A Galerkin/neural approach for the stochastic dynamics analysis of nonlinearuncertain systemsMichele Betti ∗, Paolo Biagini, Luca FacchiniDepartment of Civil and Environmental Engineering (DICeA), University of Florence, Via di Santa Marta 3, I-50139 Firenze, Italy
a r t i c l e i n f o
Article history:Received 1 February 2007Received in revised form6 September 2011Accepted 27 September 2011Available online 5 October 2011
Keywords:Parameter uncertaintiesDisordered systemsNon linear systemsNeural networkRadial basis functionsGalerkin methods
a b s t r a c t
The paper presents a Galerkin/neural approach (GNa) for the dynamics analysis of nonlinear mechanicalsystems affected by parameter randomness. In the specialised literature various procedures are nowadaysavailable to evaluate the response statistics of such systems, but a choice has sometimes to be donebetween simple methods (that often provide unreliable solutions) and other more complex methods(where accurate solutions are provided with a heavy computational effort). The proposedmethod, wherea Galerkin approach is combined with a neural one (basically an expansion of RBF for the approximationof the system response) could be a valid alternative to the classical procedures. Furthermore the proposedGalerkin/neural approach introduces an error parameterwhich can provide an effective criterion to acceptor refuse the obtained approximate solution. To validate the proposed approach several nonlinear systemswith random parameters are introduced as case studies, and the results (main moments of the responseprocess) are compared with Monte Carlo Simulation (MCS).
Engineering design requires us to compute response quantitiessuch as displacements, stress state, vibration frequencies, etc.against a given set of design parameters. These parametersdue to the complexity of structures, or manufacturing errorsand inaccuracy in measurement, may be uncertain and theseuncertainties may lead to large and unexpected excursion of thestructural response behaviour, resulting in drastic reductions instructural safety [1]. As the dynamic response of a mechanicalsystemwith uncertain parameters possesses probabilistic featureswhich depend on the probability distribution of the systemparameters, the reliability analysis for such a structure stronglydepends on the variation of the parameters itself. Therefore, thestudy of the random responses of structures with uncertaintyis of significant interest in many engineering applications. Theanalysis of such kinds of systems requires a probabilistic approachfor adequate reliability analysis, and the study of such systemshas become in the past twenty years very topical [2,3] butif the dynamic analysis of deterministic structures to randomexcitations is substantially well established [4–6] the dynamicanalysis of systems with uncertainty has not been developed to
the same extent. Several approaches have been proposed [2], andare available, in the scientific literature to solve the problem ofthe random dynamic response analysis of linear and nonlinearuncertain systems, and some of them are very powerful to assess aspecific class of problems.
The most commonly employed technique adopted to face theproblem is the well known Monte Carlo simulation (MCS) thatcan be applied to a wide range of linear and nonlinear problems.MCS allows statistical evaluation of the system response based ona large number of analyses with different values of the randomparameters. Since it lies in the generation of a large number ofsamples of the uncertain parameters and on the solution of thecorresponding deterministic (linear or nonlinear) problems thisapproach is very expensive and is usually performed as a test foranalytical approaches. Random numbers, representing uncertainparameters with given correlation structure are generated by dig-ital simulation and then the procedures adopted in a determinis-tic setting are applied to solve the mechanical problem at hand. Ingeneral, however, especially for nonlinear systems, the simulationprocedures are quite inefficient due to the large number of samplesneeded to guarantee accurate statistical results. Moreover, the im-plementation of MCS requires the complete probabilistic descrip-tion of the uncertain quantities through their probability densityfunctions, which are often unavailable. Even if MCS is the mostused among the stochastic analysis methods for structural prob-lems, as the number of degrees of freedom (DOFs) of the struc-ture and the number of uncertain parameters increase, the Monte
122 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Carlo based structural analysis becomes very heavy from a com-putational point of view and, in some cases, the computational ef-fort makes it inapplicable (especially in the case of nonlinearities).To reduce the computational effort, quite recently, new statisti-cal methods have been proposed; among them the so-called QuasiMonte-Carlo method [7] and the Latin Hypercube Sampling [8] (aconstrained sampling technique that requires a reduced order ofsample to simulate uncertainty).
