Response near resonance of non-linearly damped systems subject to random excitation with application to marine risers

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8 AU&. 1983.

_Lab.

_v.

Scheepsbouwkunde

CH I E

Tech nische

Hogescbo.011818z'o3

Ocean Engng, Vol.9, No. 3, pp. 235-257 19

Deift

RESPONSE NEAR RESONANCE OF NON-LINEARLY

DAMPED SYSTEMS SUBJECT TO RANDOM EXCITATION

WITH APPLICATION TO MARINE RISERS

J.J.H BROUWERS

Koninklijke/Shell Exploratie en Produktie Laboratorium, P.O. Box 60, 2280 AB Rijswijk ZH The Netherlands

AbstractIn connection with questions concerning the resonant behaviour of certain types of offshore structures, the response of non-linearly damped systeinsto Gaussian excitation has been studied Both a single and a multi-degree of freedom system excited predominantly at one natural frequency have been considered. Explicit descriptions of the response, partly in the time-domain and partly in terms of non-Gaussian probability distributions, have been derived for the limit in case the damping is small The solutions have been used to indicate the effect of non linear damping on the distribution of response maxima, on the expected fatigue damage and on the expected extreme response. Comparisons have been made with results obtained from equivalent lineansation methods and with results obtained from numencal time domain simulations of the non-linearly damped response of a marine riser to random waves.

1. INTRODUCTION

THE THEORY for predicting the stationary response of linear systeths to Gaussiari

excitation is well developed. According to this theory, the response is also a Gaussian process, the properties of which can be determined using existing techniques of linear

analysis (e.g. see Bendat, 1955). When the system contains non-linear elements,

however, the response distributions are no longer Gaussian and in general, closed form

solutions do not exist Some limited information may be obtained from equivalent

linearisation techniques (Caughey, 1963a)

The particular problem of a simple non-linear oscillator excited by white noise can be

solved in greater detail Here, a Fokker Planck equation can be written for the joint

probability density of displacement and velocity Exact solutions for this probability density can be found for oscillators with non-linear stiffness (Caughey, 1963b; Lin 1968)

while approximate solutions can be derived in the case of non linear damping

(Stratonovitch, 1963; Roberts, 1977). The approximate solutions are asymptoticallyvalid as the damping tends to zero and can also be applied to non-linearly thmped oscillators subject to non-white noise excitation (Roberts, 1978).

In this paper, the stationary response near resonance of non-linearly damped systems to non-white noise, Gaussian excitation is studied in some detail. For a single degree of

freedom system excited at its natural frequency, an alternative method of solution is

-given which provides further insight into the applicability of previous solutions

(Stratonovitch, 1963; Roberts, 1977). The accuracy of the solutions is assessed and extension to multi degree of freedom systems excited predominantly at one natural

frequency is established.

The solutions are of particular interest for the analysis of non-linearly damped mechanical systems where response is governed by resonance, e.g. certain types of

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To obtain a solution which remains finite when the excitation frequency coincides with the natural frequency of the system, the non-linear damping term on the. left-hand side of Equation (1) has to be taken into account As for linear systems with small damping the power density .of response will then have a finite but large value in a small region of frequencies around the natural frequency (see Fig. 1). For small damping and for the natural frequency somewhere in the centre of the excitation spectrum, the area in this (2)

(3)

236 JJH.BRduwEis

offshore structures The effect of non linear damping on the distribution of response maxima, on the expected fatigue damage and on the expected extreme response is indicated Furthermore, comparisons are made with results obtained from equivalent

linearisation techniques and with resUlts obtained from numerical time-domain

simulations of the non-linearly damped response of a marine riser to random waves.

.2. SINGLE DEGREE OF FREEDOM SYSTEM

The equation of motion of the single degree of freedom system covered in this paper is

given by

ml + g(x) + mo02x = F(t) (1)

where x is the displacement response, m is the mass of the system, g(i) is the damping

force, w0 is the undamped natural frequency and F(t) is the excitation dots denote

differentiation with respect to the time t The damping force g(x) is assumed to be odd

with respect to ± The excitation F(t) is taken to be stationary, Gaussian and of zero-mean. The power density of the excitation, S(w), defined as the Fourier transform of the autocorrelation function of F(t) (Bendàt, 1955), is zero for negative values of the

frequency .

2.1. Response near resonance

Consider the case of small damping, i.e. the damping force g(i) is small as compared

to the inertia and restoring forces By dropping g(x) Equation (1) reduces to that of a

simple linear oscillator subject to Gaussian excitation The response is also a Gaussian process, the properties of which can be determined using existing techniques of linear analysis (Bendat, 1955). The power density of the response, S(w), is

S(w)

S(ü)

₌ _w02)2

Inspection of the dçnominatorin this equation, however, shows that the power density of

the response tends to infinity as w-w0, S(wo) # 0. This phenomenon is known as

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237 Excitation force - Displacement res / /

/

n poñse S(w0) wm?e2 wo Frequency

FIG. 1. _{Schematic of the pOwer density spectra of excitation and response.}

resonance peak will be much larger than the area underneath the response spectrum

away from resonance. The overall solutiOn is then dominated by response near

resonance and the objective is to calculate this response in the case of non linear damping.

