CRANFIELD REPORT AERO No. 20
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INSTITUTE OF TECHNOLOGY
APPLICATION OF THE FOKKER - PLANCK EQUATION TO RANDOM VIBRATION O F NONLINEAR SYSTEMS
by
1
Cranfield Report Aero No„20 April 1974
CRANFIELD INSTITUTE OF TECHNOLOGY College of Aeronautics
APPLICATION OF THE FOKKER-PLANCK EQUATION TO RANDOM VIBRATION OF NONLINEAR SYSTEMS
by
Colin L„ Kirk SUMMARY
The stationary Fokker-Planck equation is applied to the determination of the probability density function of
the response of a single-degree-of-freedom system with linear and cubic stiffness terms, subject to Gaussian white noise excitation„ The statistical quantities evaluated are
(1) mean square amplitude (11) average rate of level
crossing (111) mean value of peaks» Application of (11) to a random Palmgren-Miner fatigue analysis reveals that a
hardening spring characteristic increases the expected fatigue life.
For systems with nonlinear damping proportional to
x" sign„x the equation of motion is reduced to an equivalent nonlinear differential equation for which the exact probability density function can be obtained from the Fokker-Planck equation„ Comparison is made with the equivalent linearisation method
which assumes a Gaussian response and it is shown that this assumption always over estimates the mean square responsCo
CONTENTS
Page
SUMMARY NOTATION
INTRODUCTION 1 THE THEORY OF MARKOFF RANDOM PROCESSES 3
THE FOKKER-PLANCK EQUATION 8 ANALYSIS OF S.D.F. SYSTEMS WITH NONLINEAR
STIFFNESS AND DAMPING 22 REFERENCES
FIGURES
Influence of Nonlinear Spring Stiffness on Mean Square Response
Influence of Nonlinear Spring Stiffness on Average Rate of Crossing Level 'a'»
Average Rate of Crossing Level 'a'»
Influence of Nonlinear Spring Stiffness on Mean Value of Peaks.
Influence of Nonlinear Spring Stiffness on
Cumulative Fatigue Damage for Aluminium Alloy 75S-T6
NOTATION
a amplitude level
Ai limit of first moment (eq.31) Bjj limit of second moment (eq„32) b,c constants in eq.87.
D expected fatigue damage <cs> average clump size
E total energy of system <e(x,x)^>mean square error function F(x) nonlinear spring force f(t) exciting force
kj^ stiffness constants n,N integers
Pjj,Pcn "th order probability and conditional probability density functions respectively
P(y) probability density function
p ( x ) ^ T probability density function for nonlinear system R(T) autocorrelation function
r ratio ka/ki (Cubic spring constant , ' ^linear spring constant T> . . .nonlinear>
R ratio (—=-; ) mean square response ^ linear -' "i ^ S stress level
S Q Power spectral density of white noise exciting force t, T time
T expected time to fatigue failure x,y,z displacements of random process
g,g linear and equivalent linear damping ratio e nonlinear damping coefficient
Y damping coefficient r Gamma function
öx.ö^ r„m.s. displacement and velocity respectively
v+ average number of crossings of level 'a' per second
EL
1. INTRODUCTION
This report investigates the random vibration of single-degree of freedom systems having linear stiffness and
non-linear damping. The study is of value to engineers concerned with dynamic stresses and fatigue damage in
structures subjected to randomly varying forces. Examples of such structures are T.V. masts, tall chimneys and suspension bridges excited by atmospheric turbulence and wave loading of off-shore drilling platforms.
When random forces applied to a structure are severe for example during a storm, large dynamic stresses and deflections can arise causing failure or malfunctioning of the structure. It is usual for large deflections in engineering structures to be accompanied by non-linear damping and stiffness effects which in general are thought to be beneficial in that they increase with the level of
response.
The subject of random vibration of linear systems under stationary excitation is highly developed for which there is a
large body of published literature. The power spectral analysis of linear systems requires the following major assumptions to be made, (i) stationarity of the excitation (ii) linearity of the equations of motion. Furthermore it is known that the response of a linear system to Gaussian excitation is also Gaussian. In contrast, the response of linear systems to Gaussian excitation is non-Gaussian. One of the first major studies of the random vibration of non-linear mechanical systems was made by Caughey, Crandall and Lyon (References 1, 2, 3, 4 ) . In reference 1, Caughey introduced the study of discrete
non-linear dynamic systems subjected to white noise random excitation, by way of the stationary Fokker-Planck
equation whose solution yields the joint probability density of the response. The present paper builds on Caughey's
work and extends the range of application with wider
parameter studies of mean-square response, level crossing rate, mean peak values and the Palmgren-Miner expected fatigue damage criterion. In reference 2, Crandall
examined various statistical properties of response using the joint probability density function (J.P.D.F.) obtained from the Fokker-Planck equation, the analysis being
restricted to small degrees of non-linearity. In the
present paper this restriction is removed by evaluating the various complicated integrals by means of a digital computer for a wide range of increasing values of non-linear stiffnes. It is worth remarking that in using the Fokker-Planck equation to obtain the probability density for a system with
non-linear stiffness only, there is no restriction on the degree of the non-linearity.
2
-In reference 3, Crandall introduced the perturbation technique for the study of systems which are only slightly non-linear and showed that with positive non-linear
stiffness, the mean square response is lower than for the corresponding linear system. The non-linear stiffness was also found to raise the expected frequency of zero crossings for all levels of excitation. The present paper shows that the expected rate of crossing level 'a' may increase or decrease depending on the r.m.s. level of excitation and the degree of non-linearity. In reference 4, Caughey applies the Equivalent Linearisation technique to non-linearities in displacement, velocity or of the hereditary type. The method is restricted to small
non-linearities but can be applied to non-white excitation.
For multi-degree of freedom systems having damping forces proportional to (velocity)'^, the equivalent
linearisation method can be used in which the coefficient of non-linear damping is replaced by an equivalent
quasi-linear damping coefficient which dissipates the same average energy as the non-linear damping. In references 5 and 6 the author applied this method to taxying induced vibrations of aircraft in which the damping in the
undercarriage was of the (velocity)^ type. Because the aircraft were excited by non-white runway displacement inputs and possessed more than one-degree-of-freedom, a cyclic procedure was adopted to determine the equivalent linear damping which was proportional to the r.m.s. sliding velocity of the oleo strut. A similar method has been
used by Penzien and Kaul (reference 7) to analyse the
response of off-shore towers to strong motion earthquakes. In this case the (velocity)^ damping was caused by interaction of the structure with the ocean. In references 5, 6, 7 it was necessary to assume that although the systems were
non-linear the response remained Gaussian. For an N-degree of freedom system there is no available method for
assessing the degree of error involved in assuming Gaussian response for the purposes of linearising the non-linear damping. In a private communication however Professor T.K. Caughey has described an approximate method for obtaining the J.P.D.F. for S.D.F. systems having non-linear stiffness and non-linear damping. The method combines an exact
solution of the Fokker-Planck equation for a particular
form of non-linear differential equation, with the equivalent linearisation technique. This method is described in some detail at the end of this paper together with its derivation from the Fokker-Planck equation.
From the above discussion it can be seen that the Fokker-Planck equation plays a central role in the study of random vibration and stochastic processes. It is therefore important for engineers requiring a more fundamental understanding
of random vibration, particularly of non-linear systems, to study the derivation of the Fokker-Planck equation and its solution in the stationary case. For this reason the paper begins with an introduction to the theory of Markoff random processes as a basis upon which the Fokker-Planck equation is derived.
