variations in steepness of the probability density function of beam random vibration

Eur. J. Mech. A/Solids 19 (2000) 319–341

2000 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS0997-7538(00)00166-2/FLA

Variations in steepness of the probability density function of beam random vibration

Alexander Steinwolfa, Neil S. Fergusonb, Robert G. Whitec

a Department of Mechanical Engineering, University of Sheffield, S1 3JD, UKb Institute of Sound and Vibration Research, University of Southampton, Hampshire, SO17 1BJ, UKc Department of Aeronautics & Astronautics, University of Southampton, Hampshire, SO17 1BJ, UK

(Received 23 July 1998; revised 30 November 1999)

Abstract – Dynamic behaviour of a beam, subjected to stationary random excitation, has been investigated for the situation in which the response isdifferent from the model of a Gaussian random process. The study was restricted to the case of symmetric non-Gaussian probability density functions ofbeam vibrations. There are two possible causes of deviations of the system response from the Gaussian model: the first, nonlinear behaviour, concernsthe system itself and the second is external when the excitation is not Gaussian. Both cases have been considered in the paper. To clarity the conclusionsfor each case and to avoid interference of these different types of system behaviour, two beam structures, clamped-clamped and cantilevered, havebeen studied. A numerical procedure for prediction of the nonlinear random response of a clamped-clamped beam under the Gaussian excitations wasbased on a linear modal expansion. Monte Carlo simulation was undertaken using Runge–Kutta integration of the generalised coordinate equations.Probability density functions of the beam response were analysed and approximated making use of different theoretical models. An experimental studyhas been carried out for a linear system of a cantilevered beam with a point mass at the free end. A pseudo-random driving signal was generated digitallyin the form of a Fourier expansion and fed to a shaker input. To generate a non-Gaussian excitation a special procedure of harmonic phase adjustmentwas implemented instead of the random choice. In so doing, the non-Gaussian kurtosis parameter of the beam response was controlled. 2000 Éditionsscientifiques et médicales Elsevier SAS

beam / non-Gaussian random vibration / kurtosis

1. Introduction

When a mechanical system is subjected to stationary random excitation, its behaviour, after some transientprocess at the beginning, will be also of the stationary random type. In this concern, the crucial question iswhether this response can be reasonably described by the model of a Gaussian (normal) stationary randomprocess. If this is the case, the solution is very much simplified and can be found in terms of power spectralpresentation of the excitation and the response as for a linear system, also for a nonlinear one (Roberts andSpanos, 1990; Anh and Schiehlen, 1997).

There are two basic factors influencing the response probability densityP(u) which can be of Gaussianshape defined by meanm and r.m.s.σ values according to the function

PG(u)= 1

σ√

2πexp[−(u−m)

2

2σ 2

](1)

or different from it to some extent. The first factor is internal and manifests itself if there is an essentialnonlinearity in the system characteristics. The main peculiarity of a nonlinear system is to produce a responsewith extra properties which are absent in the loading. For sinusoidal excitation, this feature results in anappearance of additional harmonic components, multiple to the driving frequency. In the case of randomexcitation, nonlinearity leads to transformation of the probability distribution making it different for the input

320 A. Steinwolf et al.

and output processes. Consequently, the probability density function (PDF) of the nonlinear system responsedoes not correspond to the Gaussian law (1) even if the excitation is Gaussian.

There is extensive literature (Lin, 1967; Sobczyk, 1991; Ibrahim, 1995) where non-Gaussian responses ofnonlinear systems are treated by analytical methods based on assumptions that the excitation is an idealised‘white noise’ and the single-degree-of-freedom (SDOF) model is applicable. These studies, whose reference listis obviously much wider than the above citations, produced important fundamental results and gave clear insightinto the behaviour of the aforementioned simplified models. However, a Monte Carlo simulation method basedon Runge–Kutta or similar numerical integration of the equations of motion is a widespread tool which is uniqueif one prefers to consider structural components under the real excitation measured in practice and withoutSDOF simplification of the model. This approach has been implemented in the paper because its objectivewas not to investigate general dependencies between input and output of the system under consideration, butto predict a response for the particular excitation and system parameters, then, to identify if it is close to theGaussian process and, if not, to describe the obtained PDF accordingly. The latter is considered as a necessarypreliminary step for further studies where the dynamic system response is likely to be involved (a widelyoccurring situation of such kind is the fatigue life estimation).

It is a typical situation for structural systems that the nonlinear behaviour may be negligible for small levelexcitations but, if the excitation intensity increases, the effect of nonlinearity becomes essential for the samesystem which behaved as linear at the beginning. Such a case was studied and reported in the first half of thepaper where dynamic flexure of the clamped-clamped (C-C) beam under random loading is considered. Whenthe excitation level increases so that non-Gaussian features of the beam random response manifest themselves,a question arises, what analytical model can be employed instead of the Gaussian law which was used forsmaller excitation levels? The theory of statistics suggests some feasible solutions, including Pearson andJohnson systems of PDF curves as well as an expansion into the Gram–Charlie series (Kendall and Stuart,1969).

There are two probability density expressions, provided by the Pearson

PIV (u)= C(a2+ u2)−ζ exp[−barctan(u/a)

](2)

and Johnson

PSu(u)= C(u2+ 1)−1/2

exp{−0.5

[a + b log

(u+

√u2+ 1

)]2}/√

2π (3)

distribution families, which are applicable to the problem under consideration. However, they are too intricateto be efficiently employed in analytical manipulations, since the argument of exponent in the relationships (2)and (3) includes further functions ofu. That is why the distributions of Pearson and Johnson have not beenimplemented in stochastic dynamics domain and the expansion into Gram–Charlie or Edgworth series is themain approach that has been used (Crandall, 1980; Melzer and Schueller, 1983; Ochi, 1986). However, thereare certain restrictions imposed on the latter method since truncation of the series often results in appearance ofmeaningless negative values at the PDF tails. To overcome this drawback the idea of functional transformationof the Gaussian random process (similarly as in the Johnson system) has been developed by Winterstein (1988).This method employs Hermite polynomials, likewise the Gram–Charlie expansion. However, Winterstein’sapproach avoids erroneous negative distribution values because the Hermite functions are applied not to PDFbut to the time history itself. Some further applications of the method have been made by Mohr and Ditlevsen(1996).