To calculate the response near resonance, one may proceed as follows For small

damping, only a narrow band of frequencies around the natural frequency determines the solution If it is assumed that over this frequency band the excitationspectrum S(w) is

substantially constant the excitation may be represented by white noise of power density

S(o0) Solutions can then be obtained by applying successively averaging techniques and the technique of a Fokker-Planck equation to the equation of motion (Roberts. 1977). In

the averaging technique it

is assumed that the solution can be represented by a

sinusoidal wave of slowly and randomly varying amplitude and phase Approximate equations for amplitude and phase can subsequently be obtained by substituting the

sinusoidal representation into the equation of motion and then integrating equation of

motion and excitation over one period of oscillation (2irIw) thereby assuming that amplitude and phase are almost constant The resulting simplified equations of motion can be transformed into a Fokker-Planck equation for the Joint probability density of amplitude and phase Exact solutions for this probability densitycan be found for the

(4)

_ 4mw0

p(a) = C0 a exp

JJ g(wz siny) siny dy

dz}

00

(7)

238 J.J.H. BRouwERs

The averaging technique was first applied by Stratonovich (1963); it is an extension of the method of Bogoliubov and Mitropolski (1961) to stochastic problems More rigorous

investigations on, the mathematical foundation of the approach have been given by

Khasminskii (1966) and Papanicolaou and Kohler (1974). 2.2. Two-scale method

An alternative method for calculating non-linearly damped response near resonance is given in Appendix A. Rather than using averaging techniques, approximate solutions are derived by means of two-scale expansions. In common with the averaging technique, the two-scale method was onginally developed for deterministic problems like non-linear oscillators subject to sinusoidal excitation (Nayfeh 1973) However, when the excitation F(t) is represented by an infinite series of sinusoidal waves of different amplitudes and frequencies With random phase angles, a description often used to simulate Gaussian seas (Borgman, 1912), the two-scale method can also be applied to the present problem of random vibration.

As shown in Appendix A, the two-scale method involves reducing the equation of motion to approximate form by expressing the response in terms of two independent

time variables and expanding the solution and the excitation (not white noise) for

frequencies close to the resonance frequency using a small dimensionless parameter c.

The solutiOn can then closely be approximated by a sinuSOidal wave of slowly and

randomly varying amplitude and phase:

x(t, T) a(T) cos {coot ± 4(T)} (4)

where a and vary with a characteristic time T, which is related to t by

TEt

(5)

The amplitude and phase are described by two coupled stochastic differential equations

of the fitst order jn T. These equations can be transformed into a Fokker-Planck

equation for the stationary Joint probability density of amplitude and phase p(a ) for

which exact solutions exist (see Appendix A)

p(a,4) = p(a) . p(4),

(6)

where

(5)

which can be expressed jn closed form for some types of the damping function

g(ozsiny) Probabilistic descriptions for the displacement and velocity may be obtained by transforming (6), (7) and (8) from the a ₄₎ variables to the x, x variables using relation

(4).

-2.3. Conditions for validity

Th.e above descriptions for amplitude and phase are the same as those obtained from

averaging techniques (Roberts, 1977). The present method of using two-scale

expansions, however, has the advantage of yielding better insight into the nature of the approximations and the order of magnitude of the errors involved

Three types of approximation can be distinguished Firstly in the two scale expansions

terms of relative magnitude 0(r) are disregarded The resulting error'of 0(r) in the

solutions is small whenever

(10)

where r represents the ratio of the magnitude of the damping force near resonance to the

magnitude of the inertia or restoring force near resonance A rigorous perturbation

analysis with the proper definition of r is gwen in Appen4ix A.

Secondly the excitation spectrum S() is taken to be constant over a frequency band of width O(w0e) around the undamped natural frequence O. The two-scale expansions indicate that only this part of the excitation generates damping-controlled response near resonance The excitation may then be approximated by white noise of power density

S(wo) when

(ds\

woS(üo)II

_dw

r41,

j

W=W

a condition which is satisfied when r 1 and S(w) varies smoothly with respect to w around (U(z0.

Thirdly the solution for response away from resonance is neglected This implies that the area underneath the response spectrum away frOm resonance must be much smaller

239 and

p(4)) = (2ir)'

04)2ir.

