Journal of Sound and Vibration (1985) 101(2), 181-192

EXTENDED DAMPING MODELS FOR VIBRATIONDATA ANALYSIS

J. A. FABUNMI

Department ofAerospace Engineering, University of Maryland, College Park,Maryland 20742, U.S.A.

(Received 5 January 1984, and in revised farm 6 August 1984)

A simple 3 degree of freedom spring-mass-damper system has been used to investigatethe response of structural systems when the damping mechanisms are more general thanviscous or hysteretic damping. The damping forces in the dashpots were assumed propor­tional to a general power of the relative velocity, which led to a system of equations havinga frequency dependent damping matrix. The numerical results obtained from this studyseem to agree with the trends in the mobility measurements made on helicopter-likestructures by other investigators. The analysis also led to a formulation for the modalseries equation for such structures, which admits damping models more general thanviscous or hysteretic.

1. INTRODUCTION

Steady state vibratory response of structures to dynamic excitations is possible becauseof the mechanisms present in those structures which dissipate the energy that wouldotherwise cause the responses to grow without bounds. The theories that are used toanalyze the vibration of structures, or interpret measured vibration data, are only as goodas the model adopted for characterizing the damping in the structure. An instructivediscussion of the role of damping in vibration theory has been given by Crandall [1].

Nowadays, it has become accepted to use various techniques of modal testing to studythe vibration characteristics of a wide variety of structures, ranging from simple beamsto complex aerospace structures [2]. These techniques have proven sufficiently powerfulto the extent of identifying the resonant frequencies and "mode shapes" of most practicalstructures. However, only limited success has been achieved in predicting unmeasuredmobilities or vibration levels. In applications such as analytical testing [3], systemsidentification [4] or force determination [5], to name .a few, where these mobilities arethe starting point for further analyses, there arises a need for more accurate dampingmodels. Poor or inconsistent damping estimates result in significant errors in the predictedvalues of structural mobilities and response at locations on the structure where directmeasurements were not made.

In current modal testing and analysis practice, two types of damping model are popular,based on viscous or hysteretic damping assumptions. They both lead to systems of constantcoefficient linear differential equations of motion, from which the derivation ofthe mobilityfunctions is straightforward [6].

This paper is intended to contribute in the quest for more comprehensive dampingmodels by presenting a re-examination of the way in which damping considerations enterthe equations of motion for steady vibrations. For single degree of freedom systems,Jacobsen [7] approximated a general velocity power damping law by a coefficient multiply­ing the first power of velocity, with the appropriate sign to ensure energy dissipation.

181

0022-460X/85/140181 + 12 $03.00/0 © 1985 Academic Press Inc. (London) Limited

182 J. A. FABUNMI

This coefficient turned out to depend on both frequency and response level, and wasdetermined by matching the work dissipated per cycle of oscillation at any given frequencyof excitation. The effectiveness of this approximation has since been established by exactanalysis and experiment for zeroth (Coulomb) and first power velocity (viscous) dampinglaws [8].

An extension of this idea to a simple 3-degree of freedom system results in a frequencydomain system of equations, with a damping matrix which depends on both frequencyand response levels. These equations are investigated numerically to study how thedepartures from linear damping models affect the shape of the mobility functions,calculated as the ratio of response to excitation over a selected frequency band.

In mobility testing practice, a structure is said to behave "linearly" when the measuredmobilities are independent of the excitation level. For such structures, it is possible toleave out the dependence of the damping matrix on the response levels. In the resultingequations, the damping matrix can have a general functional dependence on frequency,usually consisting of linear combinations ofvarious frequency powers. By suitably defininga set of complex valued vectors, termed "damped modes", it is shown that the mobilityfunction can be expressed in the usual modal series form, suitable for modal analysis.The difference now is that the modal functions now have a more general dependence onfrequency than the usual formulations. This permits the "tailoring" of the assumeddamping laws to the actual behavior of a particular structure or class of structures. Itthen becomes possible to identify modal parameters which are more consistent with themeasured data.

2. SPRING-MASS SYSTEM WITH GENERAL DAMPING

The extension of Jacobsen's approximation to multiple degree of freedom systems isdeveloped by considering a simple arrangement of masses, springs and dampers shownin Figure 1. Three lumped masses Mil M2 and M3 are connected by linear springs K l e

O~---Df------<O

I' • I

/'

Figure 1. 3 DOF mass-spring-damper system with non-linear damping. The damping law of the ith damperis fd = 'FCc(dx/dt)"'.