Alternative non-statistical procedures to the MCS have beenproposed in the literature and are nowadays accessible notonly to the research community but also, in some cases, tothe practitioner’s engineering community. These methods mainlyconsist in a direct approach using probabilistic, instead ofstatistical, theory. Most of them, like the so-called polynomialchaos expansion (PCE), are already well known proceduresuniversally adopted in a stochastic context. Originally the PCEhas been developed by the seminal works of Ghanem andSpanos [9–11] based on the Wiener concept of homogeneouschaos. The approach, that in its original form is substantiallya spectral expansion based on orthogonal Hermite polynomialsin terms of Gaussian random variables using deterministiccoefficients, has been proven to be an efficient method in randomproblems. It is noteworthy to observe that, with respect to itsoriginal form (where optimal convergence is only achieved whendealing with Gaussian stochastic processes), the PCE has beenimproved in the last few decades to assess more general cases.Being a polynomial expansion, for instance, enrichment schemeshave been proposed by a set of special non-polynomial functions(enrichment functions) [12]; Xiu and Karniadakis [13] extendedPCE into a broader framework called generalized polynomial chaos(gPC) in order to obtain optimal convergence for more generalstochastic (non-Gaussian) processes; Le Maître et al. [14] proposea wavelet bases approach. The PCE approach (considering itsextension) has been used successfully in a plethora of engineeringapplications such as soil mechanics [15], nonlinear randomvibration [16] and, for instance, recently also in the optimal controlproblem of random oscillators [17]; however, there is no universalchoice of bases for every problem.
Other alternative non-statistical procedures are the so-calledperturbation methods (PM). These methods, both in the originalform and in themodified/improved one [18,19], follow all the stepsof deterministic analyses and are therefore applicable, in principle,for arbitrarily large FE-models. Substantially these methods use,both in static and dynamic cases, expansion methods where,for instance, the stiffness matrix of the structural problem issplit into a deterministic part (obtained with the mean valueof random parameters) and into a part which accounts for thefluctuation of the random variables about its mean value. Methodsbased on perturbation of the stochastic quantities have beenpresented both for linear [20] and nonlinear problems and havebeen applied in the solutions of several engineering problemssuch as frame analysis, reliability analysis, and buckling of shellswith random initial imperfections [21,22]. Taylor expansions orNeumann expansions [23–25] are adopted in some cases toavoid inversion of matrices depending on the random parametersand they allow us to obtain approximations of the probabilisticresponse in terms of moments. The solution is obtained by solvinga set of recursive equations. In its usual forms the expansionused retains the first order terms; it is possible to include higherorder terms but it dramatically increases the computational effortsespecially in the case the number of random variables is large.Due to this the perturbation approach is used to approximatethe first two moments of the stochastic response, i.e. the meanvector and the covariancematrix. Unfortunately, these approachesshow the drawback of being less and less accurate as the levelof the uncertainty of the parameters increases. Consequently, in
its standard form, they can be applied only in the case ofGaussianity of the response, which is rarely the case, even if thebasic uncertain parameters are modelled as Gaussian. In fact, dueto the nonlinear relationship between the system response andthe basic random variables, the response is usually strongly non-Gaussian, even for linear systems. The classical PM techniquescan then be applied only to perform a second order analysis toget qualitative information on the system behaviour and for lowlevels of parameter uncertainties. Particularly, a method due to Liuet al. [23] shows how to obtain an estimation of the time historyof the first two moments for the structural response in a linear ornonlinear system. This approach has been improved by Chiostriniand Facchini [26] where a stochastic input has also been taken intoaccount. The first two moments of the response at hand have beenevaluated taking into account a Taylor expansion of the structuralresponse centred on the mean value of random parameters. Themethod is efficient when the dependence of the response on therandom parameter is approximately a polynomial of the samedegree as the expansion. It is needed, hence, to take into accountmore flexible expressions for the approximation of the response.
A recent review of uncertain linear systems in randomdynamics is reported in [2,3].