There is one more non-Gaussian theoretical distribution entering the stochastic dynamics domain fromphysics and information theory. The method based on the maximum entropy principle was extensively

Variations in probability density function of beam vibrations 321

developed by Sobczyk and Trebicki (1990 and 1993) who formulated it for the stochastic dynamics problemsin such a way that the ‘most rational’ PDF of the nonlinear system random response is one which maximisesthe entropy functionalH = ∫ P(u) ln{P(u)}du with a number of constraints relevant to moments of unknownresponse PDF. These constraints are either moments directly or have the form of equations involving momentsof excitation and system response. The latter is the main advantage of the maximum entropy PDF thatincorporates moment equations so easily as the prescribed moment values, whereas all other methods arerequired to solve moment equations before constructing an analytical PDF. However, just as being a heuristicprinciple in statistical physics, the maximum entropy method still suffers from a lack of theoretical proofs inthe stochastic dynamics domain.

There are controversial opinions on this matter where, on the one hand, the maximum entropy PDF wasregarded “as a general method of inference about an unknown PDF” (Sobczyk and Trebicki, 1993) and, onthe other hand, certain scepticism (also supported by numerical results) has been expressed (Winterstein andLange, 1995) about applicability of the maximum entropy distribution for a class of softening nonlinear systems(those whose stiffness decreases with the increase of deformation). Fortunately, the system under considerationin this paper is of the opposite hardening type. Therefore, the maximum entropy method was implementedwithout any doubts alongside other aforementioned methods and a new approach to construct a non-GaussianPDF from a few sections of Gaussian distributions with different parameters each. In so doing one can retainin non-Gaussian consideration all mathematical tools and convenience provided by the Gaussian law just byapplying it several times. It means that any further analytical manipulations, where the PDF of nonlinear systemresponse might be involved, are not more complicated than those which are well-developed for the Gaussiancase.

Apart from internal nonlinear features, there is another, external factor which may lead to non-Gaussianbehaviour because characteristics of the system response are obviously related to those of the applied excitation.If the latter, that is the input to the system, is non-Gaussian, then the output, that is the system response, willbe, in general, of similar kind. However, quantitatively the dependence between non-Gaussian properties of theinput and output may vary. It was noted in previous works of Lutes and Hu (1986), Roberts (1966) and Bucherand Schueller (1991a) that if a system is lightly damped, the output process will be more Gaussian than theinput process.

When the system under consideration is linear then there are no inside sources of non-Gaussian behaviour. Itcan be caused only by an external excitation. There have been theoretical papers (Lutes and Hu, 1986; Bucherand Schueller, 1991a) that produced some analytical solutions for such a statement of the problem, however,only for the case of white noise non-Gaussian excitation which is also artificial as a well-known concept of theGaussian white noise. There have also been a number of works that have considered non-Gaussian excitationsinherent in particular applications. The first example is quadratic Gaussian excitation (Grigoriu, 1984; Kotulskiand Sobczyk, 1981) used to describe wind pressure proportional to the square of wind velocity which is aGaussian random process. Further extension of these results is a study of linear system responses to polynomialsof Gaussian processes made by Grigoriu and Ariaratnam (1988) and Iyengar and Jaiswal (1993). Anothergroup of previous works was devoted to the problem of ocean wave excitation (Moshchuk and Ibrahim, 1996;Langley, 1987) and the third example relevant to ground vehicle dynamics is representation of the excitation asa sequence of random impulses with various probability distributions of the impulse strength (Roberts, 1966;Lin, 1965).

The above theoretical studies gave some knowledge about the relationship between non-Gaussian charac-teristics at the input and output of linear mechanical systems. However, there was no adequate experimentalsupport for these investigations because it is not an obvious task how to generate physically a non-Gaussianrandom excitation. The second half of this paper presents results of an experimental study on non-Gaussian


random excitations and responses of a cantilevered beam with a point mass at the free end. If such a structureis excited in the region of the first natural frequency it behaves as a linear SDOF system. Thus, this is the casewhen non-Gaussian features of the response are governed only by those of the excitation and the system itselfdoes not introduce any deviations from the Gaussian model.

Experimental simulation on electrodynamic shakers is discussed below as a control problem when it isnecessary to reach the specified response characteristics by making adjustment of the driving signal generatednumerically by computer. The common requirement imposed on such a simulation is to reproduce the powerspectral density (PSD), that is a frequency domain characteristic, and therefore is not sensitive to non-Gaussianbehaviour of the system response. However, there are certain vibration regimes for vehicle components(Charles, 1992; Connon, 1991; Krenk and Gluver, 1988) when simulation in the PDF domain, because it isnon-Gaussian, is also required.

The work reported upon here was restricted to the case of symmetrical non-Gaussian probabilitydistributions. This concerns both types of non-Gaussian behaviour outlined above: due to the system nonlinearbehaviour (clamped-clamped beam) or due to deviations of external excitation from the Gaussian randomprocess (cantilevered beam). To meet these objectives the kurtosis value

γ =M4/(M2)2 (4)

describing steepness and tail length of symmetric PDFs was under study.

Kurtosis, governed by moments

Mj =∫ ∞−∞(u−m)jP (u)du, j = 2, 4; m=

∫ ∞−∞

uP (u)du (5)

of the probability density functionP(u), is included as an additional parameter besides the conventionalGaussian consideration of the PSD whenγ is constant and equal to 3. Variation of the kurtosis values willbe analysed below for responses of the nonlinear C-C beam. Further sections of the paper present results onsimulation of variable kurtosis values in shaker experiments with a linear system of the cantilevered beam.

2. Clamped-clamped beam under Gaussian random excitation

2.1. Numerical prediction of nonlinear random responses

There is a geometrical nonlinearity inherent in the C-C beam because of contribution of the axial componentto total strain energy. The phenomenon is neglected by a linear small deflection bending theory and the greaterthe deflection of the beam, the weaker is this assumption. To correctly simulate large deflections of a C-Cbeam, the effect of axial stretching and increase of the tensile forces in the mid plane must be considered, i.e.,a nonlinear model is needed. The problem has been considered via a numerical approach.

This section develops a modal solution approach for large amplitude beam response and uses the normallinear modes of the beam to generate an infinite set of coupled nonlinear equations. Consider the partialdifferential equation of motion for flexurew of a beam that is axially constrained at its ends (Bennett andEisley, 1970)

ρA∂2w

∂t2+EI ∂

4w

∂x4− AE

2L

∫ L

0

(∂w

∂x

)2

dx∂2w

∂x2= F(x, t). (6)


HereρA is the mass per unit length,A the cross sectional area of the beam,E the Young’s modulus of thebeam material,L is the length andI is the second moment of area of the cross section in bending.F(x, t) isthe external transverse force per unit length acting on the beam.

