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    Journal of Sound and Vibration (1986) 105(3), 361-371

    APPROXIMATE TIME DOM AIN NON -STATIONARY ANALYSIS

    OF STOCHASTICALLY EXCITED NON -LINEAR SYSTEMSWITH PARTICULAR REFERENCE TO THE MOTION OF

    VEHICLES ON ROUGH GROUND

    R. F. HARRISON

    Department of Engineering Sciences, Oxford Universi ty, Oxford OX1 3PJ, Engla nd

    AND

    J. K. HAMMOND

    Inst i tute of Sound and Vibrat i on Research, U ni versi t y of Southampt on, Southampt on SO 9 5 N H ,England

    Recei v ed 31 Januar y 1985)

    An approximate state-space method for obtaining the time varying m ean and covarianceof non-linear systems excited by non-stationary random processes is presented . In par-ticular the class of non-stationarity associa ted with the motion of a vehicle on rough

    ground (i.e., the process is frequency modulated as a result of the vehicles variablevelocity) is of interest. The method is based on a technique of modelling the input processas a shaping filter in the spatial domain which may be linked to the vehicle dynamicequations through th e velocity function. The non-linear problem is overcome by using thetechnique of statistical linearization. An exam ple is briefly discussed.

    1. INTRODUCTION

    The dynamic response of vehicles to unevenness in the underlying surface on which theyare travelling is of obvious engineering interest. The type of unevenness of interest hereis that which admits a statistical description: i.e., it may be rega rded as a realization ofa random process and is termed roughness here. However, in the most general case asurface m ight be considered as a combination of random and deterministic processes. Ingeneral a rough surface is perceived by the vehicle as a non-stationary random processwhen regarded as a temporal (time dependent) inputdue either to inhomogeneity (spatialnon-stationarity) in the surface roughness or to variations in the vehicle velocity, or both.Analysis of the response of vehicles to such processes is further complicated by inherentnon-linearity in their dynamics, rendering exact statistical analysis analytically intractable.

    Previously, analysis of the non-stationary response of vehicles modelled by lineardynamics has been successfully accomplished (for a single input and the surface roughnessconsidered homogeneous) by classical impulse response techniques or by an evolutionaryspectral method in the spatial domain, while, to the best of the authors knowledge, thenon-stationary, non-linear vehicle problem has not been successfully tackled at all, unlessperhaps by numerical simulation. There exists, therefore, no unified approach to this verygeneral problem, and furthermore, the problems of multiple inp uts (wheels) and spectralrepresentation for the non-stationary, linear case, have been largely ignored and soit was

    361

    0022-460X/86/060361+ 11 03.00/O @ 1986 Academic Press Inc. (London) Limited

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    362 R. F. HARRISON AND J. K. HAMMOND

    felt that the whole problem should be re-addressed in an attempt to provide a unifiedapproach.

    In an earlier paper [l], the authors presented a state-space approach to the problemof the statistical analysis of the response of vehicles travelling on homogeneous (spatially

    stationary) rough ground. In that work the vehicle dynamics were modelled by linearordinary differential equations in the time domain whilst the excitation process w asmodelled by a differential equation cast in the spatial domain and driven by a spatiallywhite process, the key novel feature being the link between the time and space domainsby a formal change of variable. This enables an augmented, time variable, state-spacerepresentation to be established, to which certain resu lts from linear systems theory apply.Harrison [2] demonstrated that the formulation in reference [l] is completely general inthe sense that inhomogeneous inputs m ay also be included.

    In reference [2] the extension of the technique of reference [I] to the problems of theevolutionary spectral (time/frequency) analysis (see also the paper by Hammond eta l .

    [3]) of the non-stationary response of vehicles and to the problem of multiple wheels(previously only tackled for the constant velocity case) was presented. However, in thispaper the extension of the method to the analysis of vehicles whose dynamics are modelledby non-linear elements is proposed. The method is approximate and makes use of thetechnique know n as statistical linearization which is well suited to the present purpose.The authors have already given a brief description of this technique [4].t

    Previous work in the field is not dealt with here but reference can be made to the thesisby Harrison [2] in which a detailed account of the relevant literature dealing with boththe vehicle problem and the characterization of rough ground is presented.