(8)

The constant C0 is determrned by

C0

= j

I

L 4mwo If

J

. .1

g(zsiny) siny dy dz1 da,

(9)

a exp1

2()

J

(6)

(12)

240 JJ.H. BROUWERS

than the area in the resonance peak. The area away from resonance is approximately equal to U)0

I

S(w)

I

S(w) I 2

4dw*-

I 2

j

mw

_j

mw

0 U)0

see Equation (2), while the area in the resonance peak is O(wo3m2S(o0) r') (see Fig. 1). Hence,

w'S(wo) {

Jscoo

₊

J ()

S(w)dw} 1, (13)

a condition which is satisfied when E 1 and the natural frequency is somewhere in the

centre of the excitation spectrum. For natural frequencies WO far in the tails of the

excitation spectrum (woS(wo)0), on the other hand, the above condition may well be invalid, even for small values of the damping parameter e In that case, response away from resonance is important.

3. MULTI DEGREE OF FREEDOM SYSTEM

The solutions 4an be extended to non-linearly damped multi degree of freedom systems provided that the power density of excitation is such that excitation occurs

predominantly at one natural frequency of the system This has been demonstrated in Appendix B for a hon-linearly damped tensioned beam subjected to a randomly varying

transverse force, a problem which is of interest for the analysis of marine risers. As shown in Appendx B, for excitation at predominantly one natural frequency of the

system, the system responds predominantly in the corresponding natural mode

Couplings betweer the different natural modes, due to non-linear damping, can be

disregarded and the amplitude of the dominant mode can be described by an equation

which is analogous to that of the single degree of freedom system In this way the

solutions for the single degree Of freedom system then result in analytical expressions for the response of the multi degree of freedom system as well.

4. EFFECT OF NON'LINEAR DAMPING

To study the effect of non-linear damping, the damping force g(x) is expressed in the

form

=

c4t_i

(14)

where a 0. This description can be used to represent various types of actual damping. FOr example, Coulomb friction is represented by a0 and (linear) viscous damping by a1, fluid drag by a=2 and internal material fnction according to Lazan and Goodman

(1961) by a value for a varying from 1 3 for small amplitude motions to 7 for large amplitude motiOns. The influence of a on the distribution of response maxima, the

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expected fatigue damage and the expected extreme response using the previous solutions will be calculated below.

& 1. _{Distribution of response maxima}

If the damping force is expressed as in Equation (14), the probability density of

amplitudes or response maxima may be derived from (7) and (9) as

p(a) =

cr2

(a+1)r(1)a

_[

_I

2a2r2( 2

exp

_{[ -}

2crr(---\a±1)

\a+1

{

/ a+3 312(a+1) S(wo)f( 2 ₎ /a +2 4mcJT - 2 2

]

(15) (17) d, Xa8 , (19) where

f()

=

J

1e (16) 0

is the Gamma function tabulated by Abramowitz and Stegun (1968), and

is the mean square response. The distribution is shown in Fig. 2 for a=O, 1 and 2. Note that the solution results in the Rayleigh distribution when a=1, i.e. when the damping is linear, in which case

p(a) = -- exp

(-)

(18)

4.2. Expected fatigue damage

Consider the case that the response variable x represents a stress. Furthermore,

assume that fatigue damage accumulates according to the Palmgren-Miner hypothesis (Palmgren 1924, Miner, 1937) For the ith cycle with amplitude a, of the stress history described by Equation (4), the fatigue damage can be written in the form

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242 J.J.H. BROUWERS

0/0

FG. 2. Probability density of maxima for a damping force which is proportional to a power a of the velocity.

where A and 6 are material constants. The expected fatigue damage after n cycles is

according to Crandall (1963)

E[D] = n J Xa6p(a) cia , (20)

0

where p(a) is the probability density of the amplitudes Substitutmg the expression for p(a) given by Equation (15), we obtain

E[D]

f

2r()

(21)

where o is the root mean square stress. It can be seen from this equation that, apart from the values of the material constants A and 8 and apan from the root mean square value of

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TABLE 1. EXPECrED FATIGUE DAMAGE PER CYCLE VS THE NON-LINEARITY PARAMETER aFOR6 = 4.38 AND x = io_13 _{IN MINERS LAW AND FOR A ROOT MEAN SQUARE STRESS OF}_{30 MPA}

a=0 a=1 a=2 a=3

a4

ct5

106E[D]/n 1.b5 0.55 0.45 0.40 0.38 0.37

the stress, the expected fatigue damage pr cycle also depends on the value of the

non-linearity parameter a. The effect of this parameter has been indicated in Table 1. Here, we have given E[D]In vs a for 6=4,38, X=iO'3 and o=30 MPa; the vàlües of 8 and a correspond to the central part of the AWS X fatigue curve (Amencan Welding Society, 1979) sometimes applied to welded offshore structures

It is noted that the

expected fatigue damage decreases with increasing a.