Large deflections are possible; the third term being introduced to incorporate a restoring force due to a net in-plane extension of the neutral axis of bending. The other terms in the equation are those for the Euler–Bernoullibeam equation, i.e. neglecting rotary inertia and shear deformation effects. In addition, the curvature in bendingis assumed to be small. The response is assumed to be expressable in terms of the linear modes,8s(x) of thebeam and the generalised coordinates,qs(t), for each mode. The flexural deflection at the positionx and timet is then

w(x, t)=∞∑s=1

qs(t)8s(x), (7)

where8s(x) is the solution that satisfies the linear differential equation for free vibration, ignoring in-planeeffects,

ρA∂2w

∂t2+EI ∂

4w

∂x4= 0

and the same boundary conditions.

Substituting the assumed response, as given by equation (7), into equation (6) and applying Galerkin’smethod (Bennett and Eisley, 1970), one obtains the nonlinear differential equations in the generalisedcoordinatesqs(t):

q̈s + ω2s qs +

1

ms

∞∑i=1

∞∑j=1

∞∑n=1

AE

2LqiqjqnMs,ijn = Fs(t)

ms, s = 1,2,3, . . . , (8)

whereωs =√ks/ms is the natural frequency of thes-th linear mode;

ks =EI∫ L

0

(8′′s (x)

)2dx, ms = ρA

∫ L

082s (x)dx,

Fs(t)=∫ L

0F(x, t)8s(x)dx, and Ms,ijn =−

∫ L

08′i8

′j dx

∫ L

08′′n8s dx

are the generalised stiffness, mass, force, and nonlinear coupling constants in thes-th mode respectively; sign′ indicates derivatives with respect tox.

The set of nonlinear differential equations (8) can be solved by using numerical techniques such as theRunge–Kutta method. For the case of random excitationF(t) the procedure takes the form of Monte Carlosimulation when the system is subjected to a number of excitation time history samples with the samecharacteristics. By running Runge–Kutta integration for these excitation samples, a number of random responsesamples is obtained for subsequent analysis leading to determination of the response characteristics such asPSD, PDF and non-Gaussian parameters, particularly kurtosis in this study.

If the third coupling term is eliminated from equation (8), it degenerates into the equation of motion for alinear SDOF system. It is known (Roberts and Spanos, 1990) that the mean square value of the response ofsuch a system to broad-band random excitation of the white noise type is inversely proportional to dampingwhich is introduced by a linear term including the general coordinate derivative. Thus, if there is no dampingin the model the mean square value of the response tends to infinity. Based on simple energy considerations


one may conclude that similar behaviour would occur in the nonlinear system (8) and its random responseincreases without limit if the system is undamped. To produce realistic results, additional modal damping termsηsq̇s have been introduced into the left hand side of equations (8). For numerical simulations, the modal lossfactorηs was assumed constant and equal for all modes. It has a direct relationship to damping ratioζ in thelinear harmonically excited SDOF system and can be assumed to be twice the value ofζ .

In this study only the first three linear modes were used in the series solution assuming the contributions fromthe others to be at much reduced levels and insignificant compared to the main responses. The external forcewas applied at the beam centre. Therefore, the second mode was not excited directly. Consequently, equations(8) can be simplified and transformed into the following form

q̈s + ηsq̇s +ω2s qs +

AE

2Lms

{q3

1Ms111+ q21q3(Ms113+Ms131+Ms311)

+ q1q23(Ms133+Ms313+Ms331)+ q3

3Ms333}= Fs(t)/ms, s = 1,3 , (9)

where only two equations remain, each including several nonlinear coupling termsMs,ijn. For completenessthe first three modes, including the antisymmetric modes = 2, were included in the numerical calculations.The response in the antisymmetric mode was negligible for the excitation at mid span, and equation (9) couldhave been adopted. In general this is not the case.

2.2. Results of simulation and analysis of the response probability distributions

A clamped-clamped beam of lengthL= 1000 mm having rectangular cross-section with thicknessh= 2 mmand widthb = 20 mm was considered. The material was aluminium with Young’s modulusE = 68.9× 109

N/m2 and mass densityρ = 2720 kg/m3. The excitation was a broad-band Gaussian random stationary processwith uniform PSD up to 20 Hz. The excitation frequency range covered the first natural frequency of the beam(10.35 Hz) and was chosen to be as wide as possible, although not to excite the third natural frequency of 55.9Hz and to have the general coordinateq3 present only due to nonlinear coupling in the system.

The set of two second-order differential equations (9) was transformed to four first-order equations forvariablesq1, q̇1, q3, q̇3 and numerically integrated by using the NAG FORTRAN library subroutine DO2BBFwith double precision. Random excitation was applied to the system by the method most commonly used fornumerically simulating time history data (Rice, 1954; Shinozuka and Jan, 1972; Hu and Schiehlen, 1997). It isbased on the Fourier expansion with a large numberN of harmonics

F(t)=N∑k=1

Ak cos(2πk1f t + ϕk). (10)

According to the required uniform shape of the excitation PSD, the amplitudesAk are all equal and definedAk = σF√2/N by the prescribed r.m.s. valueσF of the excitation. If the phase anglesϕk are chosen in arandom manner, then the conditions of the central limit theorem apply and the pseudorandom polyharmonicexcitation generated has a probability distribution of instantaneous values close to the Gaussian law. The timediscretization step used for the excitation process sequence coincided with the integration time increment andwas given a value of 0.001 s.

Ahead of starting Monte Carlo simulation with random excitations, a few preliminary runs for sinusoidalexcitation were made to reveal the transient response stage which is not obvious in random response plots(figure 1a). For sinusoidal excitations the transient process is clear and its length can be clarified (seefigure 1bwhich shows the response time history for sinusoidal excitation of 10 Hz that is close to the first natural


(a)

(b)

Figure 1. Clamped-clamped beam response time histories by numerical simulation: (a) For random excitation; (b) For sinusoidal excitation of 10 Hz.

frequency). The second objective of preliminary runs was to find appropriate tolerance value for the above-mentioned Runge–Kutta routine, to have the obtained response time history stable and insensitive to fluctuationsdue to numerical procedure errors. It appeared that a tolerance of 10−7 is required and the transient processlasted less than 2 s for the loss factor valueηs = 0.06 used in the calculations. This transient vibration, precedingthe required stationary solution, should be passed before collecting data for subsequent statistical processing.

Monte Carlo simulation by making use of Runge–Kutta integration of equations (9) for random loadingwas performed for various excitation levels. Probability density function, r.m.s. value and kurtosis (4) of theresponse were calculated by time averaging for data samples of 72 000 points. The displacement at the centreof the beam was considered and the r.m.s. excitation level increased gradually from 0.0025 N to 0.32 N (see

Table I. Characteristics of C-C beam random response obtained by Monte Carlo simulation.