K 2 , K 3 and non-linear dampers (C}, n1), (C2, "2) and (C3, "3); the damping force of thejth damper is

h= -Cjldx/dtl"j sgn (dx/dt), (1)

where Idx/dtl is the magnitude of the relative velocity across the damper (see Figure 1)and sgn (dx/dt) is its sign. The approximation consists of replacing equation (1) by

h=-Cjdx/dt (2)

DAMPING MODELS FOR VIBRATION ANALYSIS 183

(3)

for harmonic motions. In Appendix I, it is shown that during such motions, i.e., withx(t) = X sin wt, the damping force in equation (2) will dissipate the same amount ofenergy per cycle of oscillation if

c; =~;(r(nj ;2)/ r(nj ;3))X"j-1W"j-\

where I'( ) is the gamma function of the argument shown and w is the frequency. Inreference [7] it has been shown that the results obtained for a single degree of freedomsystem agreed very well with exact analysis in reference [8] for Coulomb damping (n = 0)and viscous damping (n = 1) laws.

Implicit in the foregoing approximation is the assumption that only the fundamentalharmonic of the excitation force is present in the response, even though the dampingmechanism is non-linear. This is not unreasonable. In modal testing practice, duringsteady state harmonic forcing of most structures, only the responses at the fundamentalfrequency of the forcing function are measured. If the coefficients in equation (3) inconjunction with equation (2) are used to describe the dampers in the system of Figure1, the following equations are obtained:

(-w 2( M ]+ [K ]+ i[D(w )]){ji} = {N}. (4)

Here {N} = {N} e i"" is the harmonic forcing vector, {y} = {y} eiwr is the vector of displace­

ments of the lumped masses from equilibrium, and {ji} and {N} are complex valuedvectors of the amplitudes of response and excitation respectively. The matrices in equation(4) are defined as follows:

[M]=[~I :2 ~], [K]=[K~;:2 K~:~3 -~3]'o 0 M) 0 -K3 K 3

[

Dl(W)+ D2(W) -Diw) 0][D(w)]= -D

02(W) D2(w )+ D3(w) -D3(w) ,

-D3(w) D 3(w)

D ( ) - c"l- - 1"2-12 W = W 2 Y2 - YI , D ( ) - c"l- -I" -13 w = W 3 Y3 - Y2 3 ,

In equation (3) X was the amplitude of the relative motion across the dampers. Thedependence of the damping coefficients on these amplitudes appears through the quantitiesenclosed between the I I notations above; CJ is different from C; for this reason.

If the dampers obey a first power velocity law, equation (4) reduces to the familiar setof equations governing a viscously damped system. Relative to the [M] and [K] matrices,the elements of the [D( w)] matrix are small, at most frequencies. However, in theneighborhood of an undamped natural frequency, when the w 2[M ] and [K] matricesneutralize each other, the role of [D(w)] becomes pronounced. For lightly dampedsystems, this region is narrow, and the damping matrix can be replaced by the constantvalues assumed at that frequency. The resulting equations now resemble the hystereticdamping form. Although the physical arguments may be slightly different, the measuredbehavior of the structure agrees with these equations. In the study of structures withmoderate levels of damping, both viscous and hysteretic damping models have been usedwith comparable success [6,2].

184 J. A. FABUNMI

(5)

2.1. NUMERICAL STUDY

The system considered for numerical investigation of equations (4) consists of threeuniform masses M, springs K and dampers (C, n). Let

w~=K/ M, w/wo= w, {M=(I/ Mw~){N},

1 r(T) CW~-211=---= --

J-rr r(n;3) M'

[

151«(;) )+D2 (w)

[D(w)] == - D~(W)

Then, for this uniform system, equation (4) can be written as

[[

2- a? -1 0] ]-1 2-(;)2 -I

N2

+i[15(w)] {ji}={M.

o -1 l-w

Equation (5) has been solved numerically by a method outlined in Appendix II, forselected ranges of Wbetween 0 and 2. This range of values can be shown to cover all thethree possible resonances of the system. In order to simulate single point mobility testing,only one element of {iL} is non-zero. The value assigned to this element is used to specifythe excitation level. The responses computed for each co-ordinate are then ratioed to thisnumber in order to obtain the displacement mobilities. The parameters varied in thisstudy are as follows: 1-t3, a measure of the excitation level at co-ordinate 3; YJ, a measureof the damping ratio (note that when the damping exponent is unity, this quantity isexactly the ratio of viscous damping coefficient to the critical damping coefficient); n,the exponent in the assumed damping law.