Considering this background in the present paper uncertaintieshave been studied to provide an insight into the statistical responsevariations presenting a computationally efficient approach. Theproposed method foresees the approximation of the systemresponse by means of an expansion of radial basis functions (RBF).In particular the approximating RB functions are adopted tomodelthe dependence of the response generalized displacements uponthe uncertain variables of the mechanical system. This leads to aremarkable accuracy level for first and second order statistics ofthe response and, in general, for the probability density function(PDF). After formulating the problem, the approach is first brieflyreported with the aim of introducing some useful quantitiesand notations for the discussion presented in the successivesections where the approach is particularised with reference tothe structures under examination by the presenting of someillustrative examples. One useful point of the proposed method isthat it enables us to evaluate the committed error in the Galerkinprojection and, hence, it offers a control parameter, very usefulin an a posteriori analysis to verify the parameter correctness,as the number of terms in the series and other characteristics.Such expansion can be classified among the so-called radial basisneural networks with time dependent coefficients. The proposedprocedure is presented in detail, together with the iterativealgorithm employed to achieve practical results, discussing fourcase studies. Results are shown and commented; quality ofapproximations is evaluated both by comparisons with MonteCarlo simulations and by checking of the a posteriori error function.
2. Theoretical statement
Let us consider a general nonlinear multiple degrees offreedom (MDOF) mechanical system affected by randomness in itsparameters, and subject to a forcing process f(t). The governingequation of motion of the system may be written in the followingform:
M x + g (b, x, x) = f (t) (1)
where M is the generalized mass matrix, x is the n-dimensionaldisplacement vector, g is the internal forces vector, and f is theexternal forcing vector; b is, in general, a q-dimensional randomvector accounting for the system parameter uncertainties. Thisvector might include, for instance, the probabilistic distributionof material properties, assembly errors, probabilistic distributionof geometrical properties, and so on. In Eq. (1) dot superscriptsindicate differentiation with respect to time. The main idea
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 123
proposed in this paper lies in the employment of neural networksto build a satisfactory approximation of the system response, as itis described in the following.
2.1. Some remarks on neural networks
Artificial neural networks can be effectively employed to obtaina satisfactory approximation of the dependence of the structuralresponse on both time and random parameters. A wide class ofneural networks can be described by the general expression:
f (t, b) =
Nϕk=1
w(k) (t) · ϕk (b) (2)
where w(k)(t) is a set of time-dependent vectors and ϕk(b) arefunctions depending only on the random parameters b of themechanical system. Nϕ is the number of functions employed,which correspond to the number of neurons of the last hidden layerof the network.
Such ϕk(b) are assumed to be the activation functions ofthe network neurons; therefore, they can take on differentexpressions, according to the kind of network which is employed.The argument of the activation function might depend on therandom parameters b or might as well be the output of a previoushidden layer of the network. It will be assumed that the formerhypothesis holds, but the extension to the case of multiple hiddenlayer networks is straightforward. As a first example, a feed-forward back-propagation (FFBP) network is characterised byactivation functions described by the general expression:
ϕk (b) = φ (ϑkhbh + ϑk0) (3)
where the function φ is generally a sigmoid functionwhich usuallytakes one of the following expressions:
φ (r) =1 − exp (−r)1 + exp (−r)
(4a)
φ (r) =1
1 + exp (−r). (4b)
It is worth noting that in the two previous cases the first derivativeof the activation function can be expressed in terms of the functionitself: in fact, the first derivatives of expression (4a) and (4b) are,respectively,
φ′ (r) = 1 − φ2 (r) (5a)
φ′ (r) = φ (r) [1 − 2φ (r)] . (5b)
Another important class of neural networks is represented byradial basis function (RBF) networks, which are characterized bya different kind of activation functions. Such functions are usuallydefined as decreasing functions of the distance of the vector b fromsome pre-defined points called centres of the RB functions [27]
ϕk (b) = φb − ϑ (k)
/ϑk0. (6)
In Eq. (6) the vector ϑ (k) represents the centre of ϕk(b) and ϑk0 itsdecay parameter. These functions are monotonically decreasing;in the inherent literature it is possible to find a broad range offunctions that satisfy this requirement [28]. Some examples ofradial basis functions that can be found in the specialised literatureare listed below:
Gaussian bells : φ (r) = exp−r2/2
(7)
Chebyshev hat : φ (r) = exp
r2
r2 − 1
. (8)
An interesting RB function, called ‘‘Up’’ function, can be found inGotovac and Kozulic [27], and belongs to a more general class of
functions obeying the differential equation
φ′ (r) = 2φ (2r + 1) − 2φ (2r − 1) . (9)Such a function is defined on the compact support [-1, 1] andadmits no closed form expression. The interested reader shouldrefer to the original paper for details. The first derivatives of RBfunctions defined by Eqs. (7) and (8) are respectively:
φ′ (r) = −r φ (r) (10)
φ′ (r) = −2r
r2 − 12 φ (r) . (11)
In the following, it will be important to distinguish between linearparameters – the set of vectorsw(k)(t) – and nonlinear parameters,namely the ϑij, which characterize the activation functions ofthe employed network. For the sake of simplicity the nonlinearparameters will, from now on, be grouped together in thevector θ.