R.m.s. of R.m.s. of Ratio to Frequency of Kurtosis of Crest factor, i.e. ratioexcitation response previous response power spectrum the response between response peak(N) (mm) r.m.s. value maximum (Hz) and r.m.s. values

0.0025 0.032 10.35 3.02 4.04

0.005 0.065 2.00 10.35 3.02 4.05

0.010 0.13 2.00 10.35 3.00 4.05

0.020 0.25 1.92 10.46 2.82 3.42

0.040 0.47 1.88 10.84 2.66 3.41

0.080 0.87 1.85 11.88 2.53 3.15

0.160 1.44 1.65 13.88 2.34 2.75

0.320 2.18 1.51 17.18 2.10 2.52


first column intable I). Before starting these extensive calculations, the assumption of ergodicity have beenchecked and confirmed by comparing PDFs of responses to different excitation samples of the same level.

Nonlinear behaviour of the beam can be observed even before analysis of probability distributions. Sinceeach subsequent value of the r.m.s. excitation was twice as much as the previous one, the ratio (third column)between the corresponding response r.m.s. values would be close to 2 if the system is linear. However, thiswas the case only for low excitation levels less than 0.01 N. For higher excitation levels with increased beamdeflection, the ratio is decreasing and a value such as 1.5 is an evident indication of nonlinear behaviour.Another sign is that the frequency of the response power spectrum peak (fourth column) is shifting to the rightwith an increase of excitation level, as was observed in previous studies on nonlinear vibration of a C-C beam(Wolfe and White, 1993).

The results obtained in probability distribution analysis for different excitation levels are shown in non-dimensional form infigures 2and3 where the abscissa axis represents a ratio between instantaneous valuesand r.m.s. of the beam centre displacement. At first there was no substantial influence of nonlinearity and theresponse PDF (solid curve infigure 2a–c) coincided with the Gaussian law (dotted curve) and the kurtosis valuewas close to 3, as for a Gaussian process. Such behaviour remained in effect for excitation root mean squarevalues 0.01 N and less. Then the kurtosis value began reducing (2.82 for the r.m.s. force equal to 0.02 N, 2.66for 0.04 N), whereas the PDF was still not significantly different from Gaussian. However, when the excitationlevel reached 0.08 N and the response kurtosis became 2.53, a noticeable deviation from the Gaussian lawoccurred (seefigure 2d–fwhere deviations are presented separately at the distribution middle section and forboth tails shown on the logarithmic scale). Further increase in the excitation level led to more and more non-Gaussian response characteristics (figure 3).

The non-Gaussian nature of the response probability density functions presented in the above figures can beexplained as follows. The beam stiffness at large deflections is higher than at small because of the contributionof cubic terms produced by coupling in the equations of motion (9). In other words, the beam resistance tobending loads increases when its deflection grows. Consequently, high peaks in the displacement time historyare less likely to occur than for a linear system with constant stiffness under the same excitation. As a result,in a nonlinear regime, the crest factor value, that is a ratio between the time history maximum and the r.m.s.value, becomes less than for low excitation when the system is still linear and its response is close to Gaussian.The crest factor value is presented in the last column oftable I. It reduced from Gaussian values of about 4.0at excitation level of 0.01 N and less to essentially smaller values, about 2.5 for 0.32 N excitation. In general,not only the crest factor but all instantaneous values of the beam vibration are affected in the same way, andconsequently the tails of the response PDF become narrower and shorter than those of the Gaussian law. Sincean area under the PDF curve remains equal to unity, the above changes in the distribution tails result in othersections of the PDF being affected as well. To compensate for the reduction of probability at high excursionsof response (large abscissa coordinates), the non-Gaussian PDF must exceed the Gaussian law for abscissacoordinates closer to the distribution centre (this can be seen infigures 3aand3d). Further decrease of theabscissa coordinate brings the non-Gaussian PDF under the Gaussian law again and makes the experimentaldistribution peak flatter than that of the Gaussian curve (seefigures 2d, 3aand3d).

2.3. Analytical approximation of non-Gaussian probability density functions

Although the kurtosis parameter reflects changes in the PDF steepness and the length of tails, usually itis not sufficient to describe non-Gaussian deviations just by one figure. Therefore, an analytical expressionfor probability density function is desirable to make use of the response prediction results in further analysis,for example, in fatigue life estimation. The problem under consideration requires symmetrical approximating


(a) (d)

(b) (e)

(c) (f)

Figure 2. Response probability density functions (solid curves) and Gaussian law (dotted curve): (a) Distribution middle section at 0.01 N excitation;(b) Distribution left tail at 0.01 N excitation; (c) Distribution right tail at 0.01 N excitation; (d) Distribution middle section at 0.08 N excitation;

(e) Distribution left tail at 0.08 N excitation; (f) Distribution right tail at 0.08 N excitation.


(a) (d)

(b) (e)

(c) (f)

Figure 3. Response probability density functions (solid curves) and Gaussian law (dotted curve): (a) Distribution middle section at 0.16 N excitation;(b) Distribution left tail at 0.16 N excitation; (c) Distribution right tail at 0.16 N excitation; (d) Distribution middle section at 0.32 N excitation;

(e) Distribution left tail at 0.32 N excitation; (f) Distribution right tail at 0.32 N excitation.


distributions and PDF expressions of this kind provided by the theoretical models referred to in the introductionhave been implemented. Distributions of Pearson (2) and Johnson (3) could not be used because they do notexist for PDFs smoother than Gaussian (with the kurtosis less than 3) like those obtained in the numericalanalysis (figures 2and3).

A symmetrical case of the Gram–Charlie expansion is presented by the following expression

PGC(u)= 1

σ√

2πexp[− u2

2σ 2

][1+ γ − 3

24

(u4− 6u2+ 3

)], (11)

where the kurtosis valueγ is conveniently involved in an explicit form. However, again the probability density(11) does not work properly for the problem under consideration because the kurtosis values obtained are lessthan 3. The latter circumstance means that with increase of the variableu, the last term in (11) inevitablybecomes negative. Then, increasing further, it exceeds unity as well and produces negative values for thefunction PGC(u) as a whole. The result is that the probability distribution constructed has negative tails forarguments from a certain starting value to infinity and this is unacceptable in most applications. For example, theGram–Charlie PDF (11), which was composed for the case of moderate nonlinearity with the 0.16 N excitation,gives meaningless negative values for response variables even less than 3 r.m.s.