3
-2. THE THEORY OF MARKOFF RANDOM PROCESSES
A relation observat the pres the Stat probabil fundamen pertaini shown th particul exponent (a)
Markoff process is a random process whose ship to the past is confined only to the ion of the process that immediately precedes ent. Such processes are completely defined in istical sense by their transitional or conditional ity laws. These laws are obtained as the
tal solution to the Fokker-Planck equation ng to a particular dynamic system. It can be at a stationary Gaussian random process is a ar case of a Markoff process when it has an ial autocorrelation function (Reference 8 ) .
Joint Probability
To formulate the concept of a Markoff process it is necessary to first define certain joint probability density functions (J.P.D.F.). Consider therefore a random process y(t) at various sequential times ti, ta
and let
exists in the range ... t as shown in Fig.l,
Pi(yt)dy = the probability that y y to y + dy at time t
Pa(y1t1,y2tJdyidy2 = the joint probability that y exists in the range yi to yi + dyi at ti, and in the range ya to ya + dya at ta P 3 ( y i t 1 . y a t a , y 3 t 3 ) d y i d y 2 d y 3 = (1)
:f^t)
FIGURE 1
It follows from eq.(1) that the general n-dimensional joint probability density function (J.P.D.F) can be
written
P n ( y i t i .yata , ... yj^t^)dyidy2 . . . d y ^
= Prob. fyi <y(ti) ^ yi + dy i ; ... y^^ < y(t^) ^ y^ + dy 1 ... (2)
For example if n = 2 and n = 3, eq.(2) yields
P2(yiti,yat2)dyidya = Prob fyi < y(ti) 4 yi+ dyi;
ya < y(t2) < y? + dy-A
P 3 ( y i t i , y 2 t 2 , y 3 t 3 ) d y i d y 2 d y 3 = P r o b . f. . . , y 3 < y ( t 3 ) < y r ) + d y ; J . . . ( 3 )
It can be seen that pg implies both pj and P2 and that in general p^ implies all previous p, , with k <n. In order to relate pj^ to p^^, it is helpful to first consider the
simple case of the two-dimensional joint probability density function p(x,y), which forms a surface in the x-y plane. Thus we can write
p(x)dx = p(x,y)dx.dy = 1 . (4)
Differentiating eq.(4) with respect to x yields
P(x) = P(x,y)dy
> - o o
.. (5)
Extending the idea of eq.5 to the present situation gives the relation
Pi(yiti) = r P2(yiti,y2t2)dy2 ... (6) J " CO .oo
P2(yiti,y2t2) = P3(yiti,y2t2,y3t3)dy3
»0O .00
hence pi(yiti) = Ps(yiti,y2t2,yat3)dy2.dy3
i- 00'- 00 or in general
00
Pk(yiti,y2t2 ,. . . Yktk) = J J PnCyiti,y2t2,
Vn)
'>^-k dy, ,
fold ^k+1 dy
n The probability density functions p ^ are all
invariant with respect to time translation for a stationary random process.
We now recall that a Markoff process is defined as a current random process that is related only to the
immediate preceding observation. In order to formulate a Markoff process mathematically it is also necessary to
introduce the concept of conditional probability as follows.
(b) Conditional Probability
If at time t = ti , y = yi , then the conditional probability of finding y in the range y2 to ya + dy2 at a time ta - tj later, is given by the expression
Pc(yi/y2,t2 - ti)dy2
This situation is depicted in Fig.2.
(7)
5
-In order to find P2(yiti,y2t2), i.e. the joint probability that y exists in the range yi to yi + dyi at
ti and in the range y2 to y2 + dy2 at t2, the following relation is used
P(AB) = P(A)P^(A/B) ... (8)
which defines the probability of events A and B occurring together, in terms of the product of the probability that A occurs and the conditional probability that B occurs, when it is known that A has already occurred. Thus
following the example of eq.(8) we can write
P2(yiti ,y2t2) = Pi(yiti).p (yi/y2 ,t2-ti) ... (9)
Ca
Eq.(9) states that the joint probability of y existing in the range yi to yi + dyi and in the range ya to y2 + dy2 at t2, is equal to the product of the probability that y exists in the range yi to yi + dyi at
ti , and the conditional probability of finding y in the range ya to y2 + dya at time ta - ti later. Since
Pi(yiti) will be the known one-dimensional density
function,then pa is defined in terms of Pca• The latter conditional density function integrates in the same way as an ordinary density function, thus
[" Pc2(yi/y2.t)dy2 = 1 ... (10) ' _ o o
From the first of eqs.(6), by exchanging y2 for yi , we have Pi(y2t2) = = f Inserting (9) in (llT gives P2(yiti,y2t2)dyi ... (11) Pi(y2t2) = I Pi(yiti).p^2(yi/y2.t2-ti)dyi... (12) — 0 0
In addition to eqs.(10) and (12), p must satisfy the condition that p (yi/y2,t) > 0. ^
C2 ^
Now by definition a Markoff process is a random phenomenon characterised by an nth order conditional probability density function, for all n ^ 2. This defines the conditional probability that y lies in the
interval yi to yi + dyi at ti , ya to ya + dyaat
t2, y^.j to y^_^ + dy^_j at t^_^ The Markoff
process depends only on the process that immediately precedes the present, i.e. only on y at t and y , at
f ^ ' • ' • ' n n n - i
t , . Thus it is necessary for ti< ta < ..-. t . Furthermore the conditional density function is written
6
-Eq.(13) states that the probability of the random process y(t) taking on a value of yj^ at time t = t^j, assuming
that its values at some previous time are known, depends only on the most recently observed value, y(t) = y^-i at t = tn_j. Such relations are described by Y.K.Lin (Ref.9) as "one-step-memory random processes". The conditional
probability function appearing on the right-hand side of (13) is known as the "transition probability function" of the Markoff process y(t). The physical interpretation of eq.(9) is that if the probabilistic information of the
random process is available at time ti in terms of pi(yiti), and if the evolution mechanism of the random process is
known from the transition probability p^^ then the complete probabilistic nature of the process can be predicted
at a later time.
(c) The Smoluchowski or Chapman-Kolmogorov Equation
By the process of induction, from eq.(12) we can write an integral equation which governs the transitional probability of a Markoff process as
p„ (yi/y2,t)
U 2
P c 2 ( y i / y . t o ) Pc2^y/y2,t-tQ)dy
... (14)
Eq.(14) is the Smoluchowski equation and it is the basic equation of the theory of Markoff processes. For the reader who may find difficulty in following the
derivation of eq.(14) from eq.(12) the following more
physical approach may be easier to grasp. In Fig.3 let yi be known at ti and consider the probability that y exists
in the small interval dy^^ at a time t later. Then the increment of probability is given by
dP(ya) = P„ (yi/y2,t)dy2
*-2
(15)
^ Since the maximum probability of y2 lying between -00 is unity, from eq.(15) we have
dP(y2) = P„ (yi/y2,t)dy2 = 1 ... (16)
O 2
y.,t,
m
FIGURE 3 FIGURE 4
Consider next the state of affairs depicted in Fig.4 where yiti is given. The incremental probability of finding y in the interval dy at a later time t is given by
7
-Furthermore, given ytQ, the incremental probability of finding y in the interval dy2 at a time t - tQ later is
dP2(y) Pc2(y/y2'*-'to)dy2 ... (18)
Thus the probability that given yiti, y is found in the interval dy2 at time t later is given by
dP(y) = dPi(y).dP2(y)
From eq.(19) we can write
dP(y) - 1
. (19)
.. (20)
Thus inserting eqs.(17) and (18) into (19) and noting (20) it is found that
oo
P„ (yi/y,to).p (y/y2,t-t )dydy2 = 1 -- ^? o ca o _ _ ^21)
Differentiating eqs.(16) and (21) with respect to ya and equating the result yields eq.(14).