    The technique of statistical linearization is briefly presented and a pragmatic justificationfor its use is given. This technique is then combined with the linear systems theory ofreference [l] to derive the so-called Covariance Analys is DEscribing function Technique(CAD ET) of Gelb and Warren [5] and the extension to the case involving the parametrizedshaping filter representation is discussed. An example of the application of the techniqueto the vehicle problem is discussed .

    2. THE TECHNIQUE OF STATISTICAL LINEARIZATION

    It is often the case in dynamic problems (including that of vehicle motion) that one is

    faced with the problem of solving a non-linear, time varying stochastically driven vectordifferential equation having the form

    * = f(x, t) + B( t)w( t); x( to),

    where x is an n-vector of system states, and w an m-vector of white noises, with

    (I)

    E[wC)l = 0, E[w(r,)w=(h)l=O(t)@4 - fJ, (2)

    in which Q is a symmetric matrix (in the case of Q = constant this is the spectral amplitudematrix of w( t)) and S(t) is the D irac delta function;f(x, t) is a vector valued function ofthe state vector and time, and B is a coefficient matrix of appropriate order. It is possible

    to include more general input forms by modelling them as the output of white-noiseexcited shaping filters and increasing the order of equation (1). This matter is re-addressedlater.

    Here interest lies in the solution of equation (1) in terms of its statistical parameters,and in many engineering applications the first and second order mom ents constitute an

    t In reference [4] the results presented, although self-consistent, do not apply to the vehicle problem as stateddue to inappropriate choice of initial conditions.

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    NON-STATIONARY, NON-LINEAR ANALYSIS 363

    adequate description of the system response although, in general, an infinite number ofmom ents are required for a complete description.

    With a technique already devised for generating the mean vector and autocovariancematrix for linear systems (see reference [l]), it is logical to extend this to include non-linearelements as well. The technique of statistical linearization, as will be seen, fits easily intothe formulation and is ideally suited to the purpose.

    2.1. A BRIEF DESCRIPTIO N OF STATISTICAL LINEARIZATIONIn broad terms one may state the essence of the technique as the replacement of a

    non-linear element by a linear one so that their output difference is minimized in someway. These linear elements are often referred to as random input des& ibing functions inthe literature of automatic control.

    So, for the system given by equation (l), one replaces f(x, t) by n,,(t) + N,( t)x, i.e.,

    f(x, t) = no(t) + N, ( t)x, (3)where no is an n-vector and N, is an n x n matrix. The linearized elements no and N,may be shown to be given by:

    no = W (x)1 - W xl, IV1 (E[f(x)xT] - E[f(x)]pT)P-, (435)

    P=Hb-d(x-141, F = E[x]. (67)

    In order to evaluate the expectations in equations (4) and (5) the probability densityfunction (p.d.f.) of the input to the non-linearity is required and this is not in general

    known a priori. It is therefore necessary to assum e a form of probability density functionwhich is characterized by mean and variance (as yet unknown) and the natural choiceis, of course, the Gaussian distribution. This choice is by no means a requirement of themethod but since, in general, the states of the system are correlated, the assumption ofa non-G aussian, joint density function is intractable for most practical cases.

    The choice of the joint Gaussian density fuction may be justified, qualitatively at least,by the so-called filter hypothesis (see, e.g., reference [6]). This states that, althoughthe output from a non-linear element may be non-G aussian, the linear part of the systemwill provide sufficient low-pass filtering to remove high frequency distortions introducedby the non-linearity so that the signal appearing at the input to the non-linear elementis effectively made Gaussian in form. If, however, there is insufficient filtering (i.e., theeffect of the non-linearity is very severe), then, of course, considerable error may result.