To avoid erroneous negative values of PDF one can use the hardening Hermite polynomial model(Winterstein, 1988)

PW(u)= b(1+ 3au2)√2π

exp{−1

2

[bu(1+ au2)]2} (12)

and the maximum entropy distribution (Sobczyk and Trebicki, 1993)

PME(u)= C exp(−λ2u

2− λ4u4) (13)

(both expressions correspond to the symmetrical case). The first of them has an advantage that its parametersa

andb were expressed by Winterstein, De and Bjerager (1989) via kurtosis in an analytical form

a =−c4K−2/(1+ 3c4), b = (1+ 3c4)/K,

c4= (γ − 3)(35− 9γ )/192, K = (1+ 42c24

)−1/2. (14)

The maximum entropy approach requires some numerical procedures and a general scheme developed bySobczyk and Trebicki (1990) includes the iterative Newton method for a set of nonlinear algebraic equations aswell as taking integrals numerically. For the symmetrical maximum entropy distribution (13), parametersλ2,λ4, andC are related by the set of three equations

C

∫ ∞−∞

exp(−λ2u

2− λ4u4)du= 1,

C

∫ ∞−∞

u2 exp(−λ2u

2− λ4u4)du=M∗2 , (15)

C

∫ ∞−∞

u4 exp(−λ2u

2− λ4u4)du=M∗4


involving the second and fourth central moments which are defined by equation (5). In this particular case of thenonlinear random response, which is non-Gaussian only by kurtosis, one can avoid solving a set of nonlinearalgebraic equations (15) transforming it to just one equation by the following algorithm.

First, for the prescribed secondM∗2 and fourthM∗4 moments the kurtosis valueγ ∗ should be calculated byformula (4). To reach the desired pair ofM∗2 andM∗4 values is the same as to reachM∗2 andγ ∗, and there aretwo variablesλ2 andλ4 in the set of equations (15) serving this purpose (the third variableC is easily excluded,being expressed throughλ2 andλ4 from the first equation in (15)). Let us put asλ2 any arbitrary valueλ0

2. Thenif the following equation

∫ ∞−∞

exp(−λ0

2u2− λ4u

4)du

{∫ ∞−∞

u4 exp(−λ0

2u2− λ4u

4)du

}= γ ∗

{∫ ∞−∞

u2 exp(−λ0

2u2− λ4u

4)du

}2

(16)

is solved inλ4, the above objective will be met partially since the maximum entropy distribution (13) with theparametersλ0

2 (taken arbitrarity) andλ∗4 (found from equation (16)) will have the necessary kurtosis valueγ ∗,but its second moment

M02 =C∗

∫ ∞−∞

u2 exp(−λ0

2u2− λ∗4u4)du (17)

will be different from the requiredM∗2 value. In formula (17) coefficientC∗ is a constant defined as

C∗ ={∫ ∞−∞

exp(−λ0

2u2− λ∗4u4)du

}−1

. (18)

To attain the necessary value of second moment the preliminary PDF obtained should be first standardised

by the change of variablev = u/√M0

2 transforming the PDF to the non-dimensional form

P̃ME(v)= C∗√M0

2 exp{−λ0

2M02v

2− λ∗4(M0

2

)2v4}

with a second moment equal to 1. Then, if the opposite, de-standardisation procedureu= v√M∗2 is applied butnow with the necessaryM∗2 value, the new PDF

PME(u)= C∗√M0

2

M∗2exp{−λ0

2M0

2

M∗2u2− λ∗4

(M0

2

M∗2

)2

u4}

(19)

will acquire the second moment valueM∗2 prescribed at the beginning alongside with kurtosis valueγ ∗ alreadyattained.

Thus, to construct the maximum entropy PDF with the prescribed second and fourth moments one needs tosolve one nonlinear algebraic equation (16) and this can be achieved by making use of a simple bisection orsecant numerical method (Press et al., 1996). Such a procedure seems to be easier than, as in general solutionfor the maximum entropy method (Sobczyk and Trebicki, 1990), to consider the set of three equations (15) byimplementing the iterative Newton scheme with the construction of the Jacobi matrix involving differentiationof the equations’ left hand sides. The necessity of taking integrals numerically remains either in the generalscheme and the above algorithm.

The above two analytical models have been implemented for approximating PDFs of the C-C beam responsewhich were obtained in Monte Carlo numerical simulation described in the previous section. The hardening


Hermite polynomial model (12) with parameters (14), found according to Winterstein, De and Bjerager (1989),is shown by the dotted curves infigure 4a(PDF middle section to the left and PDF tail to the right), whereassolid curves represent the numerical PDF for the random excitation of 0.16 N level subjected to modelling.It was revealed that the solution given by formulae (14) is not exact but approximate. For the prescribedsecondM∗2 = 1.0 and fourthM∗4 = 2.34 moment values of the standardized PDF of the beam response underconsideration, the corresponding characters of the hardening Hermite model were 0.9 and 1.76, which arerather different from the above values. This is probably why the theoretical model was above the experimentalPDF in the middle section, and conversely, took slightly lower position at the tail (figure 4a). It appeared thatthe situation can be improved by standardising the distribution. The moments after that becameM2= 1.0 andM4= 2.39 that are close to the prescribed values. Thus, generally, the PDF fitting by Winterstein’s hardeningmodel can be regard as successful.

The maximum entropy model of Sobczyk and Trebicki (1990) was implemented in the form of equation (19)with parameters found by the above scheme, which requires solving one nonlinear algebraic equation (16). Theresults of fitting the same experimental PDF are presented infigure 4band seem to be perfect. This could beexpected because expression (13) for the maximum entropy distribution involving second and fourth powersinside the exponent coincides with a PDF of the known exact solution (Lin, 1967; Bucher and Schueller, 1991b)of the Duffing nonlinear oscillator which, as was shown by Lin (1967), is an appropriate SDOF model for theC-C beam or plate under bending loading. Thus, both methods for modelling non-Gaussian response PDF canbe recommended for the problem under consideration: one has the advantage of easier solution, another givesmore precise results. However, when the approximating probability density (12) or (13) is constructed, theyare somewhat difficult to handle in further analytical manipulations, where the PDF of the C-C beam responsemight be involved.

For instance, in relationships for theoretical estimation of fatigue life under random loading (Lin, 1967),probability density functions appear in integrals as a product with some other function, say a polynomial.Unfortunately, the both PDF expressions (12) and (13) do not allow such integrals to be taken in closedanalytical form. Therefore, a new approach has been attempted to construct a non-Gaussian PDF from afew sections of Gaussian distributions, each with different parameters. Since the Gaussian law is suitablefor integration and gives integrals in closed form when combined with many other functions (Gradshteynand Ryzhik, 1994), the piecewise-Gaussian model may be regarded for convenience for usage, providing thatprecision of such approximation is comparable to that of the above two methods.