(d) Derivation of the nth Order Probability Density Function
The conditional p.d.f. Ps(y111 ,yata,yat 3 ) , p,,(.... , y u 14)
are determined from eq.(9). Thus
P3(yiti.yat?,y3t3) = Pa(y1t1,yata)p^ (yata/yata) Ca
Also
. (22)
P2(y2ta/y3t3) = Pi(y2t2).p„ (y2ta/y3t3) Ca
or P^ (yata/ysts) = Pa(y2ta/y3t3)/Pi(yata) *-a
... (23)
Inserting (23) in (22)
P3(yiti ,yata ,y3t3) = P2 ( Y 111 , ya ta )-Pa ( y2 t2 , y 3 ta)/p 1 ( y 212 > ... (24) In a s i m i l a r manner we o b t a i n p U y i t i , . . . y 4 t 4 ) = P 3 ( . . . y s t s ) . p „ ( y a t a / Y i + t , , ) C2 . . . ( 2 5 ) I n s e r t i o n of (23) and (24) in (25) y i e l d s p U y i t i , . . . y ^ t 4 ) = P 2 ( y i t i , y 2 t 2 ) P 2 ( y 2 t 2 , y 3 t 3 ) P 2 ( y 3 t 3 , y a O P i ( y 2 t a ) P i ( y 3 t a ) . . . ( 2 6 )
Eq.(26) shows how the 4th order J.P.D.F. is
obtained from the product of three 2nd order distribution functions. In practical situations there are few
examples of such higher order Markovian processes. More often it may happen that a non-Markovian process is
combined with another dependent variable, Z = dy/dt (say), to produce a Markoff process. In this case we have two random variables, say y and z, and the Smoluchowski equation, eq.(14), becomes the double integral equation
.00 .00
P^ (yiZi/y2Z2,t) = Pc (yiZi/yz.to)p (yz/y2Z2,t-t )dydz
J— ooJ — o o 2 ^ 2
... (27)
In the general case of an N-dimensional phase space. eq.(27) becomes
r°° (• N
P^^(Yi/Ya,t) = J_^^....J_u^ p^^(Yi/Z,t^)p^^(Z/Y2,t-t^)dZ. ... (28)
where Y is a position vector of a point in N-dimensional phase space and the N-fold integral extends over all the phase space.
3. THE FOKKER-PLANCK EQUATION
(a) Solutions to the Smoluchowski integral equation (14) are frequently obtained by solving an equivalent differential equation, referred to as the Fokker-Planck equation. The solution of this equation yields the transitional probability density function of the Markoff random process. The Fokker-Planck equation is derived following the method of M.C. Wang and G.E. Uhlenbeck (Ref.8, p.121).
From the elementary theory of probability in random processes it is well known that the nth order moment of a process is given by
00
< y'^> = f y".p(y)dy ... (29) J - 00
Thus for example when n = 2, (29) yields the mean square value of the process. In the case of Markoff
processes we are concerned with the transitional behaviour of a random process, and in particular with the change of various moments of the process with time. Fig.5 shows a
random process which has a given displacement y at time t and which changes by a small amount to a value z at time t + dt.
1 *" '1
kn/Mïï
W \l
L«3(tfrrVl
V ^
/\/i/iAi/^
\J\A/\I ]f
r
h .
FIGURE 5^ ^
9
-The moments of the change in the process in time dt, by analogy with (29) are defined for one-dimensional phase space by
a (z,dt) (y-z) p(z/y,dt)dy n
J - oo
n = 1, 2, 3, ...
(30)
As a result of the central limit theorem (see
Ref.lO, p.97) only the first and second order moments are assumed to exist in the limit as dt-* 0. This assumption implies that in a small time interval dt, the space
coordinates of a random process only change by a small amount, hence the higher order moments, for n > 2, can be ignored. Thus the rates of change of the first two
moments of increments in y(t) are For n = 1, For n = 2, A(z) = B(z) = lim. ^ Uj (z,t)J dt -.- 0 lim. ^ la^ (z,t)J (31) (32) dt ^ 0
where a, (z,t) and aj (z,t) are given by (30). We next consider the Smoluchowski integral equation (14) in the one-dimensional form
p^(x/y,t + dt) =
J — o o
p^(x/z,t).p^(z/y,dt)dz ... (33) Let R(y) be an arbitrary scalar function of the variables yj , yj , ... y^, which approaches zero
sufficiently fast as y approaches + oo or - oo . For example R(y) could be a decaying exponential function.
Multiplying (33) by R(y), and dropping the suffix c, integration over all the phase space yields
R(y).p(x/y,t)dt = R(y)dy p(x/z,t).p(z/y,dt)dz
oo
... (34)
Since the basic aim of the Fokker-Planck equation is to describe the variation of p(x/y,t + dt) with time, we require to determine 8p(x/y,t)/9t. Thus from (34), omitting the integration limits,
R(y)|£(x/y.t)dy = lim ^ dt O dt
j R(y)dt p(x/z,t).p(z/y,dt)dz|
(35)
J
Since y and z are close together, R(y) can be developed in a Taylor series expansion about the point y = z, i.e.
R(y) = R(z) + (y-z)R'(z) + è(y-z)2R"(z) + ...
10
-higher order terms being ignored on account of (31),(32). Eq.(36) is next inserted in (35) and by incorporating the limits of (31) and (32) we obtain, noting that as
dt -^ 0, y -^ z,
JR(y).|f(x/z,t)dy = |p(x/z,t)[R'(z)A(z) + èR"(z)B( z)J dz ... (37)
The right hand side of (37) is now integrated twice by parts to yield
R(y)|fdy = p.A(y)R(y)l -
R(y)^[A(y).p]dy
+ è.R'(y).p.B(y)r -R(y) |-rB(y).p1r
R(y).-g|^(y).p] dy (38)
Recalling that R(y) vanishes at the limits of integration, then by rearranging the terms in (38) it is found that
Since R(y) is arbitrary, (39) must hold for any R(y). Thus the expression in braces must be zero. From this condition the Fokker-Planck equation is obtained, for the one-dimensional case, as
If = - ^ A ( y ) . p . i ^ |i[B(y).p] ... (40)
In the case of N-dimensional phase space the Fokker-Planck equation is written
3£ 3t
i = l i J ^ J
(41)
where Aj^(y) and Bj^j(y) are the limits of the first and second moments defined by eqs.(31) and (32). It is obvious that as y approaches x,t approaches zero, so that p_ approaches unity. Thus for each value of i in
(41), as t->>0,
p(x/y,t) ^ 6(y-x) as t ^ 0 (42)
where 6(y-x) is the Dirac delta function. Eq.(42) implies the rather obvious fact that no transition from one state to another can take place in zero time. The solution of (41) must therefore satisfy the initial conditions (42).