    Beaman [7] has offered a criticism, by counter example, of the filter hypothesis forclosed loop systems and showed that for a number of non-linearities, notably those withdiscontinuities at their operating point, the input to the non-linearity actually becomesless Gaussian as the amount of low-pass filtering is increased, which m ay lead toconsiderable error in the approximate result.

    It should be noted that the technique of statistical linearization is perfectly general inthe sense that non-stationary responses are admitted, since the linearized elements depend

    only on the instantaneou s statistics of the response process (see e.g., reference [8]).2.2. THE SCALAR APPROACH

    In the preceding section, a general, vector/matrix formulation for the statisticallinearization of a non-linear system was presented. Fortunately, for a large number ofsystems, whose elements are of the single input, single output type, under the assumptionthat the states are jointly Gaussian, it is possible to linearize each element separately (see

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    364 R. F. HARRISON AND J. K. HAMMOND

    reference [9]). in a scalar sense, according to the rule

    ~~=ElI~(~i)l-~~E[~~l~ nl = IE[x fj(xi)l~ll~~l~~~~~~>ll>l~~~xflE[xi12) @,9)and each non-linear element is replaced independently, enabling a linearized state differen-

    tial equation to be set up.

    3. THE DERIVATION OF CADET

    In this section the so-called CAD ET equations are developed in a different way fromthat of Gelb and Warren [5].

    Consider the non-linear system (1). If one considers the state vector, x(t), as consistingfo a zero-mean random vector,r( t ), superimposed upon a mean-value vector, p(r), andthen replaces the non-linear function f(x, t) with its describing function, given by

    f(x, t) = n,(p, P, r) + N,(I.I., P, r)(r(t) +p(r)) (10)then one may rewrite equation (1) as

    ~+i=n,+N, p +r)+Bw. 11)

    Taking the expectation of this, remem bering that E[r( t)] = 0, and E[w( t)] = 0 gives

    ri==n,+N,p(t), cL(fcJ). (12)

    Subtracting this from equation (11) yields:

    i=N,r+Bw, r t0). (13)

    Notice that II,,+ iV,p is simply the mean value of the non-linear function given byE[f(x)] = a(p, P, t ) . Equation (13) is of the form required in the paper by Ham mond andHarrison [l] to enable the equation for the zero-lag autocovariance matrixP( t ) ( =E[rrT])to be written down. Th is is a standard derivation and may be found in many references(e.g., the book by Bryson and Ho [lo]). So, a coupled set of deterministic, non-lineardifferential equations is obtained, whose solutions approximate the means and covariancesof the system states, given by

    ti = a(p_, P, 0, p( to), (14)P=N,(p, P, t )P+ PN T(p , P, f ) .B ( l )Q BT( f ) , P (h )* (1%

    These equations must, of course, be solved simultaneously, in general numerically,requiring only the provision of suitable initial conditions. Finally, note that if a stationarysolution exists the left-hand sides of equations (14) and (15) become zero and a coupledset of non-linear algebraic equations is obtained which may, in general, be solvednumerically.

    4. THE APPLICATION OF CADET TO THE VEHICLE PROBLEMIt is well know n that systems excited by non-white random processes may often be

    written in the form of equation (1) by augm enting the state-space. For instance, allstationary Gaussian processes possessing rational power spectra may be considered asthe output of linear time invariant filters operating on white noise. Here, however, it ismore convenient to consider the excitation process in the space domain: i.e., to assum ethat the rough ground is amenab le to shaping filter representation in the space domain.

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    NON-STATIONARY, NON-LINEAR ANALYSIS 365

    No .attempt is made here to justify this assumption but in Chapter 6 of reference [2]strong evidence can be found in support of this.

