a faulty ith

component can be simulated by applying an excitation source y at the ith component in the lumped-mass

model (Figure 27.8). In order to represent all frequencies in the excitation source, y can be selected to

consist of unit amplitude sine waves of all frequencies. The displacement of various components in the

travel path due to an excitation exerted at the ith component can be obtained for a typical N-mass path as

(James et al., 1994)

a11 a12 0 · · · 0

a21 a22 a23 · · · 0

.. .

.. .

.. .

· · · .. .

0 · · · 0 aN N21 aNN

2

66666664

3

77777775

x1

x2

.. .

xN

2

66666664

3

77777775

¼

y

0

.. .

0

2

6666664

3

7777775

ð27:15Þ

where ½x1; x2; · · ·; xN 􀀉T represent the displacements of the N components in the path, y denotes the

magnitude of excitation at the first component, and the coefficients aij are defined as

ann21 ¼ 2jvcn21 2 kn21 ann ¼ 2mnv2 þ jvðcn21 þ cnÞ þ ðkn21 þ knÞ

annþ1 ¼ 2jvcn 2 kn

ð27:16Þ

In the above equation, mn denotes the mass of the nth component, kn and cn represent the stiffness and

damping coefficients between the nth and (n þ 1)th components, respectively, and v denotes frequency.

In SBCN, the average vibration representing the overall vibration transferred from the component to

the accelerometer is characterized by the RMS value of vibration across all frequencies. RMS values of

vibration are readily obtained from Equation 27.15 by numerical integration of the square of

displacements across all frequencies. In these calculations, to avoid unnecessary numerical problems at

the natural frequencies of the components with negligible damping, the integration is carried out

excluding the natural frequencies.

For the purpose of assigning structural influences, the RMS values are scaled so that the component

directly adjacent to the accelerometer has the highest influence. Different functions can be used for

defining the influences. For example, Ii can be defined as

Ii ¼

logðriÞ

logðrN Þ ð27:17Þ

where ri represents the RMS value of vibration with the excitation source at the ith component, and rN

denotes the RMS value when the excitation is at the Nth component. In both cases, the accelerometer is

considered at the Nth component. The influence for the other components is obtained in a similar

fashion, by moving the excitation source to them in the travel path.

The RMS values, ri, are only approximate estimates due to the simplifications made for their

computation. Such approximate RMS values would, in turn, result in approximate influences. To

yi Sensor

k0 k1 k2 k3 kN-1

c0 c1 c2 c3 cN-1

m1 m2 mi mN

FIGURE 27.8 Illustration of the lumped-mass model of a vibration travel path consisting of N masses.

27-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

characterize the approximate nature of influences, they can be defined as fuzzy variables (Zadeh, 1975) by

mapping the influences into the range associated with fuzzy variables, such as nil: (0, 0.1), low: (0.1, 0.4),

medium: (0.4, 0.6), high: (0.6, 0.9) and definite: (0.9, 1).

Another body of knowledge commonly used by diagnosticians is that of the type of fault represented by

a feature. To incorporate this knowledge, fuzzy featural influences can be defined according to the

relation between the frequency content of each feature and the rotational frequencies of various

components (McFadden and Smith, 1986; Stewart Hughes Ltd., 1986). For example, a feature such as

envelope band energy (BE), which represents the energy at the bearings rotational frequencies and

harmonics, is assigned a featural influence of “high” in relation to bearing faults.

The SBCN is designed to provide fault possibility values for gearbox components without any prior

training. However, its design does not preclude the possibility of training when confronted with

misclassifications, which are in the form of undetected faults, false alarms, and misdiagnoses. Among

these, undetected faults are safety hazards that should be avoided at all costs, and false alarms and

misdiagnoses, although not as crucial as undetected faults, should be minimized so as to improve the

reliability of the diagnostic system. One of the features of the SBCN is its ability to benefit from

connectionist learning (Hertz et al., 1991) to improve diagnostic performance after each misdiagnosis.

For this purpose, an error-minimizing adaptation algorithm can be considered for adapting the fuzzy

influence weights of SBCN so as to avoid reoccurrence of misdiagnosis. This algorithm reduces the error

between the outputs of the SBCN ck(t) and the binary target Tk(t) obtained after inspection. The binary

target takes the value of zero for all the normal components and one for the faulty components.

Sequential update rules for adapting the fuzzy influences in SBCN have the form

lik ¼

lik þ h3ðTkðtÞ 2 ckðtÞÞð1 2 fiðtÞÞfiðtÞ if 0 , lik , 1

lik otherwise

(

ð27:18Þ

uik ¼

uik þ h3ðTkðtÞ 2 ckðtÞfiðtÞÞ2 if 0 , uik , 1

uik otherwise

(

ð27:19Þ

where h3 represents the learning rate. In this method, in order to allow uniform interpretation of the

trained fuzzy influences with respect to their original values, adaptation is stopped when the weight

values reach the bound of zero or one.

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