11
-(b) The stationary solution
With the lapse of time it is possible that the conditional probability Pc(x/y,t) will approach a limiting stationary p.d.f. which is independent of
subsequent time, or the initial conditions (42). In this case we can write
Pc^'^/y't) = p(y)
^ = 0 Eq.(41) then becomes
2 and l , j = l •>'i ^ j L -> (43) (44) N
-I
' - ^ 8 y . | A i ( y ) - p ( y ) 1 = 1 • ' i '-i
-(45)(c) Application of Fokker-Planck Equation to Single-Degree-of Freedom System
Consider the non-linear system defined by the equation
k- + Y* + F(x) = f(t) (46)
where f(t) is a stationary Gaussian exciting force/unit mass having a white power spectrum and the non-linearity
is present only in the spring stiffness term. F(x)
denotes the spring force per unit mass and y = 2Ba) »
Eq.(46) is replaced by two first order equations by writing yi = x and y2 = * which leads to
9i = y2 (a)
ya = -Yy2 - F(yi) + f(t) (b)
(47)
In the present example of a single-degree-of-freedom system, the phase space is two-dimensional corresponding to the dimensions of yj and y . Thus in deriving the
corresponding Fokker-Planck equation (45), N = 2 and i,j, = 1,2 The first step is to evaluate the average values of the
first and second order moments for N = 2, corresponding to eq.(30) which relates to N = 1. The moments are
a^(z,t) = b,j(z,t) = J... j(y.-z.)p^(y/z,dt)Tr.dz. N-fold 48) (yj^-Zj^)(y-j-Zj)p^(y/z,dt)7r.dz^ i,j = 1,2,. ..N
12
-In the limit A. and B. . are derived from (48) as follows, "^ A = lim. i d t ^ O 'da. dt „ _ lim. ij d t ^ O i^dt db.
Furthermore in the limit as d t ^ O , then y. -»-z. and p ->1. Noting also that y^-z. = dy. , it can be seen that
C^ 1 1 1
ai = <dyi>, aa = <dy2>
bii = <(dyi)2>, bi2= bzi = <dyidya>, baa = <(dy2)^>
Thus _ lim. <dyi> d t ^ O dt -, . <dv2> lim. ^^ = <y2> = <±> (50) B
dt->-'o dt~ ••• *° ^^ evaluated from eq.47
11 = lim. ^ x ^ = dt which is neglected since it is of the second order of smallness,
B 12 = B
•j^j^jj, < d y i d y 2 > < d x . d x >
21 - d t ->• 0 dt dt = 0
since for a stationary random process x and x are uncorrelated.
T- < ( d y 2 ) ^ > l i m . v^^j^y d t ->• 0 d t
B 22 to be evaluated from eq.47.
To determine the average value of A2 and B22 , we integrate (47b) over a short time dt to give
<dy2> = -<Yy2>dt - <F(yi)>dt +
rt+dt
:f(C)>d^ (51)
Since <f(t)> = O, eq.(51) becomes <dy2> = -<Yy2>dt - <F(yi)>dt
or <dy2> = -<Y*>dt - <F(x)>dt (52)
Note that all variables are now defined by their mean values Hence from (50) and (52)
A2 = -<Y*> - <F(x)>
To determine B22 , eq.(47b) is used thus,
rt+dt
<(dy2)^> = <-Y.y2.dt - F(yi)dt +
<- Y.y2.dt - F(yi)dt + f(C)d^ > * • t+dt f(n)dn> -'t (53) (54)
13
-Now F(yi) is the non-linear force/unit mass characteristic expressible in the form
F(yi) = kiyi + k2yi^ + k3yi^+ . . . ... (55)
For a small non-linearity in stiffness it is assumed that the response to a stationary random excitation is also stationary. In that case it is known that <x Z x'^> = 0. Furthermore for stationary white noise excitation the autocorrelation function is given by
R ( T ) = 2-n.S^.6(T) ... (56)
where So denotes the constant power spectral density for white noise and 6 ( T ) is the delta function. Thus since R ( T ) = 0 when T ^ 0, then
R(0) = <f(t)2> = 2ITSQ ... (57)
Finally from (54)
B22 = lim*^^^^^'^= Y<:y2>dt + 2TrS^ = 2ÏÏS^ ... (58)
Eq.(58) states that the time rate of change of the mean square acceleration is constant.
Insertion of Ai, A2 and B22 from (50), (53) and (58) in (45) yields
^•^o 0 - alv^P-y^^ ^ alrit^y^ ' ^^y^^J -P) = 0 ... (59)
2Eq.(59) is known as the stationary form of Kramer's equation. The form of solution to be adopted for this equation is due to T.K. Caughey and T.Y. Wu (Ref.ll). This equation can be factorised into the following form
(Y.9/9y2- 9/9yi)(p. +ySQ.3p/8y2) +
+ 3/9y2 [( p F ( y i ) + 2 s ^ 9 p / 8 y i ] = 0 . (60) It can be seen that p is a function of yi and ya (i.e. a function of displacement and velocity). One solution of eq.(60) is the pair of equations
p.ya + yS^.9p/9y^ = 0 ... (61)
and p . F ( y i ) + y S ^ 9 p / 9 y i = 0 . . . (62) In e q u a t i o n s (61) and (62) y, and y^ r e f e r t o a v e r a g e or e x p e c t e d v a l u e s . I n t e g r a t i n g (61)
14
-where K i s a c o n s t a n t of i n t e g r a t i o n . E q . ( 6 3 ) can be w r i t t e n
p ( x ) = p ( y 2 ) = A . e x p . (-Yy2 V2TTSQ)
which can be seen to be a Gaussian distribution. Integration of eq.(62) yields the displacement probability density
-yi P(x) = p(yi) = B.exp
(64)
T T S .
F(n)dn
(65)Eqs.(64) and (65) show that the density functions
for displacement and velocity for a system with non-linearity in the stiffness term, are uncorrelated. Thus
p(yi,y2) = p(yi)p(y2) = p(x,*) = c.exp.J^[j2+ F(n)dn|
... (66)
Eq.(66) can also be written
P(x,*) C.exp. (-YE/TTS ) (67)
where E = *^/2 +w4F(n)dn, the total energy of the system per unit mass. It should be noticed that since the
probability density functions are uncoupled, this implies that the velocity and displacement terms are statistically independent. It is clear that the probability distribution for displacement, when non-linear stiffness is present, is in general non-Gaussian. Thus if the spring force per unit mass is given by the polynomial
n
F(n) =
I
k x" ..: (68)
n=l
representing a hardening spring, then eq.(65) yields n+1 1 P(X) N.L, = B.exp, -Y . TTS V k .X" (n+1) (69)
It can be seen from (67) and (69) that the strain energy for a given amplitude increases with the degree of non-linearity, causing an increased attentuation of p(x) as X increases. Thus a hardening spring stiffness
characteristic reduces the large displacement peaks in a random process thereby reducing the r.m.s. response.
Clearly the presence of such non-linearities in structures subject to random excitation is beneficial, and there is a case for introducing them at the design stage wherever possible.
To demonstrate the effect of spring stiffness non-linearity on the mean square response of a S.D.F. system with linear damping, we will consider linear and cubic terms in eq.(69). Thus the spring force/unit mass is
15
-F(n) = k m + kan' •.. (70) Then the density function, eq.(69) becomes
P(x, = B.e.p.{^(^.>^)} ...(.1)
For a S.D.F. linear system subject to white noise excitation the mean square response is given by
TTS
^x. " 26kimüj • • • ^"^^^ L * n
where B denotes the damping ratio. In eq.(46), f(t) is the forcing function per unit mass, also we must replace Y by the term 23w for consistency. Thus in (71)
Y ^ 23a)„
TTS TTS ••• ^'^^^ o o
But from (72), writing m = 1, we see that
- i ï s ; = (^>^x) • • • ^-"'^
Substituting (74) in (71) yields the probability
density for the non-linear system as
p(x) = B . e x p . - ( 2 f ^ . f ^ ) ... (75)
N.L. \ X X '
L.
t-where r = ks/ki.