A symmetrical piecewise-Gaussian PDF with variable kurtosis value was composed by Steinwolf (1993 and1996) on the basis of two Gaussian relationships shifted with respect to the coordinate origins and joined at thepointsu=±σ1

PpG(u)=P1(u)=C

{exp[− u2

2σ21

]−H

}at |u|6 σ1,

P2(u)=C exp{−[|u|−(σ1−ν)]2

2σ22

}at |u|> σ1.

(20)

The first section in formula (20) is a middle part of the Gaussian function with a varianceσ 21 . The section was

clipped to be restricted within the interval−σ1 6 u 6 σ1. Then it was displaced vertically by the valueH tomeet the second section, which is also Gaussian but with different varianceσ 2

2 . This second section describesboth tails and was subjected to horizontal shift (σ1− ν) only. The last point is essential because it means that,as this is the case for the classical Gaussian model, the tails of the constructed PDF are strictly positive for anyof its parameters. Thus, the piecewise-Gaussian distribution, along with the hardening Hermite and maximumentropy models, overcomes the aforementioned disadvantage of the Gram–Charlie series.


(a)

(b)

(c)

Figure 4. PDFs of the C-C beam response (solid curve) and analytical models constructed (dotted curves): (a) Approximation by the hardeningHermite polynomial model (Winterstein, 1988); (b) Approximation by the maximum entropy PDF (Sobczyk and Trebicki, 1993) with modification

to equation (19); (c) Piecewise-Gaussian approximation.


Parameterν (which is an abscissa of the point where the second Gaussian function has been clipped) waschosen in such a way that a derivative of the piecewise-Gaussian function (20) is uninterrupted at the breakpoints u = ±σ1. The smoothness of the distributionPpG(u) itself was secured by a proper choice of theaforementioned vertical shiftH as function ofν. In order to impart to the relationship (20) the nature ofprobability density the scale coefficientC was determined from the condition

∫∞−∞ PpG(u)du= 1 and expressed

through other parametersσ1, σ2, ν andH of which the last is not independent but, in its turn, is a function ofthe first three parameters.

A specific non-Gaussian characterβ was introduced in such a way that the remaining independent parametersσ1, σ2, ν of the probability density function (20) were expressed in terms ofβ and r.m.s. valueσ , which, atthis stage, is a resulting entire r.m.s. of the proposed distributionPpG(u) and not initial r.m.s. valuesσ1, σ2 ofits sections. The complete set of equations, describing all parameters of the theoretical PDF (20) in terms ofβ

andσ , took the form

H(β)= exp(−1/2)− exp(−β2/2

), σ1= Y (β)σ2, ν = βσ2,

Y (β)= β−1 exp[(β2− 1

)/2], σ2= B(β)σ, C = [2σD(β)B(β)]−1

,

D =√π

2−8(β)+ Y [8(1)−H ], 8(β)=

∫ β

0exp(−x2/2)dx, (21)

B =√D{(2Y − β)exp

(−β

2

2

)+[√

π

2−8(β)

][1+ (Y − β)2]+ Y 3

[8(1)− exp

(−1

2

)− H

3

]}−1/2

.

Thus, the piecewise-Gaussian distributionPpG(u) derived contains one parameter more than the Gaussianlaw. It makes it possible to describe PDFs of random data obtained numerically or experimentally not onlyin terms of second momentσ 2, but also in terms of kurtosis valueγ , which, in the case of bi-sectionalapproximation (20), depends onβ as follows

γ (β)= B4

D

{Y 5[38(1)− H

5− 4exp

(−1

2

)]+[√

π

2−8(β)

][(Y − β)4+ 6(Y − β)2+ 3

]+ exp

(−β

2

2

)[4Y(Y 2+ 2

)− β(6Y 2+ 5)+ β2(4Y − β)]}. (22)

All relationships in (21) as well as equations (20) and (22) correspond to the hardening case, such as theresponse of the C-C beam, when kurtosisγ less than 3 is required. i.e. the PDF has narrower tails than those ofthe Gaussian law. The opposite case ofγ > 3 has been also developed by Steinwolf (1993 and 1996).

To construct the distribution starting from the prescribed kurtosis value, one should consider (22) as anequation inβ and solve it by the bisection or secant numerical method (Press et al., 1996). When the result issubstituted into the relationships (21), all parameters of bi-Gaussian PDF (20) obtain their numerical values.For example, the probability distribution of the C-C beam response to the random excitation of 0.16 N level(solid curve infigure 4), already treated above by the hardening Hermite and maximum entropy models, wasapproximated by the following analytical function

PpG(u)=

1.83σ√

2πexp[− (u+0.57σ)2

0.8σ2

]atu6−1.96σ,

1.83σ√

2π

{exp[− u2

7.7σ2

]− 0.52

}at − 1.96σ < u6 1.96σ,

1.83σ√

2πexp[− (u−0.57σ)2

0.8σ2

]atu > 1.96σ

(23)


shown infigure 4cby the dotted curve. Closeness between the obtained PDF and the data from numericalsimulation is similar to that of the hardening Hermite and maximum entropy methods and much better thanthat of the Gaussian approximation shown infigure 3a–c.

The final expression (23) for approximating PDF confirms that the subsequent use of the constructedanalytical law is nothing else than a recurring manipulation with the Gaussian model which is representedactually in all three lines of relationship (23). It means that all convenient and effective Gaussian solutionsobtained by now can be employed for problems involving the C-C beam response in such a way that non-Gaussian behaviour caused by nonlinearity and predicted by the above numerical procedure is described.

3. Cantilevered beam under non-Gaussian random excitation

3.1. Method for digital generation of excitations with variable kurtosis value

As was discussed in the Introduction, the non-Gaussian nature of random structural responses may occur notonly in the previously discussed case of nonlinear behaviour but also in a linear system, such as cantileveredbeam, if the applied excitation is different from Gaussian. The problem has been studied experimentally andkurtosis values greater than 3 as for the Gaussian process have been considered. This case is opposite to thatdiscussed in previous sections of the paper.

Modern digital control systems for generating random excitation on shakers are capable of simulating anyshape of power spectral densityS(f ) which is a frequency domain characteristic. The procedure is based onthe Fourier expansion (10) with a large number of harmonics. Their amplitudes are not equal as was the case insection 2.2 during simulation of a uniform spectrum. Each amplitude is related to the power spectrum level atcorresponding frequencies

Ak =√

21fS(k1f ). (24)

By proper choice of amplitudesAk , one can simulate any PSD shape prescribed via a large number of frequencylinesS(k1f ). The phase anglesϕk of the polyharmonic excitation (10) are chosen as random variables withuniform distribution. It means that at certain time points each harmonic in the expansion (10) produces anindependent random variable. When they are taken in a large numberN and summed, the central limit theoremconditions apply and the excitation PDF becomes close to a Gaussian distribution. Thus, ordinary shaker controlsystems using the above procedure are not able to produce non-Gaussian excitations.