A useful concept in connection with the expected fatigue damage is the idea Of an "equivalent" sinusoidal stress history with frequence w and amplitude A such that the damage after n cycles, i e nXA8 is equal to that given by Equation (21) (Crandall, 1963) The amplitude of the equivalent sinusoidal stress history must then satisfy

A=[

(.

ri

ii&

\ a±1 j

2

\a+1J

''

/

The values of A/cr calculated from this equation when 6=4.38 and when a=O, 1, 2, 3, 4 and 5, respectively, are given in Table 2.

TABiE2. RATIO OF THE AMPLITUDE OF AN "EQUIVALENT" SINUSOIDAL STRESS HISTORYTO ThE ROOT

MEAN SQUARE STRESS VS THE NONrUNEAPJTY PARAMETER aFOR6=4.38IN MINER'S LAW

aDO -

a1

a=2 a=3 a=4 a=5

2F(i)

₁

r() F

(22)

4.3. Expected extreme response

Consider a response history of n cycles. Assumingthat the amplitudes of the cycles are mutually independent the probability density of the largest amplitude or extreme a is according to LonguetHiggns (1952)

-

IdP"(a)l

p(a)

-da Ja=a (23)

(10)

I

2r()

E[a] = a I ( 4 "-

a+1

}

fi

+

/

-_1nF(--)

'\

(a-1)

_k21fl JK

(a+1)

+

a±1

/

a 1 a+i KlflK KlflK + + 0(k2) , (28) 244 J.J.H. BROIJWERS

where P(a) is the cumulative, probability density of amplitudes, defined by

P(a)=JP(a)da.

(24)

The expectation E[a] of the extreme is

E[a]

=

J

OdIa)da

₍₂₅₎

Integrating by part and assuming

.a{1 - - 0 when a , (26)

this reduces to

E[ã] =

J{i

r(a)}da.

(27)

After subsitution of the expressiOns for P(a) given by Equations (15) and (24), it is probably impossible to write the righthand side of Equation (27) in terms of known

functions for general values of n An asymptotic approximation however can be derived when n is large. The derivation is analogous to that of Longuet-Higgins (1952) for the hnear problem (p(a) equal to the Rayleigh distribution) and involves the reduction of the function under the integral for large n When n'1 P'(a) shows a rather sudden increase

from 0 to 1 in the neighbourhood of a=a0, a01. For a4a0, r(a) - 0 and for aa0,

r(a)

1, while the values of P'(0) around a=ã0 and the value of a0 itself may be

calculated using asymptotic expansions for P(a) when a is large. The approximation for the expected extreme response then obtained is

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TABLE 3. RATIO OF THE EXPECTED EXTREME RESPONSE TO THE ROOT MEAN SQUARE RESPONSE VS TER NON-LINEARITY PARAMETERaFOR RESPONSE HISTORIES OF 1000 CYCLES(n= 1000) AND2000 CYCLES,.

RESPECTIVELY

Response of non-linearly damped systems to excitation

where 'y is Euler's constant (=0.57722) and

in n (29)

The effect of the non-linearity parameter a on the expected extteme response using the

above equations has been shown in Table 3 Here we have given E[a]Icr versus a for

n=i000 and n=2000. It will be noted that E[ã]!cr decreases with increasing a.

5. EQUIVALENT LINEARISATION

The technique of equivalent linearisation is often used as an approximate method of solution for non-linear problems. In this method, the non-linear term is replaced by a linear one such that the mean square value of the difference is a minimum (Caughey i963a). The equivalent linear form of the non-linear damping term

C (30)

thus obtained, is

c(2Iit)"2

2J2 r

(a + 2)

cr'

(31)

where cr is the root mean square velocity. Replacing g(x) by (31) in Equation (1), the mean square displacement may be calculated as, to 0 (e),

2

2 1

7eq

= 2w02 (32)

a=0 a=1

a2 .

a=3 a=4 a=S

n=1000 5.60 3.87 326 2.95 2.76 2.63

Ea1Icr

(12)

2

a+1

a

I

2 fa+3.

a+1)

(33)

The values of aeq2/02 obtained from this equation when a=0, 1, 2,3,4 and 5 are given in Table 4 It can be seen that the equivalent hneansation method underpredicts the mean

square response. The deviation is small for moderate values of a, a> 1, but increases

with increasing a, a> 1, i.e. with increasing degree Of non-linearity.