The density function P T ( X ) for the linear system is obtained by writing r = 0 in ( 7 5 ) . Thus the ratio of the mean square response of the non-linear system to the mean square response of the linear system is given by
f°°2
^x .p(x)jj L.dx
R = ° -^^^-^ ... (76)
,00 7 , ^ |x .p(x)j^ dx
Inserting (75) in (76) the integrals are evaluated
numerically on a digital computer choosing an upper limit
of X equal to 10.o , where a is the r.m.s. value or
standard deviation. The value of R is determined for various values of r and a the results being plotted in fig.(Al)
'^»-It is noted that the integral in the numerator of (76) can be evaluated analytically in terms of parabolic cylinder functions (see for example R.H. Lyon Ref.l2). However it was considered that the numerical solution in this paper avoids the complications of using tables of these functions and yields results more readily.
16
-It is seen that R decreases with an increase in non-linear spring stiffness ratio r and the r.m.s.
response o^, where from eq.(72), a
(d) x^ " *L o
Average number of crossings oi level 'a'/sec
It has been shown that a hardening spring decreases the mean square level of response. However it is known that a hardening sping produces an increase in the
expected frequency V Q of the system. From the view point of fatigue damage a reduction in stress level is beneficial but this is not so for an increase in frequency The average number of crossings of level 'a' with
positive slope is given by
N.L.
* p(a,*)d* 0 N.L.
(77)
where p(a,*) is the joint probability density function of velocity and displacement, given by eq.(66). Thus from (66) and (75) r-P N . L . ( X , * ) | " C.exp. (-*V2aJ)exp x = a
2ÏÏI
... (78) Inserting (78) in (77) yields v_ =" C expN.L.
= Coi exp. 2 T X -a" ra" X a^ * exp(-*2/2a?)dx 0 ^ ra^4^r"
.. (79)The constant C is found from the requirement
p(x,i) dx d* =- 1 . (80)
yields
Insertion of (78) In (80) and noting that a^ = w_o.^, X o x
/2TJ W a
o X
M ^ m
dx (81)Finally inserting (81) in (79) gives the expected rate of crossing level 'a' for the non-linear system as
V "o^x ^^^
f
af_ r a - \
. 1 " 2a2 " 40^
J
a
N . L .2/5?
expl<k *
^)]
. ( 8 2 ) dx17
-When r = O eq.(82) gives the expected crossing rate of level 'a' for the linear system as
w, o
^L = 2? exp(-aV2a^) (83)
Clearly a = 0 corresponds to the zero crossing rate or the expected frequency co /2Tr
From (82) and (83) the ratio (crossing rate of the non-linear system/crossing rate of the linear system) was evaluated for a^ = 0.25, 0.5, 1.0 and 2.0, for various values of the ratio a/a,
A2, A3, A4 and A5„ '
The results are shown in figures
From these figures the crossing ratios were replotted against (a/a^j ^ )• "^be results are shown in Figures A6 and A7. It can oe seen that for high values of level crossing, increasing the non-linearity factor reduces the level
crossing rate. For low level crossings an increase in non-linearity factor causes an increase in crossing rate.
In the foregoing analysis it was tacitly assumed that the response of the non-linear system to a stationary
random excitation remained stationary. For large levels of response this assumption may not be valid due to the large non-linear spring forces present and an analogue computer simulation of the system would provide valuable information in this respect.
(e) Mean Value of Peaks
The probability density function of the peaks of a stationary random process is given by
dv" P (a)„ T - 1 vo N.L. a. N.L. da (84)
and the mean value of the peaks is given by
< a > N.L. ^ Pp(^>N.L.^^ o Insertion of (82) in (84) yields (85) P (a) „ T (a+ra') -'^ 2 ^^P a X
K^
a2 ra*» ^ 4 ^ (86)18
-The ratio of the mean value of the peaks of the non-linear system to that of the non-linear system was determined for various values of r and a , the results being shown in figure A8. It is seen that tSe non-linear spring stiffness reduces the mean value of the peaks more than in the linear case and thus the effect is beneficial from a fatigue point of view. In assessing fatigue damage due to random
vibration it is necessary to take account of expected
crossing rates as well as the mean value of the peaks, but it is difficult to determine which effect has the greatest influence in producing fatigue damage. It is hoped that figures A 2 to A8 will assist the experimentalist in
resolving this difficulty. (f) General Discussion
The results given so far have been obtained from an exact solution of the Fokker-Planck equation for a S.D.F. system with a hardening spring characteristic. Thus the analysis is not restricted to the assumption of small non-linearity r, made by other investigators (Ref.2,3) in order to achieve analytical solutions.
(g) Fatigue Analysis of Non-Linear System
Crandall and Mark (Ref.l3) have considered the fatigue of aUnear elastic cantilever beam subjected to a stationary Gaussian random excitation applied to the base of the beam The total expected fatigue damage of the beam is determined on the basis of the Palmgren-Miner hypothesis of incremental damage.
The purpose of this section is to evaluate the effects of a hardening spring characteristic on the expected
fatigue damage of a S.D.F. system subjected to Gaussian white noise excitation applied to the mass. Following Crandall and Mark it is assumed that the S-N curve for the material can be approximated by the equation
NS^ = c ... (87) where S is the stress amplitude and b and c are constants determined from sinusoidal fatigue tests. Eq.(87) does not contain the mean stress level used in the fatigue tests since it has been found to have relatively little effect on fatigue life, providing the mean stress is less than about 1/3 of fluctuating stress.
The dynamic system under consideration is the same as that considered earlier in this section, and is described by the equation
X +Y* + kix + kjx^ = f(t)
19
-It is assumed that the response x(t) is a narrow-band process as in the case of a lightly damped system. A typical time history of the randomly varying stress S(t) is shown in figure 6, from which it can be seen that the output of a narrow band system oscillates predominantly at the natural frequency of the system, but the envelope
varies in a slow random manner. The frequency of fluctuation of the envelope increases in proportion to the bandwidth or damping in the system.
S(t)
•. t
Figure 6 - Output of narrow-band system
According to the Palmgren-Miner criterion, if N(S) is the number of cycles to failure at stress amplitude S, then 1/N(S) is the damage produced by one such cycle.
Thus when n cycles of stress amplitude S have occurred the accummulated fatigue damage equals n/N(S) of the fatigue life. In the case of a randomly varying stress resulting from the output of a narrow band system we are concerned with the variation of peak levels of stress. The
probability distribution p(S) of the peak values of stress fluctuation for the non-linear elastic system is given by eq.(86). The probability of a peak value existing in the stress interval S(S+dS) is given by
6v p(S) dS
N.L.
.. (88)
where v.-, is the average rate of crossing stress level S = 0 and
S(S+dS).
öv^ is the fraction of peaks lying in the interval
Thus 6v V p(S) dS
N.L.
(89)
In time T the average number of stress cycles having amplitudes in the interval S(S+dS) is
6v„.T = v_ T p(S) dS
N.L. (90)
If dD denotes the fraction of expected damage caused by stress amplitudes in the range S(S+dS), then
20
-dD = 6v^ T/N(S) = v'!'.T.p(S)dS/N(S)
s o (91)
Substituting N(S) = cS~ from (87) in (91) and integrating over all positive stress levels gives the total expected damage as
D N.L.