3. MODAL ANALYSIS

Let a set of "damped" modes of the system be defined by the characteristic equation

([K]+i[D(w )]){¢} = A(w)[M]{¢}. (6)

{4>} = {4>R}+i(4)I} is the complex characteristic vector, which is here termed a "dampedmode" in analogy to "flutter modes" in aeroelasticity where similar equations are encoun­tered (see, for example, reference [9]). For combinations of(Aj(w), {4>U and (AK(w), {4>L)which satisfy equation (6), it follows that

{4>}~([K]+ i[D(w )]){¢}j = Aj(w){4>}~[M]{¢ }j,

{4>}!([K]+i[D(w)]){4>h = Ak(w){4>}J[M]{¢}K' (7)

where {4>f denotes the transpose of {4>}. By virtue of the symmetry of the [K], [M] and[D(w)] matrices, equation (7) leads to the following orthogonality relationships:

{4>}J[M]{ cjJ}K =mAk' {¢ }J([K] + i[D(w )]){¢}K = (kj + idj(w) )Ojk' (8)

mj == {¢ }![M]{ 4>h kj ;;; {cjJ }![K]{ 4> b,

dj(w)s{¢}J[D(w)]{cjJh, 8jk={~: ;:~}.

DAMPING MODELS FOR VIBRATION ANALYSIS 185

From equations (6) and (8), it follows that

{¢ }J([K]- w 2[M]+i[D(w )]){¢h = (kj - w2mj+idj(w ))8jk = (Aj(w) - ( 2)mAk' (9)

It has been shown in reference [3] that the frequency dependent characteristic numbersare of the form

Aj(w) =n;(1 +igj(w)), (10)

where 12; is a non-negative real number associated with the jth mode, and by consideringthe form of the [D( w)] matrix, the function giw) can be expressed as

Nd

gj(w) =: aOj+ L akjw"k,k=l

(11)

(14)

(15)

where, in general, there may be Nd different damping exponents in the system, andaOj, akj, k = 1, 2, ... , Nd , are damping coefficients associated with each damping power law.

If [¢] denotes a complex valued modal matrix, such that its jth column is the jthdamped mode, equation (4) can be rewritten as

{y} = [¢ ]([¢ ]T([K] - w2[M ]+ i[D(w )])[¢ ])-l[¢Y{1V}. (12)

Equations (9) and (10) can then be substituted into equation (12) to yield

{Ji}= [¢][1/ mjnJ(1 - (w 2/ nJ) + igAw ))][¢]T{N}. (13)

The matrix of displacement mobilities [ Y( w)] is defined by the relationship

{y}=: [ Y (w )]{ N}.

Therefore, by comparison of equations (13) and (14),

[Yew)]=: Nf,<X) [{¢}"{¢}~J 2 2 12

, •

" = I m" n "(1 - (w /12") +1g" (W ) )

Equation (15) expresses the mobility functions as series of modal functions, summedover all the important modes in a given frequency band. The difference between thisequation and existing modal series formulations is that the function g,,(w) admits moregeneral damping assumptions than just viscous or hysteretic damping.

In the case of general damping, the term g,,(w) in equation (15) (or gj(w) in equation(13)) is of the form shown in equation (11) with the coefficients akj being somewhatdependent on the response level (a non-linear effect). In the case of zero exponent nj = 0,for which equation (1) models the Coulomb damping law, g" (or) will contain terms whichare inversely proportional to the response level. Resonance testing of such structures willtend to show damping levels which decrease with increasing response amplitude. Distor­tions of polar plots of frequency response curves from circles into pear-shaped loops dueto Coulomb damping were reported in reference [10]. In order to fit equation (15) tomeasured data, some considerations need to be given to the physical nature of the structureand its environment in assuming the most appropriate form for gn(w). It is probablyeasier to assume an exponent for the damping law a priori, rather than attempting toextract the exponent from a curve fitting algorithm.