- TABLE 4. RATIO OF THE EQUIVALENT LINEAR MEAN SQUARERESPONSE TO THE GENERAL MEAN.SQUARE RESPONSE VS THE NON'LINEARITY PARAMETER

FOr an equivalent linear representatiOn of the damping force and in the event of this damping force being small, the probability density of amplitudes or maxima is equal to

the Rayleigh distribution given by EquatiOn (18) with a=aeq The expected fatigue damage then calculated from Equation (20) and the expected extreme response then

calculated from Equation (27) when n is large are E[D]eq = ii

r(i

+

4-) X (V2 creq)&, (34)

EIã]eq =- \/2 aeqln"2n {1 + ½yln n + 0(ln2n)} (35)

where -y is Euler's constant (=0.57722). A comparison of these results with the general solutions given by Equations (21) and (28) using (33) has been shown in Table 5 Here

we have given E[D]eqIE[DJ when 6=4.38 and E[ãlea/EEdl when n=1000 and n=2000, for

a = 0, 1, 2 3, 4 and 5, respectively It can be seen that the equivalent lineansation method underpredicts both the expected fatigue damage and the expected extreme response

when a<1 The method overpredicts the expected extreme response when 1<a5

overpredicts the expected fatigue damage-when 1<a3 and underpredicts the expected fatigue damage when a>3.

246 J.J.H. BRouwERs

To compare this result with the general solution for the mean square displacement gwen by Equation (17), ëonsider

0.85 0.96 0.89 0.82 0.7-5

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247

TIE.5. RATIO OF THE EQUIVALENT LINEAR EXPECTED FATIGUE DAMAGE TO THE GENERAL EXPECFED

FATIGUE DAMAGE E[D] q/E[D] AND RATIO OF THE EQUIVALENT LINEAR EXPECTED EXTREME RESPONSE TO THE GENERAL EXPECTED EXTREME RESPONSE E[aJ q/E[a] VS THE NON UNEARITY PARAMETER a

6. APPLICATION TO MARINE RISERS

The given analytical sOlutions are of interest for the analysis of non-lineatly damped

mechanical systems where response is governed by resonance, e.g. certain types of. offshore.structures. An example of such a system is the 4 rn dia. riser of the SALS (Single

Anchor Leg Storage) system for oil production in water depths of 140 m shown in Fig 3 The response of this riser to unidirectional random waves has been calculated using the

above analytical methods, supported by the numencal time-domain simulation

techniques descnbed by Harper (1979) in typical cases The calculations showed that the response of the SALS nser is governed by resonance near the first natural frequency

The first natura frequency ( _{1.4 rad sec) is well in the centre of typical wave spectra}

while the damping is small The excitation energy at the higher natural frequencies (the second natural frequency is approximately 5 5 rad sec) is small and the nser responds

predominantly in the first mode of vibration (half a sinusoidal wave with respect to

vertical distance).

The following modes of fluid damping could be considered when studying the response

neat resonance of the riser:

Quadratic damping by drag forces In the analytical model, this corresponds toa

damping fOrce g() of the form

= c1ii

(36).

Linear viscous damping, reflected by a damping force g(k) of the fOrm

g(±)=c2x.

(37)

Whether quadratic or linear fluid damping governs depends on the ratio of the amplitude of thedeflection to the diameter of the riser (Stuatt and Woodgate, 1955). For the SALS nser, where this ratio is small (' 0 03), fluid damping can be expected to be linear The magnitude of this damping force, however, is very small Further investigations have shown that, rather than fluid forces, damping can also be due to Colomb fnction in the

a0

a1

a2

a3

a=4 a=5

E[DL1/E[DJ 8=4.38 0.37 1 1.13 1.05 0.93 0.79

n=1000 0.64 1 1.16 1.24 1.27 1.28

E[a]./E[a]

(14)

Mean

Un lye rso I joint

12m sea level 132n Sm I Gravity - I base

//'/////7 /7/i

Riser Yoke-Buoyancy chamber Buoyancy chariibe Vessel Longitudinal section of riser Flow iñes

FIG. 3. Schematic of SALS system.

universal joints at top and bottom of the riser. In the analytical model, this corresponds to a damping force of the form

g(±) _c3 (38)

In Figs 4 and 5, plots are presented of the cumulative probability density of

mstantaneous response and the cumulative probability density of response amplitudes

obtained from the analytical solutions in the case of Coulomb fnction and quadratic damping Furthermore in these figures the corresponding numerically calculated distributions of bending moment in the nser mid-section for a significant wave height of 1 m and a mean zero-up crossing period of the Waves of 4 sec have been shown. The

2m

31mm

4m 3lrnm

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BROUWERS-Probability less than

FIG. 4. Numerically calcu1ate distribution of instantaneots response for the SALS riser and the analytical distnbutions for Coulomb fnction and quadratic damping

distributions are normalised with respect to standard deviation (= root mean square value in the case of zero mean response) and are plotted on a Gaussian scale and a

Rayleigh scale, respectively. It should be noted that the Gaussian distribution and the Rayleigh distnbution correspond to the solution for linear damping From the figures it can be seen that the analytical and numerical results are in good agreement.