    Following other authors, the vehicle dynamics are considered as time invariant andfor this discussion the ground profile is assum ed homogeneous (although this need notbe so (see reference [2])). Hence the vehicle dynamics may be written as

    x = f x) + Bi t); x to), 16)

    where i(t) = L( t)i( t), in which (T) denotes a function of space regarded as a function,oftime: i.e., G(t) = u(s(t)) where s is the space variable. u(s) is given by

    where v satisfies

    u(s)=Fv(s)+Ew(s), (17)

    v(s) = Cv( s) + D w( s), V(Q). (18)

    x is an n-vector of states andi

    is an m-vector of inputs. f, B, I,, u, F, E, v and w are ofappropriate dimension and denotes differentiation with respect to s. L(t) defines thecoupling between the vehicle dynamics and the ground profile model.

    At this stage a formal change of variable is made in equations (17) and (18). Thisenables the crucial link between the two models to be made via the vehicles (variable)velocity i(t) and is the basis of both the work here and the spectral and multiple inputtechniques reported elsewhere. So, substituting s = s(t), so = s(t,) and converting thedifferentiation operation accordingly, i.e., d/ds(.) = (l/i(t)) (d/dt(.)), one obtains

    i(t) = IG+ Ew(s(t)), %=S(t)C~+S(t)Dw(s(t)). (19,20)

    Finally, it is necessary only to replacef by its describing function as defined by equation(3) in order to write down the augmented system as

    This may be separated into its random and deterministic parts (upon noticing tha t ifE[w(s)] =0 then E[w(s(t))] =0 also), and then equations for these components may bewritten down as (with the variables redefined)

    ri = A (t)p+n,, IL(to)r (22)

    i=A t)r+B t)w s t)), r to). 23)

    It may be show n (see reference [2]) that the zero-lag autocovariance matrix P(t) =E[r(t)rT(t)] satisfies an equation similar to equation (15), given by

    P=A t)P+PAT t)+B t)QBT r)/S f), to), 24)

    for s(t) such that i(t) 3 0. Note that for the vehicle problem the combination BL( t)E inequation (21) is always at least linear in S(t) thus avoiding singularities at 3(t) = 0. Thisneat result is obtained by using a result from the theory of generalized functions and hasbeen dealt with in detail by Hammond and Harrison [l].

    5. A N E X A M P L E S Y S T E M W I T H A S Y M M E T R I C Q U A D R AT I C D A M P I N G

    Following other workers (e.g., Sobzcyk. and Macvean [ 111) a single degree of freedommodel for the vehicle dynamics has been chosen (see Figure 1) and for this example therestoring force element is assum ed linear, wh ilst the dissipative element is represented

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    366 R F HARRISON AND J K HAMMOND

    Figure 1 Simplified undercarriage model

    by an asymmetric quadratic damper w hose characteristic is given by:

    (25)

    where z = y - h is the relative displacemer? or closure, JJ( ) is the absolute mass displace-ment (referred to static deflection) and h(t) is the ground height profile, a function ofspace regarded as a function of time. This type of element is commonly found inoleopneumatic shock struts (oleos) in aircraft undercarriages and so is of interest in theaircraft taxi-ing problem.

    5 1 THE VEHI CLE EQUATIONS

    If point contact is assum ed (a common assumption among authors although oftenunwarrantable as discussed by Harrison [2]), the equation of motion for the system ofFigure 1 is

    mj; = -k (y - i ) - c ,( i - I i ) - c f (j - 6 ). (26)

    Here is the mass, k is the spring stiffness, and c and c1 are the damp ing constants. c1is a (small) linear d amping component included to ensure the presence of some dampingat very small closure velocities (this may be regarded as a linear m odel for friction).

    By defining a new variable, x = (y - h), it is possible to re-write equation (26) as

    P + 2 5w ,i + y f ( i ) + o ,z x = - (27)

    where y = c/m, 5 = cr/m and w,, = k / m is the undam ped natural frequency of the system.Furthermore, it is convenient to normalize the system states with respect to the standarddeviation of the ground profile, a, by defining x, = x/u and x2 = /a.