The excitation signal fed to the shaker input could be modified to be of a non-Gaussian form by somememoryless polynomial transformation (Smallwood, 1996; Merritt, 1997). However, in so doing, the PSDshape may be affected to some extent. There is another approach (Steinwolf, 1996) allowing to preserve theoriginal power spectrum and to make the pseudo-random polyharmonic process (10) non-Gaussian. The lattercan be done without any disturbance of the process frequency content if we operate with the phase anglesϕkand prescribe some of them not randomly but by making use of their relation to the kurtosis parameter. In sodoing, the PSD of the modified excitation signal (10) will not change because power spectrum linesS(k1f ),being related, by equation (24), only to the amplitudes of harmonicsAk , are not affected by any change ofphasesϕk .

To develop a mathematical model for proper phase adjustment, the kurtosis magnitude defined byrelationship (4) must be considered. The secondM2 and fourthM4 central moments (5) of the pseudo-random


polyharmonic processx(t) can be determined by time averaging

Mj = limr→∞

1

r

∫ r

0

{x(t)

}jdt, j = 2,4, (25)

in a similar way to that used for stationary ergodic processes. Such an approach has been applied by Bendat(1958) to define the variance (second moment) of polyharmonic vibration

M2= 1

2

N∑k=1

A2k =

1

2

N∑k=1

(a2k + b2

k

). (26)

The fourth moment can be given by the expression

M4= 1

T

∫ T

0

[N∑k=1

(ak cos 2πk1f t + bk sin2πk1f t)

]4

dt (27)

taking into account that integration in (27) over a periodT = 1/1f of the polyharmonic process (10) isequivalent (Steinwolf, 1993) to averaging over an infinitely long time history in accord with equation (25).Although in formulae (26) and (27), cosineak and sinebk harmonic components are used

ak =Ak cosϕk, bk =−Ak sinϕk (28)

instead of the amplitudesAk and phase anglesϕk , the phase dependence of the kurtosis parameter remainsunder consideration in an implicit form.

Integral (27) has been taken in closed analytical form. The solution obtained by Steinwolf (1996) yields theformula for kurtosis

γ = 3+{

N∑k=1

(a2k + b2

k

)}−2{−3

2

N∑k=1

[(a2k + b2

k

)2]+ 6∑

j=k+2nk 6=n

[(aj ak + bjbk)(a2

n − b2n)− 2(aj bk − akbj )anbn]

+ 6∑

j+k=2nj<k

[(ajak − bjbk)(a2

n − b2n

)+ 2(aj bk + akbj )anbn]+ 2∑j=3k

[ajak

(a2k − 3b2

k

)− bjbk(b2k − 3a2

k

)]+ 12

∑j+k=n+m

j<k,n<m,j<n

[(aj ak − bjbk)(anam − bnbm)+ (ajbk + bjak)(anbm + ambn)]

+ 12∑

j+k+n=mj<k<n

[(aj ak − bjbk)(anam + bnbm)+ (aj bk + akbj )(anbm − ambn)]

}, (29)

where the summation should be carried out only for those combinations of indicesj , k, m andn that satisfyequalities and inequalities, written in expression (29) under the summation signs. The indices may take anyinteger values from 1 toN .

3.2. Shaker simulation procedure and the results of experiments

On the basis of analytical formula (29) connecting the kurtosis parameter of a pseudo-random polyharmonicprocess (10) to its amplitudes and phase angles, a technique has been developed for simulation of excitation time


history data with the prescribed power spectral densityS(f ) and controlled kurtosis valueγ of the probabilitydistribution. Initially, the amplitudesAk are determined according to expression (24) and the phase anglesϕkare chosen randomly, as with the simulation of Gaussian excitation performed in the first part of the paper.Then, keeping all amplitudes and most the phases fixed, a few phase angles are rearranged from random valuesto such a deterministic group that maximise or minimise one of the summations in expression (29). This actionleads to some increase or decrease of the kurtosis value defined by (29).

A similar operation can be performed for another summation in (29) and a corresponding group of phaseswhich will also be eliminated from the initial random phase set. In so doing the kurtosis of the excitation process(10) can be gradually increased or decreased from a starting value, when all phases were random, to the desiredmagnitudeγ∗ greater or lower than that of a Gaussian process. The procedure is performed by making use ofequations (24), (26), (28) and (29). It is clear that the above manipulations do not affect the power spectrumS(f ) since it is not affected by phase changes.

It is possible to numerically generate time history data making the number of frequency componentsN inequation (10) large enough to include the initially prescribed random phases as well as deterministic phases,adjusted to provide the required kurtosis value. However, to go further and actually run experiments undernon-Gaussian excitation is not an obvious task because there is always a difference between the characteristicsof test item vibration and the driving signal generated in a computer and used as the input to the shaker system.An experimental study of the problem was carried out for a cantilevered aluminium beam 145 mm long with auniform rectangular cross section of 20× 2 mm and pointed mass of 31 g at the free end. The beam was bolted

(a) (b)

Figure 5. Characteristics of random processes in the linear system of a cantilevered beam (first iteration): (a) Excitation signal (kurtosisγ e1 = 8.0);

(b) Beam response (kurtosisγ r1 = 3.9).


at one end to the table of an electrodynamic shaker excited from a PC via a D/A conversion board and a poweramplifier. The beam response acceleration signal was converted to digital form and acquired by the same PC.

To compensate for the influence of the shaker and test item dynamics, affecting the kurtosis value of theresponse as well as other its characteristics, an iterative technique was used. Kurtosis of the driving signal wascorrected on each iteration according to changes in kurtosis of the measured acceleration feedback. It shouldbe emphasised that the classical procedure for Gaussian simulation of the response PSD is also performed byiterations. Thus, the objective was to achieve simultaneous control of both amplitude and frequency domaincharacteristics. The experimental results obtained are given below.

The left plots in figures 5–7present PSDs, PDFs and time histories of excitation signals. The samecharacteristics of the beam acceleration response are displayed on the right. The excitation bandwidth coveredonly the first natural frequency of the system (cantilevered beam with a point mass) which, in this case, behavesas a SDOF model. The PSD shape required for the system response is shown in the top plots offigures 5b, 6band7bby the dotted curve. In addition the kurtosis value ofγ∗ = 5, making the response distribution wider andsharper than Gaussian, was prescribed.