7. CONCLUSIONS

The main conclusions are summarised below.

Explicit descriptions of the response can be derived from the technique of

two scale expansions, a technique originally developed for deterministic

problems, and the technique of a Fokker-Planck equation.

The descnption consists of a sinusoidal wave of slowly and randomly varying amplitude and phase with the amplitude anl phase being expressed in terms of probability disttibiitions

This solution is valid for a lightly damped single degree of freedom system excited

at its natural frequency and a lightly damped multi degree of freedoth system

excited predominantly at one of its natural frequencies. The conditions fOr

validity can be specified in terms of the system parameters.

The solutions show that for a damping force which is proportional to a power a of the velocity, the probability density of large response amplitudes, the expected fatigue damage and the expected extreme response decrease with increasing a at constant. mean square response (see Figs 2 and 5 and Tables I and 3).

0 4'₀ >. a, V V_C a 4-a, -S a, a, C 0 in tfl 0 0 a, C a a 4. C H 4 3 0 -2 -3 -4

-

. . .

Numerical results for SALS riser

solutions aussian-' Coulomb friction -Quadratic damping Analytical Quadratic damping Coulomb friction aussia 0.0001 0.001 0.01 0.1 0.5 0.9 0.99 0.999 0.99%

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250 J:J.H. Baouwas 4 0 0 > w 0. 0 C 0 E 01 C 0 5 0

Numerical results for

SALS riser - Analytical Solutions Coulomb friction QuOdratic damping 0.01 0.5

9--

0;90.999

Probability less than

-FIG. 5. Numerically calcdlãted distribution of response amplitudes for the SALS riser and the

analytical ditributions fOr Coulomb friction and quadratic damping.

The accuracy of equivalent linearisation methods in the. mean square. response,

the expected fatigue damage and the expected extreme response can be

estimated and differences can be significant (see Tables 4 and 5)

The solutions compare favourably with results obtained from numerical

time-domain sirliulatioris Of a marine riser in resonance (See Figs 4 and 5).

AcknowledgementThe author wishes to thank Dr W. Visser for critical discussions. APPENDIX A

SOLUTION PROCEDURE FOR THE SINGLE DEGREE OF FREEDOM SYStEM For application of the two-scale method it is convenient to represent the excitatiOti F(r) in Equation of motion (1) by an (in)finite senes of sinusoidal waves of different amphtudes and

(17)

F(t) = lim {2LwS(w1)}"2 cos(w1t + p..,)

Here, S is the (one-sided) power density spectrum of excitation, and Lw, is the (in)finitesimal

width of the cells into which the power density spectrum is subdivided The random phase angles , are mdependent and umformly distributed over the interval (0 2ir) It can be shown by a central

limit theorem that F(t) converges to a Gaussian process when wjO (see PiersOn, 1955). Dimensionless rep resentätions and two-scale expañsIöns

In the calculation of the response near resonance using the two-scale method, the solution is assumed to depend on two time variables. In non-dimensional form these time variables can be Written as

T=W0t ,T=W0Et (a2)

where e is a small dimensionless parameter which will be defined below and.u0 is the undamped

natural frequency In association with the above fast and slow time scales we select frequency

components of the excitation near resonance according to

-we wo(1 -+ ev,) (a3)

with the dimensionless frequencies v such that v 0(1). The response variable x is expanded as

x = pE'{xo(T,T) + £x1(r,1) + 0(E2)} (a4)

where xo(T,T) and xl(-r,T) are dimensionless response variables which are of unit order of

magnitude and

'w0312S"2(

The magnitude of p corresponds to that of displacement response away from resonance for w0 in the centre of the excitation spectrum The factor E_U2 in expansion (a4) thus models the

arnplificatioh near res5tiaiwe. The dimensionless parameter E represents the ratio of the typical magnitude of the damping fOrce near resonance tO the t)ipical thagnitude of the inertia force or

restoring foite near resoOaOce:

X(x=woP"2)

2 -i/2 mw0 pe

The parameter can be expressed in terms of the system parameters as

C {pwo/Gy=inwo3 p2)}2,.

where C is the inverse function of .tg(i), defined by

yg(i),±=G(y),'0

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252 JJ.H. BRouwr.is

The inverse function an be expressed in closed-form for some types of the damping function g(i)

e g the damping function given by Equation (14) As will be seen below the form of the above epansions and the value of as defined by Equation (a7) lead to a balance of forces which is appropnate for response near resonance and which results in meaningful solutions for xo(T T) x(T,T), etc., aS.

Substitute (a2),. (a3) and (a4) into (1) and (al), and use the relatibjis

d

Ia

a 1 d2 2 1 a2 a2 2

= (oOt ₊

_?