V T r~ o
^ P(S>N.L.dS (92)
Inserting (86) in (92) gives the expected damage for the
non-linear system x ^N.L. = where r' = ra^ V T o /"„b+l ^ cb+3') Js^ ^ rS** (S + rS ; exp. - ^ + - ^ dS . . (93) For the linear system r = 0, in which case (93) gives
... (94) D
V T o
S^"^-^. exp.(-SV2aM dS
We now use the following standard integral given by Crandall and Mark (Ref.l3).
x" exp.(-x^/2a^)dx
è(/5a)-ir(2ii)
.. (95) Since n = b+1 (94) becomes D. + r„ V T ^ ( / 5 a )^r(i . | ) .. (96)Eq.(96) was first derived by J.W. Miles (Ref.l4).
From eq.(93) the expect^ed number of cycles to failure, V T can be obtained writing D,.. T = 1 . The calculation
o ^ N.L.
was performed on a digital computer using values of b = 6.09 and c = 7.58 x 10^' (tons/in^) for aluminium alloy
75S-T6. The results are shown in figure A9 for various values of non-linearity r and the r.m.s. stress level (J .
An upper limit of integration in eq.(93) of 45 (i.e. five times the maximum value of r.m.s. stress) was used.
' In (93), non-linear stresses are neglected in comparison with linear stresses„ For a plate this is equivalent to
neglecting membrane stresses and only considering bending stresses. It can be shown that o = a ^j(or stress = a"" x
(displacement)), a being a constant. Eq.86 can then be transformed to yield the distribution of peak stresses used in (93).
21
-(h) Average Clump Size for Hard Spring Oscillator
Justification for using the Palmgren-Miner linear cumulative damage theory in section 3 was based on the assumption that small groups or clumps of cycles, which in themselves could cause an appreciable proportion of damage, do not occur frequently. Figure 6 illustrates how cycles of random vibration can occur in clumps. A preponderance of such clumps in fatigue tests could lead to erratic test results and are thus to be avoided.
Consequently it is of interest to determine the average clump size of peaks which exceed a given level.
The average number of exceedances of the envelope of a narrow band process of level 'a'/sec is given by
Na ^ X p(a,x)dx .,. (97)
Note that N denotes also the average number of clumps/sec exceeding 'a'. If <cs> denotes the average clump size, i.e. the average number of peaks/clump, then
''a
<cs> = J P ... (98) a
where v+ is the average number of cycles which exceed level 'a'/sec. We first consider the linear oscillator for which
^ 0 _ o 2 / o „ 2
^^ = 2? ^ ' ^ ••• (99)
In order to determine Ng^ we require the J.P.D.F. of the envelope 'a' and the response velocity x. This is given
by
p(a,x) = p(a)p(x)
where p(a) is the Rayleigh distribution of the envelope and p(x) is the normal distribution of the response velocity. Thus
p(a,x) = M e
'_AS^ '_^
... (100)
X X
Substituting this expression in (97) and performing the integration yields
XT atün - a ^ / 2 a ^ , -, r^-l x
N^ = "— e ' X ... (101) ^L /2¥
22
-The average clump size for the linear system is then given by
<cs>^ = v+JN^^ = a^//2? a ... (102)
Thus the average clump size is proportional to a
and inversely proportional to 'a'.
For the non-linear spring a similar procedure to that for the linear spring yields
<cs>jj L = (^x )/»/2i a ... (103) N. L.
It can thus be seen that the ratio of clump sizes for the linear and non-linear systems is equal to the
ratio of the r.m.s. responses, o ,/a which can be
^^ '^NL
obtained from fig.Al. Clearly the effect of a hardening spring characteristic which is typical of many
structures, is to reduce the average clump size which should ensure more consistent results in random fatigue tests.
4. ANALYSIS OF S.D.F. SYSTEMS WITH NON-LINEAR STIFFNESS AND DAMPING
(^) The solution of the Fokker-Planck equation for the case of linear damping and non-linear stiffness was given in section 3(c), yielding an exact P.D.F. for displacement and velocity, (eq.66). This expression shows that x and x are uncorrelated when the non-linearity is present only in the displacement term, since the density functions are separable. In contrast to this situation, when non-linear damping exists, e.g. cx'^, n > 1, examination of the Fokker-Planck equation reveals that the density functions of x and X are inseparable and form therefore a J.P.D.F.. To analyse this more complicated problem we consider the Fokker-Planck equation, eq.59, relating to the general case of non-linear damping and stiffness
where
g(yi ,y2) = By2" + F(yi) ... (105)
= 6x^ + (kix + kgx') (say)
Substitution of (105) in (104) yields
^ ^ o l ^ - y^^lfi^-^t-^y^ ^ p^^y^ll = ° . . . ( 1 0 6 )
"^o 0 - yalf,- B^P3^-1 . y2". | f > F(yi)|^^ = O
23
-A study of eq.l07 (see Y.K. Lin, ref.9, p.263) reveals that a solution of the form p(yi,y2) = P(yi)p(y2) is only valid when n = 1 , i.e. for linear damping which corresponds to eq.66. This leads to the conclusion that for n > 1 the P.D.F. of X and x forms a J.P.D.F. For example if it is
assumed that the J.P.D.F. corresporiding to eq.l05 is of the form
P(n,yi ,y2) = C exp •26 TT(n+l)S o ^Vi'
t
•Jo m + i -> F(ri)dn (108)It can be seen that for n = 1, eq.108 corresponds to eq.67 For n > 1 the form of eq.108 indicates that the J.P.D.F. curve becomes narrower and taller as n increases. The physical explanation for this is that non-linear damping
attenuates the higher displacement peaks in x(t) more than the lower ones.
We now solve the inverse problem of finding the value of g(yi,y2) corresponding to p(yi,y2) in eq.108. This is achieved by integrating eq.l04 with respect to ya to
give
•rrS, o
i£ _ f
9y2 J y2 9yi dy2 + p.g(yi ,y2) + R(yi) = 0
... (109) where R(yi) is
taken as zero.
an arbitrary function of yi which can be Eq.l09 then gives the following equation for the sum of the non-linear damping and spring forces
I \ / « 1^ r\\7 ^ _ TT.'S 9y:
g(yi,y2) = - y2
If 1 ^y^ -
^^o ^
(110)Before substituting (108) in (110) the J.P.D.F. is written in the form
P = C.expj- ^g^(„^i) I
where E = ^^ + y i o F(n)dn (111) (112)(the total mean energy of the system)
It is next noted that
UP
= ÏS. ^ =
F(v,)
^
9yi 9E 9yi ^ ^-y i >'• gg
y2 9p _ 9p 9E 9y2 3E 9y2 9E TTS o •9E (113)
24
-Substituting from (113) in (110) gives
r
g(yi,y2) = - - 3_ TTS o >J F(y,)p.E^(''-^^y2dy 2 - Bpy2E ... (114) i(n-l)Noting that dE = y2dy2 , (114) becomes
BF(yi)
g ( y i , y 2 ) =
-TTS^P C exp 1 - 2 B 4 ^ ! ^ 1 E * ( " - 1 ) ^ ^^^ TTS ( n + 1 ) ^ ^ dE + 6y2E
Kn-1)
( 1 1 5 ) Performing the integral in (115) yields
X
g(n,x,x) = 3x x^ F(Ti)dn
o
è(n-l)
+ F(x)... (116)
As an example consider the case where n = 1, then (116) reduces to
g(x,x) = 6x + F(x)
which corresponds to eq.46 When n = 2 we have
cx
g(x,x) = Bx
t.