2.2. Employment of neural networks in the proposed approach
Neural networks in the general form (2) are employed in thepresent work to approximate the dependence of the structuralresponse on both time and random system parameters b: a hat isoverwritten on the symbols to denote the approximation given bythe network:
x (t, b) =
Nϕk=1
w(k) (t) · ϕk (b)
⇒
˙x (t, b) =
Nϕk=1
w(k) (t) · ϕk (b)
¨x (t, b) =
Nϕk=1
w(k) (t) · ϕk (b) .
(12)
Since Eq. (12) is only an approximation of the exact solution of theproblem, its substitution into Eq. (1) leads to an error which can beexpressed as
e (t) = M ¨x + gb, x, ˙x
− f (t) (13)
which can no longer vanish, as in the case of the exact solution.The training of the network will therefore be devoted to theminimization of the error function e(t) in a proper sense.
The minimization of the expected value of the square errorleads to theminimization of both themean and standard deviationof the error, so that this criterionwas chosen for the training phase.The introduction of a variation on the parameters of the neuralnetwork leads to a variation of the approximation of the systemresponse δx and therefore to a variation of the error, δe. If we wantthat the mean square error be minimum, its first variation mustvanish and therefore
δEe2 (t)
= E [e (t) · δe (t)] = 0. (14)
2.3. The proposed approach
The main idea of the proposed approximation lies in the switchfrom the original mechanical variables of the finite elementmodel,to a proper set of generalized variables. This is achieved by meansof radial basis functions (RBF). Deriving with respect to time theequations of motion (1) and introducing the tangent stiffnessmatrix Kt and the tangent damping matrix Ct as follows:
124 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Taking into account the approximation (2), and by substitutingin Eq. (16), the original problem of Eq. (1) is converted into thefollowing one:Nϕk=1
9(k) (t) · ϕk (b) = F (t) ;
9(k) (t) = M...w(k)
(t) + Ctw(k) (t) + Ktw(k) (t) . (17)
The original problem of Eq. (1) is so converted in the problem of(17) where one has to estimate the value of 9(k) time-dependentvectors. This estimation could be done by minimizing the errorby means of a Galerkin projection scheme approach. For such apurpose, the error is defined as follows:
ε (t, b) = F (t) −
Nϕk=1
9(k) (t) · ϕk (b) (18)
and it is subsequently minimised – according to the Galerkinapproach – imposing that:
E [ε (t, b) · ϕk (b)] = 0 ∀k = 1, . . . ,Nϕ . (19)
The expected value operator is defined by:
E [f (b)] =
Df (b) pB (b) db (20)
where D is the domain of the uncertainty parameters vector b. Bysubstituting Eq. (18) in Eq. (19) one obtains:
E9(k) (t) · ϕh (b) · ϕk (b)
= E
F (t) · ϕh (b)
. (21)
The final coupled system of equations to be integrated in orderto obtain the properties of the structural response is obtained byinserting Eq. (17) in Eq. (21):
E [M · ϕh · ϕk] ·...w(k)
+ E [Ct · ϕh · ϕk] · w(k)
+ E [Kt · ϕh · ϕk] · w(k)= E
F · ϕh
∀h = 1 . . .Nϕ . (22)
Eq. (22) forms a set of Nϕ vectorial equations (the number of thenetwork neurons) with N components [26]. After integration it ispossible to obtain an approximation of the probability distributionof the structural degrees of freedom by means of
pX (x, t) = |J| pB (b) (23)
where J is the Jacobian of approximation (12). In particular, themean value and standard deviation of the system response can beevaluated by the following:
µx (t) = E [x (t, b)] =
Nϕk=1
w(k) (t) · E [ϕk (b)]
σ2x (t1, t2) =
Nϕh,k=1
w(h) (t1) · E [ϕh (b) − E [ϕh (b)]]
· E [ϕk (b) − E [ϕk (b)]] · w(k) (t2) .