° + aTaT + 0(E)

Furthermore, assume that the damping force can be expanded as

g(k) = poi0s"2 axo) {1

Then multiplying all terms by mw2p1 and using (a5) and (a6), we obtain as dimensionless

equation of motion

-ä2x0

+ xo + Xi + +

g0(!i)

f(rfl}

- _{0(E2) = 0.}

In this equation.

go(-°)

=g(± pwoE_u1t)/g(ipwoi_h12) (a12)

is dimensionless damping fOrce, go((äIäT)xo)) = 0(1), and

f(T,T) = lirn {2vjS(w0(1 + Ev))/S(w0)} cos(T ± vT + ) (al3)

is dimensionless e*cithtion force. The expression for the excitation force can be simplified by expanding the excitation .spectruth in a Taylor series arouid w = (or v = 0) accord iig to

S(w,) = S(1) ± E1fwo(

)

+ (a 14)

This shows that for the frequencies which are of interest to response near resonance. i.e. v = 0(1).

the excitation spectrum can be approimated by its local valUe at w=w0, i.e.

S(wI) = S(wo), . (a15)

when condition (11) is satisfied. Noting that frequency components away from resonance, i.e. v =

0(e') are unimportant for response near resonance (these components generate the solution

indicated by Equation (2) a solution which can be disregarded when condition (13) is satisfied) we

can extend approximation (al5) to the frequency range - v, + Equation (al3) can then

be written in the form

(a9)

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Response Of non-linearly damped systems to excitation

23

f(T,i) = f1(T)cosT - f2(flsinT, . (a16)

where

N

N..

f1(T) = lim_N-. _:=i> (2Av1)2cos(v,T + ,) f2(T) _N-.clim (2Lv,)"2sin(v,T + )

j

are mutually independent stationary white noise processes of zero-mean. The pOwer densities of f1(T) and f2(T) are tWo-sided and unity.

-Time-domain solutibns

Let e*O, equation of motion (all) then reduces to

(a 17)

(a18)

= rsinT. TCOST (a23)

th solu.tion of which is given by

x0 = d(1)cos{T + 4)(T)}, 0 a

<7l.

(a19) The slowly varying functions a and 4) in this solution follow from a balance of the tei-fns with

coefficient in equation of motion (all) Therefore substitute (a19) and (a16) into (all) Dividing all terms by E and letting e-0, We have

+ X1 2(a cos4))sinT + 2j(a siri4) cost +

-go(-a s.in(i + 4))) + fi (7) çosr - f2(fl SiflT. (a20)

Here, the damping term can be expanded as a Fourier series. Noting that g (a sin (T ± 4))) s

odd, we can wfite

go(a sin (i- + 4))) = ho(a) éos4) SIr1T + h1(a)cos24) sin2i + h0(a) sin4) cost + h!(a)sin24 cds2'r +

....

(a21)

where

h(a)

=

g(a sinii) sind1 . (a22)

Consiler now the terms on the right-hand side ofEquation (a20). Hete, terms Of the type sinr

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where

(a26)

p(4)) (2Tr)_i, 0 4) 2ir, (a28)

and

p(a) = C1 a

exf_

fho(a)da} , o a (a29)

The constant C1 is determined by the nornialisation condition

J

(a30)

On transforming the above solutions from the dimensionless vaiiabls a and4)defined by (a4) and

(a19) to the dimensional variables a and 4> defined by (4), we obtain (6), (7) and (8) and (9).

254 J.J.H. BROUWERS

Comparison with (a19) shows that, for these solUtions, x1/xo is unbounded as r°° and that the

expansion for x given by (a4) breaks down at T = O(E) To obtain an expansion which is uniformly valid for all r the coefficients of the terms sin'r and cosT on the right hand side of Equation (a20) have to be set equal to zero This condition yields equations for the slowly varying functions a( 1) and (1) in solution (a19), being

ho(a) cos4 + 2 (a côs)

f(1) ,

(a24)

ho(a) sin + 2 (a sin4)

f(7)

(a25)

Probabilistic descriptions of response

According to Equations (a4) and (a19), for E4l, the solution for response near resonance can closely be approximated by a sinusoidal wave of slowly and randomly varying amplitude and

phase The amplitude and phase are described by two coupled stochastic differential equations of first order in T given by Equations (a24) and (a25) To solve these equations for general

nonlinear forms of hó(a) is a cumbersome task. Rather than solving (a24) and (a25), we can determine the statistical properties of amplitude and phase from the Fokker Planck equation associated with (a24) and (a25) The Fokker Planck equation is a partial differential equation for

the Joint probability density whose coefficients are obtained from ensemble averages based on the equation of motion and the excitation process (Caughey 1963b Lin 1968) The stationary Fokker-Planck equation ãssociatéd with (a24) and (25) is given by Roberts (1977).