4. DISCUSSION OF NUMERICAL RESULTS

The numerical results presented here were obtained by using equation (5) to simulatethe process of single point mobility testing of the structure depicted in Figure 1. Steadystate excitation at co-ordinate # 3 was simulated over selected frequency bands, and

186 J. A. FABUNMI

although the responses at all other co-ordinates were computed, in each case, only typicalmobility plots for one co-ordinate are presented for the purposes of the present discussion.

Figure 2(a) shows the plot of Y31 displacement mobility over the range of w between0·0 and 2·0 with 60 computation points, for different damping ratios 'Y) = 0,02, 0,1 andO· 5 with constant forcing level /-L3 =0,1, and damping exponent n = 1·2. Y31 displacement

Real

30·942

(0)

Imaginary

~. \ RealJ

\S.~) t>'" j-.\, \

..... \......,\.~

!1\

il'I'IIII.

illfoooo~1r-9029 e......--_.......... --.J

5·545-A0000

1Ivl

ImaginaryIiII'I~

31'829 r0000 Frequencyradio 2-000

(b)

Imaginary

/ ....

I<,

'\\I

1\ /\ /"''' /

<, L..- 1/ Real!\1\I \i .. \, .

- \I .. \A-~

-' / '",-< <' .....- .....

. -,> -....-.

4.50 ,-------------",----------,

228

-2-56 '--- -..:.."--- ----1

Real

Imaginary

Frequency rat /0 1-490

Figure 2. (a) Broad band plot of Y,l mobility, for different damping ratios; (b) narrow band plot of Y'1mobility near the second mode for different damping ratios. n=1'2, ""3 =0'100. - . -, ." == 0·02; - -, ." =' 0'10;-, .,,=0'50.

DAMPING MODELS FOR VIBRATION ANALYSIS 187

mobility means the displacement response of co-ordinate # 1 per unit forcing at co­ordinate # 3. Due to the symmetry of the mass, stiffness and damping matrices, both Y31and Y13 will yield the same results. As expected, increasing the damping ratio reducesthe peak responses at resonance and broadens the band of influence of a given mode. Abetter definition of the behavior of the response around a given mode is achieved by

!5.30 ,-----.--------------,

Real

000 ~--l----___:"....,.::=~===__l

-5-45 L-__-!- -'

100r:=::==~==~~=:;~==::J000 I-

(al

Imaginary

I maginory

2·00Frequency ratio

-102'-- --'

000

Real

(bl

1-490Frequency ratio

Real

o 721 ,------------~~:::---,

OOOOI----------,!~---__I

-1040'-- --'-"-_~ _'

1151 ,.---------.,.....---------,

00544 oc- --'

0820

Imaginary

Figure 3_ (a) Broad band plot of Y31 mobility for different force levels; n '" 1,2, 1'/ :0 0'10; - --, 1J-3 = 1-0;-- --, 1J-3 :O 2-0; --, 1J-3 = 10-0; (b) narrow band plot of Y 3 1 mobility near the second mode for differentforce levels; n'" 1-2, 1]:0 0-10; --, 1J-3 :O 0-1; -1:>-, 1-'3 = 0-5; --, J1-3 :O 1'0; --0-, 11-3:0 2-0; - --, 1J-3 = 10·0.

188 J. A. FABUNMI

increasing the number of computation points per frequency spacing. Thus, Figure 2(b)shows a narrower range of to between 0·82 and 1·49 centered near the second resonantmode with 100 computation points, for the same Y31 mobility. Here it is seen that thedamping ratio, for the exponent of the damping law being considered, affects not onlythe level of response at resonance but also the shape of the modal loop.

In Figures 3(a) and 3(b), the effect of different forcing levels is shown: J.L3=0·1, 0,5,1,0, 2-0 and 10'0, while the damping ratio TJ = 0·1 and damping exponent n = 1-2 wereheld fixed. On the broad band plot, slight decreases in the peak mobility values withincreasing force level are observed in regions close to the resonant modes. In mobilitytesting practice, considerable judgement about the linearity of the structure is based oncomparisons of the measured mobilities at different force levels. If this is done with broadband data, among other things, the differences in the mobilities for different force levelsmay escape detection due to poor frequency resolution. However, Figure 3(b) shows thata narrow band plot, around a selected mode (here, mode # 2), reveals that the forcinglevel affects not only the response level, but also the shapes of the curves at resonance.It is interesting to note that the decreasing mobility peaks with increasing force level havebeen reported during actual testing of helicopter-like structures [11]. When these calcula­tions were repeated with an exponent of unity in the damping law, no differences inmobilities were observed at different force levels.