(24)
In the numerical examples the statistical moments defined above,Eq. (24), are evaluated and compared with the correspondingmoments obtained via Monte Carlo simulation; furthermore theability of the a posteriori error function to predict the correctnessof the RBF approximation is investigated.
3. Numerical applications
To demonstrate, and illustrate, the feasibility of the proposedmethod two examples of nonlinear SDOF systems and twoexamples of MDOF systems are next introduced and explained.The case studies have been selected to cover a wide broad
Fig. 1. The mechanical nonlinear system.
spectrum of benchmark mechanical problems. Results of theproposed approach are compared with MCS. The numericalexamples presented in this paper were obtained by implementingproprietary MATLAB routines.
3.1. Mass pendulum
As a first case study a mass pendulum system subjectedto a time–space dependent load acting on the system mass isconsidered (Fig. 1). The mass pendulum system, in its relativesimplicity, is a strongly nonlinear SDOF and offers the possibil-ity to discuss the drawback of existing methods to solve dynamicstochastic problems (mainly the perturbationmethods) in compar-ison with the proposed one. The system parameters characterizingthe problem are the pendulum length l and the system mass m.Naming θ(t) the rotation, the equation of motion can be written asfollows:
mlθ + mg sin (θ) = F (t) cos (θ) (25)
where g denotes the gravity acceleration and F(t) is the externalstochastic forcing load (‘‘sin’’ and ‘‘cos’’ denote the standardtrigonometric functions); dot superscripts indicate differentiationwith respect to time.
It is assumed that the pendulum length is an uncertainparameter; consequently the rotation θ is dependent on the lengthl, on the time t and on the forcing process F(t). As a first application,the term F(t) in Eq. (25) has been assumed as a harmonic loading:F(t) = F0 sin(ωt) so the equation of motion can be written asfollows:
m l θ + mg sin (θ) = F0 sin (ωt) cos (θ) . (26)
The stochastic dynamic problem described by Eq. (26) has beensolved firstly by means of a perturbative approach (PA) and resultshave been compared with both the proposed Galerkin/neuralapproach (GNa) and with the Monte Carlo Simulation (MCS)assuming a set of 500 simulations. Based on the perturbativeapproach, the system response is approximated by means ofa series expansion with respect to the random parameters;the stochastic dynamic problem has been solved assuming thefollowing second-order Taylor series expansion of the solutionabout the mean values of the uncertain parameter:
θ (t, l) ∼= θ (t, µl) +∂ [θ (t, µl)]
∂ l(l − µl)
+12
(l − µl)t ∂2 [θ (t, µl)]
∂ l2(l − µl) . (27)
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 125
Fig. 2. Mean value (rotation θ ): comparison between MCS and Perturbative Approach (PA) results (up); standard deviation (rotation θ ): comparison between MCS andPerturbative Approach results (down).
Substituting Eq. (27) into Eq. (26), and collecting terms of the sameorder, will yield a set of deterministic equations. In the case oflinear problems, this equation set has identical homogeneous partssubjected to different forcing terms and great efficiency is achievedby the method as the system can be solved sequentially. In thecase of nonlinear problems, as in the present case, this advantageis generally lost due to the resulting coupling of nonlinear terms;therefore this set of equations has been solved simultaneously.