f{t(a_{2_)) +e}+t(ho(a)P)O

where p(a,4) is the stationary jOint probability density of amplitude and phase. The solution of Equation (a26) has been given by Roberts (1977) and can be expressed as

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Response of nonlinearly damped systems to excitation 255 APPENDIXIB

SOLUTION PROCEDURE FOR A MULTI DEGREE OF FREEDOM SYSTEM The solutions derived in Appendix A apply tO a system wth only one degree of freedom. In certain cases however these solutions can also be used in the descnption of the response of non hnearly damped multi degree of freedom systems Consider for example the problem of a tensioned beam subject to a randomly varying axially distnbuted transverse force F(z t) The

deflection x(z, t) of the beam is descnbed by

{E1

-

Te-}x:

F(z,t), (hi)

where t is the tithe, z_{is the axial distance along the beam, mis the mass,}_g(ax/at)_{is the damping}

force E is the modulus of elasticity I is the moment of inertia and Te is the tension applied at both ends The damping force g(3x/t3t) is assumed to be odd with respect to ax/at The transverse force

F(z,t) is taken to be stationary, fully coherent, Gaussian and of zero-mean; the power density

S(z w) of the force is zero for negative values of the frequency w The ends of the beam are simply supported:

x (32/äz2)x = 0 at z=0 and z = L, (b2)

where L is the length of the beam.

The above problem is of relevance to the analysis of marine nsers in random seas A marine riser can be represented as a tensiOned beam, the hydEodynamic forces being described by Monson s equation (Harper 1979) For a linear two dimensional Gaussian model of the sea and for the excitation being dominated by inertia forces the equation of motion of the nser can be reduced to that given b' Equation (bi).

The deflection can be expressed in terms of the natural modes of the beam as

x(z,t) > x1(t) (ilrz) (b3)

Equations for the amplitude x(t) of the natural modes can then be constructed by substituting

expansion (b3) into equation of motion (bi), multiplying all terms by sin (JTrz/L) and integrating with respect to from z=0 to z=L. Using the orthogonality relation

L I . /ilTz\ J

Sin--) S

0

(1)

di 0 for i#j ½L for i=j

and dividing all terms by ½L, we have

L

m11

J

sin(1) g(

tjsin(-))

dz + mw12x1 F,(t), 1=1,2, . (b5)

Here, F.(t),

j=

1,2..., are fully coherent Gaussian processes, the power densities of. which are

given by S1(w) {JL ) sin(I:-) dz} 2 0 (b4) (b6)

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256 - J.J.H. BROUWERS

while

mL)

mL)J

/

j2i/2

(b7)

j=1,2,. .. ,., are the tmdamped natural frequencies of the beam.

Consider now the case where the damping force represented by the secànd term on the left-hand

side of Equation (b5) is small When this term is neglected Equation (b5) reduces to a simple

linear equation for x1(t) the solution of whicb is inapplicable because of the infinite power in the

resonance peak of the response spectrum then obtained. As for the single degree of freedom

system, the non-linear damping term has to be taken into account. An additional difficulty in the

present problem is that the inclusion Of the non-linear damping term will lead to a coupling

between the amplitudes of the various natuial modes. This coupling, however, can be disregarded when the power density of the excitation is such that

Sk(wk) ' S(w1), (b8)

where k is some fixed positive integer and Since for small damping the power in the resonance peak ofx1is primarily determined by the local value of S1(w) at wo1, j1,2..., in

this case,

kIj'

jrrk. SOlutiOn (b3) may then be approximated by

x(z,t) = Xk(t) Sfl

I

kirz_L (b9)

while for t=k, and j=k, the equation forXk(t)is obtained from (b5) as

mx + gk(xk) + 171(IJkXk = Fk(t), (blO)

where

gk(xk) =

_fg(±ksin)sind

(bil)

It is thus demonstrated that when both the damping is small and excitation occurs mainly at the

kth natural frequency of the system, the system responds predominantly in the kth natural mode (cf Equation b9) The equation of motion for the amplitude of this mode (cf Equation blO) is similar to the equation of the single degree of freedom system (cf Equation 1) Replacing o by

0k, g(x) bygk(xk)and S(wo) by Sk(wk) the solutions given in section 2 2 also apply as descriptions

for Xk(t). These descriptions and solution (b9) are valid when conditions (10), (11) and (13),

modified for wk,gk(x)and Sk(w), and condition (b8) are satisfied. For significant excitation at more

than one natural frequency on the other hand the system will respond in more than one natural

mode. The behaviour of the amplitudes of these modes will then be subject to couplings when the damping is iion-linear.

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BoGouuBov N andMITROPOLSKI A 1961 Asymptotic Methods in the Theory of Nonlinear Oscillations 2nd edn. Gordonand Breach,New York.

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