    The system may now be linearized by substituting the statistically linearized coefficients,do + d,x,, for f(xZ) where the expressions for do and d, are given in the Appendix. Finally,equation (27) may be written in a form required for covariance analysis, given by

    0

    -co; ] [ : : ]+ [_yO_i , l+ [_4 ]B lu .-2gbo- yud , (28)It is now necessary to include the excitation term i/u in state form.

    5.2. THE EXCITATION EQUATIONS

    It is immediately apparent from equation (28) that a shaping filter of at least secondorder is required to provide the excitation and the use of such a filter must be justified.

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    NON-STATIONARY, NON-LINEAR ANALYSIS 367

    It is well documented (see references [ 1 ] an d [ 123) that a process whose autocovariancefunction is given by R,,,,(t) = a2 e-u5is often an adequate model for a class of roughground processes. 6 is the spatial lag variable. A homogeneous process of this type maybe considered as the output of a white-noise excited, first-order, shaping filter in the spatialdomain. It is possible to justify the introduction of a second filter in cascade by notingthat when a tyre is in rolling contact with a rough surface, some filtering operation (mostconveniently considered in the spatial dom ain) takes place: i.e., a filter in cascade maybe regarded as a model for rolling contact. This idea has been discussed more fully byHarrison [2]. For the purpose of illustration, an additional first order filter is introducedto overcome the problem. The variable h(s) is not therefore the true ground height butmay be considered as the filtered version which is then tracked by a point contact. In thespace domain this second order filter model may be written as

    h+((Y+p)h+aph = kw(s), (29)

    where (Y is the cut-off w avenumber of the ground profile spectrum, 13 is the cut-offwavenumber of the rolling contact filter and k( =& pa) ensures that the variancethe true ground profile is equal to 02, so that the variance of h(s) is @ /(CT +p).

    5.3. COUPLING THE EQUATIONS1

    The input to the dynamic system i(t) = h( t)/a. There is no explicit equation forhowever, it is easily show n that

    of

    h;

    ;:h=f t)h+42 t)i (30)

    and since an expression for fi exists( =d2/ds2(h(s))) regarded as a function of time),

    i.e., equation (29), i(t) is given by

    i(t)=[i(t)i2(t)]P/a

    [ /a (31)

    and upon recalling thati t) may be written asi t) = L t)ii t) (where L(t) is now a vector)this fixes L and a. Finally, by inspection of equation (29), u(s) is given by

    0

    --a/3

    which fixes both F and E (see equation (17)).It is now possible, by putting equation (29) into its state form and transforming from

    the space to the time domain , to couple the equations of motion and the excitation:

    -0; -2500-yad, a@ (t)

    (33)

    where x, = c/u and x4 = fil/ u. Equation (33) is now of the form (21) so that equations(22) and (24) apply, yielding a coupled set of non-linear differential equations for P(t)

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    368 R. F. HARRISON AND J. K. HAMMOND

    and p(t). The equations for P and ~1are not given explicitly here but may be found inreference [2]. All that is required for the solution of these equations is a set of initialconditions, a detailed discussion of their specification again being given in reference [2].

    5.4. RESULTS AND DISCUSSIONHere results are presented for one variable only, in this case the relative displacement

    of the mass, normalized with respect to the standard deviation of the rough g round profile.In the particular case where the damping force is asymmetric, it is necessary to study notonly the variance of displacement but also the mean value, since this is no longer zero.It should be noted, however, that all other covariances and mean values are computedsimultaneously by the CAD ET method. In the absence of any exact solutions, MonteCarlo simulation was undertaken to validate the method. Details of the simulationprocedure including the problem of numerically specifying the parametrized white processw(s(t)) have been presented by Harrison [2].

    Two situations are considered here, firstly the constant velocity, non-stationary (finiteoperating time) response, and secondly the variable velocity case (starting from rest) andwhere the response is always non-stationary. Both symmetric and asymmetric dampingforces are discussed.