X -I 2
F(n)dn + F(x) (117)
and the J.P.D.F. is obtained from (108) as ^x 1 3
p(x,x) = C.exp{ -^^ \% + 1 F(n)dn
3TrS 12
V2
(118)
The general conclusion to be drawn from the above analysis is that for a S.D.F. system whose non-linear equation of motion is of the form
X + Bx
r^
F(n)dni(n-l)
+ F(x) = N(t)... (119)
or X + Bx.E*^'^ ^^ + F(x) = N(t)
the exact J.P.D.F. is given by
p(n,x,x) = C.exp \ j ^ ^ ^ ^ x^ F(n)dn
o
è(n+l)
... (120) This form of solution was first derived by T.K. Caughey (Ref.15, see also Ref.9, p.266).
25
-(b) Method of the Equivalent Non-Linear Differential Equation
We first consider the non-linear differential equation of the form
X + Yx".signx + F(x) = N(t) ... (121) where n is an integer.
To obtain an approximate solution to (121) the method of the 'Equivalent Non-Linear Differential Equation'
developed by T.K. Caughey is applied. The author is indebted to Professor Caughey for revealing the basis of this method in private communication.
The technique is to write the original non-linear differential equation in the form
X + Bx E^*^"~-^^+ (Y^^.signX - &k E*^""-^^ + F(x) = N(t) ... (122) The term is parenthesis constitutes the equation deficiency or error term, thus
f:(x,x) = Yx" signx - 3x E^^"""^^ ... (123)
In order to reduce eq.l22 to the form of eq.ll9 it is necessary to choose B in eq.l23 such that the mean square value of the error term is minmised, i.e.
<G(x,x)^>^.^ = <(Yi" signx - Bx ^^^''~^^)^>^±n ••• ^^^^^ Differentiating (124) with respect to B and equating the result to zero yields the following expression for the optimum value of g
^.n+1 . ^è(n-l)^
This value of 3 is then substituted in eq.l20 to yield the approximate J.P.D.F., from which the mean square and other statistical properties of the response can be found.
To illustrate the procedure consider the equation
X + Yx^. signx + WflX = N(t) ... (12 where N(t) is the exciting force per unit mass.
26
-The quadratic damping in (126) for example corresponds to the damping produced by a structure
vibrating in water, i.e. an offshore drilling platform.
For n = 2 the J.P.D.F. is given by (118) as
- B
p(x,x) = C . e x p l g ^ From (125) TTS, (x^ + wo'x^) 'kJ
Y<x; [i(x^ + Oio' x^)]^> <f(i==+ Wo' x2)> (127 (128) or B = Yi 0 . p ( x , x ) x ^ rj(x^ + tüo'x^)1 d x d x «. as 0, p ( x , x ) 2 ( x ^ + (JJQ X ) d x d x (129)
The integrals in (129) are evaluated by transforming to polar coordinates to yield gamma functions. The final result for n = 2 is
8/5 B = 03— Y 3TT
The mean square response is given by
:|>(x,x)dxdx <X'> oo 00 X 2 > = 4 03 0 (130) (131)
0^ r/'^^)
\ i A [
Y,
2(^0
YC^-^b/sl
-h
1.6417ü)"7
^So^ /' \ Y (132)If Caughey's equivalent linearisation technique
(Ref.4) is applied to (126), assuming a Gaussian response, the mean square response is
<x^> = TT 2a), Si Y J
.'l^
(133)Comparison of (132) and (133) shows that the more exact method based on the Fokker-Planck equation yields an r.m.s. displacement which is 2.2% greater than the more approximate method. Thus the assumption of a Gaussian response in the case of quadratic damping is for practical purposes a valid one.
27
-(c) Analysis of System with Linear and Non-Linear Damping
Consider the following non-linear differential equation in which the non-linearity exists only in the damping term,
x + Yx(l+ e |x|") + Wo^ = N(t) ... (134)
where n is an integer,
Referring to (122), it can be seen that (134) may be written
X +
3 X E ^ ( N - ^ )+ (.oSc +/Yi(l+e|xr - BiE^^N-^H = N(t)
... (135) where N = n + 1The mean square error function is then
<e(x,x)^> = <(Yx(l+e|x|" - BxE*^^""^^) ^ > ... (136) Minimising (135) with respect to 3 yields
„ ^ Y < X ^ ( 1 + C | X | " E ^ ^ ^ - ^ ) >
The J.P.D.F is then (see eq.lll)
p(x,x) = C.exp •2BE^^N-^^n
(N+1)TTS^ J ... (138)
The normalising constant C in (138) is obtained from
rOO ^00
p(x,x)dxdx = 1 ... (139)
Recalling that E = è(x^+ Wo^'), substitution of (138) in (139) yields
C = (N^Do^oA^/^"^ ... (140) 2 . r ( ^ )
where A = ^ — ( N + 1 ) / 2 ••• (^^^
(N+l)TrSo2^^*^^/'^
Insertion of (138) in (131) and carrying out the lengthy integration yields
28 -<x^> = — 2 1 Wo
r
r
N+1 2 [N+1 (N+l)Tr T N + 1 fSo' B N + 1 (142)Evaluation of (137) gives the result N-1 N-1 B = where b =
- X ^
N+3 r(2) ^^ N+1 cos'^ -^Q.dG N+1 N+1 + 4 b ^ r(2) (143) (144)To determine the mean square response from (142) it is necessary to evaluate^ from (143). The first term in
square brackets in (143) relates to the linear damping and the second term to the non-linear damping. It can be seen that when the linear damping term is included, (143) is a non-linear algebraic equation in B. If the linear term is set equal to zero with N = 2 and
writing fc=y, then b = 2/3 and B = 8/2Y/3Tr which agrees
with (130). Insertion of this value of 3 in (142) gives the same result as (132).
It can be seen that i for 3 when the linear dampi certain practical problems neglected in comparison wit example a welded steel offs to random wave forces will damping providing that the is well below the yield str fluid-structure interaction proportional to X Ix| , i.e.
t is difficult to solve (143) ng term is present. In
the linear damping term may be h the non-linear term. For hore drilling platform subjected possess small linear material stress level in the structure ess. In this case, the
will produce damping forces N = 2.
When linear damping is absent (143) yields
N+3 r7.
6 = Yr2r2. f J ;
Va, N + 1 ^ ,„ '^'cos G.dO X£ N+1W
r
r
fN+2" 2 N+32 J
(145)(d) Comparison with Equivalent Linearisation Technique
Application of Caughey's equivalent linearisation technique to the equation
X + Yx signx + (JOO^ = N(t)
29
-^•n+1^ Y<x >
2tü(,<x^>
... (146)
If a Gaussian distribution is assumed then
'°° .n+1 f X exp -0 /2TJO' *• <x"^l> = 2 (/2a" X X n+2 • 2 ^ dx /Tn a. n+2 X hence B, 1 n-1 2 " - ^ I!x J, fn+2 /2Tr 0), ..» (147)
But Ö. = cQoa^ thus
2 ^ ,, n-2 ^ n-1 „ fn+2l , . . ^ .
- — - Y w a r - ^ .„„ (148)
/2¥ ^ ^ "^ J
The mean square response for the equivalent linear system is ^'^ - 23 Wo' (149 Substitution of ( 1 4 8 ) i n ( 1 4 9 ) y i e l d s t h e r e s u l t 2 < x 2 > = ^ w, % TT gn-è p fn+2" _ 2 n + 1 /"c! i n + l £ 0 Y , . . . (150) when n = 2 , (150) r e d u c e s t o (133)
-
So
-(e) Discussion
Figure AIO gives a comparison between the r.m.s. responses calculated from eqs„ 142 and 150. The assumption of a Gaussian response in the equivalent linearisation method can be seen to underestimate the r.m.s» response for all values of
the non-linear damping exponent N. Thus for N = 2 the difference is about 2% whereas for N = 6 it is about 43%. Conclusions
The Fokker-Planck equation has been applied to the determination of the response probability density functions of single-degree-of-freedom nonlinear vibration systems and to fatigue analysis under random excitation.