In Figures 4(a)-(c), the effect of different damping exponents on the displacementmobility Y33 are presented. The exponents considered were n=1'2, 1,5,2,0, and 3,0,while the damping ratio 'T/ = 0·1 and force level /-L3 = 0·5 were kept the same. Again it isseen that the broad band plot does not reveal much about the differences in the mobilities.Figures 4(b) and (c), however, show that the different exponents result in different shapes

134,..------,-------------,

Imoginary

(a)

-

Reel

000 t-----jr------=::::::::'--=====--I

-113'--__....... ---'

0-00 1---.~-;:..---::::::::::=::7"'"--_,

Imaginary

200Frequency ratio

-221'--__-"--- -'0-00

Figure 4. (a) Broad band plot of Y33 mobility for different damping exponents; 'TJ =0·10, J.L3 = 0·50; -,n '" 1·5; - -, n = 2'0; _. -, n = 3·0; (b) narrow band plot of Y33 mobility near the first mode for differentdamping exponents; 7/=0·10, J.L3",0·50; --, n=1'2; -, /1=1'5; --. n=2'0; ---, n=3·0; (c) narrowband plot Of Y33 mobility near the second mode for different damping exponents; 'TJ = 1·0,_ JL3= 0·50; - .. -,n = 3·0; --, n =2·0; - . -, /1 = 1·5; - _., n = 1·2.

DAMPING MODELS FOR VIBRATION ANALYSIS 189

0·640Frequency ratio

151

Real (b)

0·00

,-

-129

00

- 26 31::-:-:-:---::-------,--"------:--:--::-'0170

Imaginary

0295 ,--------;r-<..___--------,

Imaginary

(c)

1490Frequency ratio

Real

0000 r------,-ry4'7"--....:...:,r------I

-0·937'-- --'--::..._.-J

-0-023 ,-~=--------------,

Imaginary

Figure 4 (cont.).

of the mobility functions around the resonant modes. The modal loop departs from itscircular form as the damping exponent is increased. In fact the "flattening" of the circleresembles a behavior that has often been blamed on signal processing errors. Althoughthere is no doubt that signal processing errors can be very significant it is neverthelessimportant to recognize that, in some cases, the structure can be revealing more of its ownnature than the model being used to analyze its responses.

190 J . A. FABUNM I

5. CONCLUSIONS

A simple 3 degree of freedom spring-mass-damper system has been used to investigatethe response of structural systems when the damping mechanisms are more general thanviscous or hysteretic damping. A formulation of the modal series expression for themobilities for such generally damped structures was also presented. This formulationallows the modelling of general damping behavior by means of a frequency dependentfunction, the form of which is indicative of the velocity power laws prevailing in thedamping mechanism.

A numerical simulation of mobility testing of this simple structural model revealedtrends in the mobility functions which are similar to those reported in the testing practiceof damped complicated structures such as helicopters and the like. It was found thatdepartures of the modal loops from pure circles may be due to the existence of non-lineardamping mechanisms within the system. Linearity checks of broad band mobility datamay not be sufficient to establish the degree to which a structure is exhibiting non-lineardamping behavior. When the exponent of the velocity power law prevailing in the dampingbehavior of a structure is larger than unity it was found that the peak mobilities atresonance decrease with increasing excitation level.

Further research is needed to validate new modal analysis techniques for handlingstructures with damping behavior which departs significantly from viscous or hystereticdamping models.

ACKNOWLEDGMENTS

This work is part of ongoing research funded by the United States Army ResearchOffice, through the Center fo r Rotorcraft Education and Research at the University ofMaryland. The author also wishes to acknowledge the computational facilities providedby the Computer Science Center of the same University.

REFERENCES

1. S. H. CRANDALL 1970 Journal oj Sound and Vibration 11, 3-18. The role of damping invibration theory.

2. VARIOUS AUTHORS 1982 Proceedings oj the First International Modal Analysis Conference.Orlando, Florida.

3. W. G. FLANNELLY, J. A. FABUNMI and E. J. NAGY 1981 NASA CR 3429. Analytical testing.4. N. GIANSANTE, A. BERMAN, W. G. FLANNELLY, E. J. NAGY and J. A. FABUNMI 1981

Kaman Aerospace Corporation Report No. R-1631. Structural systems identification technologyverification.