    5.4.1. Constant vehi cle veloci ty non-stationar y response)This case corresponds to the situation of a vehicle, travelling with constant velocity

    ( V, on a smooth surface, suddenly encountering homogeneous rough terrain. The varianceof response for both sym metric (r = 1) and asymmetricr = 0.1) damping is violentlynon-stationary in early time, settling eventually to a steady state (stationary) value (seeFigures 2 (a) and 2(c)). It is clear that for the asymmetric case the non-stationarity ismore vigorous than for the symmetric one and that this result settles to a considerablyhigher stationary value, all other parameters remaining equal. This is due to wha t amountsto less damping per cycle for the asymmetric case. As to the mean values (see Figures2(b) and 2(d )), clearly, in the symmetric case the mean value is zero, as expected, but inthe asymmetric case there is a negative shift which again exhbiits transient behaviour and

    0.01 I I I I / I 0.01 J 1 I I 1 J0.0 25.0 00 25 0

    / I

    0.0 I I I - 00

    b )

    r: , A ,=0-I-0.4 I I I I -10 I 1 1 1 1

    o-o 25.0 0.0 25.0I /

    Figure 2. Normalized mean and variance of response, constant velocity. CADET vs. simulation. y@=O.l,c z V / o = 0.8, p/a = 10.

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    NON-STATIONARY, NON-LINEAR ANALYSIS 369

    settles to a constant bias. This shift is expected due to the unequal dissipation of energyin the system which results in the system operating about a different position.

    The important feature to note from Figure 2 is how closely the CADET results (smoothcurve) match the simulated (true ) results. The simulation results were obtained froman ensemble of 2000 realizations, and 99% confidence limits on the mean an d varianceare show n. From these it is clear that, for these particular conditions, CAD ET yieldsresults of accuracy comparable to that of simulation but at approximately one two-thousandth of the cost.

    5.42. Var i ab l e veh i c l e ve loc i t y

    Only the case of constant acceleration (a) from rest is considered here for reasons ofcomputational economy. It should be noted, however, that no special problems arisefrom the inclusion of a more complicated velocity history subject to one small restriction:that the vehicle should not reverse its direction.

    Figure 3 depicts the non-stationary behaviour of the mean and variance of normalized,relative displacement for both symmetric and asymmetric configurations. Notice that ineach case the variance of response (see Figures 3(a) and 3(c)) exhibits similar characteris-tics, although in the symmetric case the amplitudes are lower and the non-stationarybehaviour slightly less vigorous: i.e., the peak is broader and the decay less steep.Naturally, the mean value for the symmetric case is zero (see Figure 3(b)) and as wouldbe expected for the asymmetric case (see Figure 3(d)), the mean value becomes increas-ingly negative as time (and hence velocity) increases.

    0 a9 CL ICI

    , :0.10.0

    o-o 10.0

    39 L

    i i

    -0-4 - ,:I _

    0.0 10.0t I

    Figure 3. Normalized mean and variance of response, constant acceleration. CADET us. simulation. ya = 0.1,w0 = 10, 5 = 0.01, a = 8, I = 0.2, p = 2.

    As velocity increases, one would expect that the variance of response of the relativedisplacement should become asymptotic to a constant value, reflecting the underlyinghomogeneity of the ground profile (since the mass becomes almost motionless) and indeedthis appears to be happen ing for the symmetric case. However, due to the couplingbetween the equations for mean and variance and the fact that the mean value appearsto be monotonically decreasing, this effect is not apparent in the asymmetric case.The decreasing nature of the mean value is due to the lack of end stops and a

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    370 R. F. HARR ISON AND J. K. HAMMON D

    hardening spring characteristic, present in a real oleopneumatic system which wasconsidered in reference [2].

    As for the constant velocity case, it is clear that the CA DET results lie close to thesimulated values (200 0 realizations) and that the approximate results are of accuracy

    comparable to that of the simulation . It is also apparent that as time (velocity) increases,the accuracy of the CADET results decreases.