In the case of the hardening cubic spring stiffness characteristic, which is typical of many real structures (e„g„ membrane effects in plates) an exact distribution function was determined from which various statistical properties of the response were found» It was observed that stiffness nonlinearity of the hardening type was
beneficial in reducing the r.m„s„ level of response, but that the expected level crossing rate could either increase or decrease depending on the level and magnitude of excitation. For fatigue analysis the nonlinear stiffness effect was
found to increase the expected fatigue life and this method is currently being applied to the acoustic fatigue of
aircraft panel structures„
With nonlinear damping the equation of motion is reduced to a form for which there is an exact solution to the Fokker-Planck equation, from which the joint probability density function of response is obtained»
REFERENCES
31
-1. CAUGHEY, T.K. Derivation and Application of the Fokker-Planck Equation to Discrete Nonlinear Dynamics Systems Subjected to White Random Excitation»
The Journal of the Acoustical Society of America, Vol.35, No.11, paper SI, November 1963»
2. CRANDALL, S»H, Zero Crossings, Peaks and other Statistical Measures of Random Responses.
The Journal of the Acoustical Society of America, Vol.35, No.11, paper S2,
November 1963.
3. CRANDALL, S.H, Perturbation Techniques for Random Vibration of Nonlinear Systems
The Journal of the Acoustical Society of America, Vol.35, No.11, paper S3,
November 1963»
4» CAUGHEY, T„K, Equivalent Linearisation Techniques The Journal of the Acoustical Society of America, Vol„35, No.11, paper S4, November 1963»
5» KIRK, C»L. and
PERRY, P.J»
Analysis of Taxiing Induced Vibrations in Aircraft by the Power Spectral Density Method.
The Aeronautical Journal, Vol.75, March 1971. 6. KIRK, C»L. 7. PENZIEN, J. and KAUL, M.K» 8. UHLENBECK, G.E, and ORNSTEIN
The Random Heave-Pitch Response of Aircraft to Runway Roughness»
The Aeronautical Journal, Vol.75, July 1971. Response of Offshore Towers to Strong Motion Earthquakes»
International Journal of Earthquake Engineering and Structural Dynamics. Vol.1, No.l,
July-September 1972»
On the Theory of the Brownian Motion Selected Papers on Noise and Stochastic Processes.
Edited by Nelson Wax, Dover Publications, Inc. New York»
9» LIN, Y.K.
10. BENDAT, J»S,
Probabilistic Theory of Structural Dynamics McGraw Hill, p.95»
Principles and Applications of Random Noise Theory»
References ctd.
32
-11» CAUGHEY, T„K. and
WU, T»Y»
Response of Nonlinear Systems to Random Excitation»
California Institute of Technology, Report 84 (1956) (unpublished)
Sled-Track Interaction and a Rapid Method for Track Alignment Measurement»
Aeronautical Engineering Research Inc» Report 114, Part 2 (30 June 1958)
12. LYON, R.H»
13. CRANDALL, S»H and
MARK, W.D. 14» MILES, J»W»
On the Vibration Statistics of a Randomly Excited Hard Spring Oscillator
The Journal of the Acoustical Society of America, Vol.33, No.10, October 1961» Random Vibration in Mechanical Systems Academic Press, New York, p.120.
On Structural Fatigue under Random Loading, Journal of Aeronautical Sciences,
21:753-762 (1954)
15. CAUGHEY, T.K, On the Response of a Class of Nonlinear Oscillatiors to Stochastic Excitation, Proc» Colloq» Intern» du Centre National de la Rech rcher Scientifique. No.148, pp»393-402, Marseille, September 1964»
INFLUENCE OF NON-LINEAR SPRING STIFFNESS ON MEAN SQUARE RESPONSE OF S.D.F. SYSTEM
0-3
0 2
-O-l
k| •> lintar spring itiffntss oonstant
1(3 > nofl-lintar spring stiffn«ss eonstont
tfw 3 mtan squor* rtsponscs of non-linior systtM
dt^ • mton squart rtsponst of lintar sytttm, whieh it prtptrtionil to SQ. tht p.s.d. of tht txciting forct
O O I 0 2 0-3 0-4 O-S 0 6 0-7 0-6 0 9 lO
I N F L U E N C E O F N O N - L I N E A R SPRING STIFFNESS O N AVERAGE RATE OF CROSSING L E V E L 'a'
OL a r.m.s. volut of rtsponst of lintar systtm
^^ = ovtrogt rott of crossing Itvtl 'o'/ste for non-lintor systtm
9^ z. ovtrogt rott of crossing Itvtl 'o'/stc for lintor systtm
INFLUENCE OF NON-LINEAR SPRING STIFFNESS ON AVERAGE RATE OF CROSSING LEVEL a'
V
+o« oe
INFLUENCE OF NON-LINEAR SPRING STIFFNESS ON AVERAGE RATE OF CROSSING LEVEL a"
• e
i>'
1-4 1-2 l O 0 8 0 6 0 4 0 2 -AVERAGE r - i 0 : f—'• r = 0-4 . ^ .RATE OF CROSSING LEVEL 'a'
r « 0 - 6 ^ ^ ^ ^ ^ ^ . ^ r « 0 2 ^ ^ ^ \ ^ ^ s N v r = 0'2
^V\\
''°'^
= 0-5 \ \ V \ r«0 6 \ \ V r • I-O >v 1 1 1 ^— O 0-5 1 0 I-S 2 0 2-5 3 0 FIGURE A.6AVERAGE RATE OF CROSSING LEVEL 'a'
1-4 r 1-2 *r>-10 0-8 O-é 0-4 0-2 rsO-4 r>0-2 — I • „ . . • I I — O-S 10 I I »• I I. IS '/(T, 2 0 2'S 3 0 «ML FIGURE A.7
INFLUENCE OF NON-LINEAR SPRING STIFFNESS ON MEAN VALUE OF PEAKS OF S.DF SYSTEM
lOi 0-9 oe rt.T 0 6 •J V o 4 II 2 0-3 0-2 O-l O ^ ' • i ^ \ ^ \ - — - -— ^ . ^ ^ _ - - - _ ^ -—a J
~~^''-i
1—r-tfx; 0 25 5 1 0 <^x,-2ci _ _ _ _
k| • lintor spring stiffntss constant k3 ' non-lintar spring stiffness oonstant <<lo^^_ * iMan volut of ptoks of non ^ lintar systta <a> = mton valut of ptoks of lintar systtm
• 1 1 1 1 1 1 1 1 1 O-l 0-2 0-3 0 4 0-5 K| 0-6 01 0-8 0-9 10 FIGURE A.8
INFLUENCE OF NON-LINEAR SPRING STIFFNESS ON
CUMULATIVE FATIGUE DAMAGE FOR ALUMINIUM ALLOY 75S-T6.
«n -0|X>
et
X
R.M.S. RESPONSE
WITH NON-LINEAR DAMPING 1 3 1 2 I I I-O 0-9 0-8 0 7 0-6 0-5 0-4 0-3 0-2 O-l k NON-GAUSSIAN RESPONSE 1