5. R. JONES, W. G. FLANNELLY, E. J. NAGY and J. A. FABUNMI 1981 Kaman AerospaceCorporation Report No. R-1625 . Experimental Verification of Force Determination and GroundFlying on a full scale helicopter.

6. D. J. EWI NS 1979 (September) Journal oj the Society of Environmental Engineers, 1-13. Whysand wherefores of modal testing.

7. L. S. JACOBSEN 1930 Transact ions oJ the American Society ojMechan ical Engineers 52,169-181 .Steady forced vibration as influenced by damping.

8. J. P. DEN HARTOG 1931 Transactions oj the American Society oj Mechanical Engineers 53,107-115. Forced vibrations with combined Coulomb and viscous friction.

9. A. V. SRINIVASAN and J. A. FABUNMI 1984 Journal oj Engineering for Gas Turbines andPower, Transactions of the American Society ofMechanical Engineers 106,35-43. Cascade flutteranalysis of cantilevered clades.

10. G. R. TOMLINSON 1980 Journal of Sound and Vibration 71, 443-451. An analysis of thedistortion effects of Coulomb damping on the vector plots of lightly damped system.

11. D. J. EWINS 1983 Vertica 7, 1-8. The effects of slight nonlinearities or of modal testingof helicopter-like structures.

DAMPING MODELS FOR VIBRATION ANALYSIS 191

APPENDIX I

The energy dissipated per cycle of oscillation by the damping force given in equation(l) is _ 12

.../W (dX) .+1

Wn - Cn d dr,o t

For harmonic motions, given by x = X sin (wt),

f7T/2w

w, =4Cnxn+lwn+1 0 [cos (wt)]"+1 dr,

It can be shown that

1'TT 12

o [cos (O)]2m+1 dO = (.J:;/2){f(m + I)/f( m +3/2)}.

Hence, for n = 2m,

(Al)

(Al)

(A3)

(A4)

When n = 1, WI = 7rC1X2

W . Therefore the required coefficient of a first power velocitydamping law that will make the dissipated energy equal that of a general power velocitydamping law is

(A5)

APPENDIX II

. Let the response of coordinate j at a selected frequency ratio be Yj= yjR + iyJ, and thejth element of the forcing vector be IJ-j = IJ-jR + iIJ-f· The solution to equation (5) can besought as roots of the following non-linear system of equations:

( -):: [[E(W)] -[D(w, x)]]{_}_{ '}=Oz x, W -[D(w, x)] [E(w)] x f.L , (A6)

{X}T == [yf, yf, yf, y[, y~, yj],

[2-2-w

[E(w)]= ~1

{ '}T _ [R R R I I 1].IJ- = J.LI,J.L2,f.L3,!-I-1>!-I-2,IJ-3,

-1 0]2-w 2 -1 .

-1 l-w2

[D(w, x)] here is the same as [D(w)] defined earlier, with the elements of the {x} vectorplaced appropriately.

Let a perturbed function be defined as

{z(J, x, w)} == {z(O, x, w)} + ({z(x, w)} - {z(O, x, w)})J/ N,

_ _ [[E(W)] -[Do(w)]] _ _,{z(O,x,w)}= -[Do(w)] [E(w)] {x}-{J.L},

[2-I 0]

[Do(w)]~21Jwn -1 2 -1.

° -1 1

(A7)

(A8)

192 J. A. FABUNMI

N is some selected integer, and J is also an integer which varies from 1 to N. It isapparent that {zeN, X, £lin ={z(x, wH. The root of {z(J -1, X, w)} = 0, {i}l-I, is a suitablestarting vector for seeking the root of {z(J, i, w)} =0, by using the well known secondorder convergent Newton-Raphson scheme, whereby

{ik+I}J ={ik}J -[a{z}k / a{x}]:fl{z(J, xk, Wn. (A9)

The matrix [a{z}k/ c3{x}] is the Jacobian of the {z} vector with respect to the {x] vector,evaluated with the {ik}J values .

With {.i0}J = {.i}l-I> equation (A9) is iterated until suitable convergence is achieved.The final solution, for J = N, is the desired root of equation (A6).