    6. CONCLUDING REMARKS

    In this paper a novel method has been established for the approximate statisticalanalysis of the non-stationary response of non-linear systems. This has been achieved bycombining an existing technique, the so-called C AD ET method of Gelb and Warren [5]which enables non-linear systems to be treated approximately via the method of statisticallinearization, with the representation for non-stationary random processes established inan earlier paper [l] for linear system s. A major feature of this technique is that thelinearization does not rely on the small perturbation assum ption: i.e., there is a dependenceon input magn itude-an important aspect of non-linear systems. The new technique offersa unified approach to a very general problem (having particular application to the problemof vehicles on rough ground) which although approximate is highly tractable and, forthe example studied here, of acceptable accuracy when compared with simulated results.In reference [2] Harrison has considered a more realisitic oleo model including apneum atic (hardening) spring with end stop-a very severe non-linearity-and achievedreasonable accuracy for this more demanding example. The results of the present methodrequire on the order of l/N of the computational effort of the simulation, where iV isthe ensemble size.

    REFERENCES

    1. J. K. HAMMOND and R. F. HARRISON 1981 Transact i ons oft he Am eri can Societ y ofM echani calEngi neers, 103, 245-250. Non-stationary responseof vehicles on rough ground-a state spaceapproach.

    2. R. F. HARRISON 1983 Ph.D. Thesis, Uni versi t y of Southampt on. The non-stationary responseof vehicles on rough ground.

    3. J. K. HAMMO ND, Y. H. TS O and R. F. HARRISON 1983 Proceedings of the Znt ernati onal

    Conf erence o n A coust i cs, Speech and Signal Pr ocessin g, Bost on, U .S.A. Evolutionary spectraldensity m odels for random processes having a frequency modulated structure.

    4. R. F. HARRISON and J. K. HAMMOND 1981 Proceedi ngs of t he I nsti t ut e of Acoust i cs, Spri ngConf erence, N ew castl e upon Tyne, Engla nd. Th e response of vehicles on rough ground.

    5. A. GELB and R. S. WARREN 1972 AZ AA Conference on Guidance and Control , St anford,Cali forni a, AZ AA Paper No. 72-875. Direct statistical analysis of non-linear systems-CA DE T.

    6. A. GELB and W. E. VAN DER VELDE 1968 M ult ipl e Input D escri bing Functi ons and Non-li nearSyst em Desi gn. New York: McGraw-Hill.

    7. J. J. BEAMAN 1980 n SIA M New Approaches t o Non-L inear Roll ers in Dy namics (editor P. J.Holm es). Accuracy of statistical linearization.

    8. W. D. IWAN and A. B. MASON 1980 nt ernat i onal Journal of N on-Li near M echanics 15,71-82.Equivalent linearisation for systems subjected to non-stationary random excitation.

    9. R. J. PHANEUF 1968 Ph.D. Thesis, M assachuset t s I nsti t ut e of Technol ogy. Approximate non-linear estimation.

    10. A. E. BRYSON and Y-C. Ho 1969 Appli ed O ptimal Control . Ginn.11. K. SOBCZYK and D. B. MACVEAN 1976 n Symposium on St ochast i c Problems in Dynam ics.

    Uni versi t y of Southampt on, England, editor B. L. Clarkson). Non-stationary random vibrationof road vehicles with variable velocity.

    12. I. G. P ARKHILOV SKII 1968 utom obil naya Promy shlennost 8,18-22. A study of the probabilitycharacteristics of widely used types of road surface.

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    NON-STATIONARY, NON-LINEAR ANALYSIS 371

    APPENDIX: D ESCRIBING FUNCTION FOR AN ASYMM ETRIC QUAD RATICFUNCTION

    It is straightforward to show that the functions do and d, are given by

    4&, PI = -(1 + r)p*P2*p+ (p - p2)

    J27T 2 I-r)+(l+r)fierfblJG) ,CL I

    d,(p,p)=(l+r)2J~;22p+ {IC?l

    (I-r)+(l+r)fierfWJ%) ,c1 1

    where erf (z) is the error function.