The first run was made with a flat spectrum of the excitationfigure 5a. At the same time, the kurtosis wasincreased toγ e

1 = 8, that is higher than the required level ofγ∗ = 5 because the cantilevered beam under test wasclose to a linear system, and consequently, acted as normalising filter that diminishes non-Gaussian featuresof the response compared to those of the excitation. In the first test run the PSD of the signal measured byan accelerometer was very much different from the prescribed spectrum (figure 5b) since the test beam had a

(a) (b)

Figure 6. Characteristics of random processes in the linear system of a cantilevered beam (second iteration): (a) Excitation signal (kurtosisγ e2 = 9.5);



(a) (b)

Figure 7. Characteristics of random processes in the linear system of a cantilevered beam (third iteration): (a) Excitation signal (kurtosisγ e3 = 14);


resonance within the excitation frequency interval. The response kurtosis valueγ r1 = 3.9 showed that further

increase of the excitation kurtosis was needed.

The second iteration was performed withγ e2 = 9.5 and the power spectrum (figure 6a) corrected according

to the system resonance. The kurtosis valueγ r2 = 4 of the second response appeared to be less than expected,

whereas the PSD was much improved (figure 6b). To prescribe the next iteration, a slope of the dependencebetween input and output kurtosis values was evaluated from the results of the first and second iterations (moreprecisely, the difference was considered between the current kurtosis value and the Gaussian kurtosis value of3). As a consequence, the third iteration was excited by a signal withγ e

3 = 14 (seefigure 7a) and the responsekurtosis value obtainedγ r

3 = 4.8 (figure 7b) differed from the desired value ofγ∗ = 5 no more than the scatterof kurtosis estimation. Further iterations gave some minor improvement of power spectrum keeping the sameprecision in kurtosis. Thus the objective of non-Gaussian response simulation for a cantilevered beam structurewas achieved.

4. Conclusions

Bending vibration of a clamped-clamped (C-C) beam due to a force applied at the midpoint was consideredin the case of random excitation. A numerical model, extended beyond the bounds of linear theory, was usedin the study, and therefore, large deflections of the beam were simulated. The method was based on a modalexpansion for the nonlinear response and made use of the orthogonality of such a series formulation. In so


doing, the number of coupled equations to be solved was far reduced. The procedure was realised in the formof Runge–Kutta integration and a Monte Carlo algorithm was invoked to calculate statistical characteristicsof the beam random responses. According to the main topic of the paper, analysis of response time historiesobtained by computer simulation was concentrated on transformation of the beam response probability densitiesaway from the Gaussian law.

The nonlinear numerical model used for C-C beam response prediction described the effect that the beamstiffness at large deflections is higher than at small. Consequently, high peaks in the displacement time historyare less likely to occur than for a linear system with constant stiffness under the same excitation. As a result, thetails of the response PDF become narrower and shorter than those of the Gaussian distribution. It is the kurtosisparameter that describes the length of distribution tails and, in compliance with basic statistical principles, theresponse kurtosis tended to values smaller than 3, which is inherent for the Gaussian law. The non-Gaussianbehaviour appeared when the r.m.s. deflection at the beam centre reached a quarter of the beam thickness.Further increase of the deflection led to more and more non-Gaussian response processes. When the r.m.s.deflection was about half of the beam thickness, the response was considerably non-Gaussian with a kurtosisvalue of 2.5 and the length of tails shortened from 4 r.m.s. values, as for a Gaussian process, to 3 r.m.s. values.When the r.m.s. deflection became comparable to the beam thickness, the kurtosis decreased to almost 2 andthe tails were no longer than 2.5 r.m.s. values from the mean.

Analytical approximation of non-Gaussian probability distributions with reduced kurtosis values issomewhat difficult because the widespread approach of the Gram–Charlie series presentation inevitablyproduces distribution tails with negative values, which are meaningless for probability estimation. Thisdrawback has been overcome in other methods which were developed recently and can be implemented forconstructing a theoretical non-Gaussian PDF with various kurtosis values obtained in numerical simulations orexperimentally. The hardening Hermite model (Winterstein, 1988), the maximum entropy distribution (Sobczykand Trebicki, 1993) and the piecewise-Gaussian model (Steinwolf, 1996) all showed good matching of thebeam nonlinear response PDF both near the distribution peak and at the tails. The Hermite polynomial modelprovides analytical formulae for determination of PDF parameters, whereas the other two methods requiresome numerical procedures. The maximum entropy method gives the best precision for the particular case ofC-C beam response but the solution is most intricate, involving a set of nonlinear algebraic equations. The lattercan be reduced to just one equation by using an algorithm suggested in this paper, however, the necessity ofnumerical integration remains. The piecewise-Gaussian model, composed in the symmetrical case from twosections of different Gaussian laws, has an advantage of simplifying further analytical manipulations, wherethe C-C beam response PDF, after it is constructed, might be involved. For instance, if the PDF appear inintegrals like those for theoretical estimation of fatigue life, the Hermite and maximum entropy models leavenothing else to do as numerical integration. To integrate the piecewise-Gaussian function means just to repeatthe procedure several times with the Gaussian law which is suitable for integration and gives integrals in closedform when combined with many other functions.

In the second half of the paper the case studied was when there is no nonlinearity in the beam structure andthe non-Gaussian response behaviour is caused only by excitation of such kind. A linear cantilevered beammodel was considered and the tendency for the kurtosis to increase was introduced, which is the opposite tothat which occurred for the C-C beam. A methodology has been developed for controlling the response kurtosisvalue by making use of pseudo-random excitation in the form of a Fourier expansion with a large number ofharmonics. Non-Gaussian simulation was achieved on the basis of an analytical expression derived for thekurtosis parameter of the polyharmonic process in terms of its amplitudes and phase angles for any number ofharmonic components. The amplitudes should be fixed for PSD simulation in the frequency domain. However,


the phase angles alone, if they are adjusted by the proposed method, are capable of making the probabilitydensity function non-Gaussian.

An experimental study has been undertaken to verify a theoretical conclusion that the change of kurtosis,representing non-Gaussian behaviour, does not violate power spectrum simulation. Although the kurtosis valueof the system output (vibration of the cantilever beam tested) appeared to be less than the kurtosis of theshaker input signal, it was possible to control this non-Gaussian parameter by a few iterations of driving signalcorrection. Three iterations were required to achieve a kurtosis value of 5. Since the test specification was setup with a realistic spectrum profile and a kurtosis value typical of ground vehicle vibration, the results obtainedcan be used in test houses when non-Gaussian dynamic responses must be reproduced in shaker experiments.

Acknowledgements

This study has been supported by the Royal Society and the Leverhulme Trust. The authors are very gratefulto the reviewers for their valuable comments.

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