Fig. 2 shows the system response in terms of mean valueand standard deviation as obtained by the perturbative approach;these solutions are compared with the solution obtained byMCS. Analysing Fig. 2 it is possible to observe that the solutionobtained by the perturbative approach is able to match theactual system solution only up to about 20 s; afterwards theapproximation becomes unacceptable. This is due to the fact thatthe accuracy of the results relies on the accuracy of the quadraticapproximation adopted for the dependence of the response onthe random parameter. Since the degree of fluctuation within thisrange is generally not known in advance (especially when systemnonlinearity and time factors take effect) themethods can producemisleading results. The failure of the perturbative approach, in thepresent case, can be better explained analysing the dependency ofthe system response (rotation θ ) on the pendulum length l. Fig. 3shows that the pendulum response depends approximately in alinear manner on the length of the pendulum up to about 5 s. From10 s onwards, the quadratic approximation becomes unacceptableand this causes the highlighted errors; the approximation (27)could provide acceptable results only when the dependency ofthe response process by the random parameters of the system isroughly quadratic, which results in a restriction on the typologyof nonlinearities that affect the system behaviour or into thevariability of uncertainties.
The same problem is next solved with the proposed approach;based on the assumption on deterministic loading the expansion(2) is rewritten as follows:
θ (t, l) =
Nϕk=1
wk (t) ϕk (l) (28)
where w(t) is a time-dependent vector, and ϕk(l) are functionsdepending only on the random parameters l of the mechanicalsystem that are assumed as Gaussian-shaped RBF [27]:
ϕk (r) = exp−
r2
2 σ 2
. (29)
With these assumptions Eq. (24) becomes:
µθ (t) = E [θ (t, l)] =
k
wk (t) E [ϕk (l)]
σ 2θ (t1, t2) =
h,k
wh (t1) E [(ϕh (l) − E [ϕh (l)]) (ϕk (l)
− E [ϕk (l)])]wk (t2) .
(30)
Fig. 4 reports themean value and standard deviation of the rotationθ respectively, comparing the solution of the proposed approach(in the case of Nϕ = 2) with MCS; the proposed approximationis able to follow the system response even in the range wherethe perturbative approach fails. Furthermore, if the response ofthe pendulum is analysed at several time instants (Fig. 5) theproposedmethod shows its ability to approximatewith confidencethe actual system behaviour.
3.2. Duffing oscillator
As a second case study an SDOF Duffing oscillator (metalband driven by an oscillating magnetic field) subjected to a timedependent forcing process is considered. The equation of motionof the Duffing oscillator is written as follows:
m x + c x + kx + β x3
= f (t) ,
with f (t) = a · sin (ω t) (31)
where the forcing process is assumed to be a harmonic load.Uncertainties of the system are assumed to be concentrated
in the stiffness K of the system: b = {k} so that the nonlinearrestoring term in Eq. (1) can be written as follows:
g (x, x, t; k) = c x + kx + βx3
(32)
126 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Fig. 3. System response (rotation θ ) at different instants (5, 10, 25 and 50 s).
and consequently the equation of motion in general form can berewritten accordingly:
m x + g (x, x, t; k) = f (t) . (33)
Based on the envisaged procedure, the first step of the solutionmethod is represented by the derivation of the equation of motion(33) with respect to time:
m...x + ct (x, x, k) x + kt (x, x, k) x = f (t) (34)
The second step of the solution is the RBF approximation accordingto:
x (t, b) =
Nϕk=1
wk (t) ϕk (b)
x (t, b) =
Nϕk=1
wk (t) ϕk (b)
...x (t, b) =
Nϕk=1
...wk (t) ϕk (b) .
(36)
Finally, the last step is the evaluation of the weight functionsw bymeans of the training of the net:
9(k) (t) = m...w(k)
(t) + ct w(k) (t) + ktw(k) (t) (37)
by assuming the following error function:
E
k
ψk (t, b) ϕk (b) − f (t)
2 = min . (38)
Eq. (38) constitutes a system of coupled nonlinear equations thatcan be solved in the deterministic field with the usual numericaltechniques. For the specific case, the following parameters havebeen assumed: a = 300 N; ω = 15 rad/s; Ek = 800 N/cm;
β = 0.021/(cm2); c = 2.85 N s/cm; m = 200 N s2/cm.For comparative purposes the problem has been preliminarilysolved by means of MCS where the response of the system hasbeen obtained by direct numerical integration of the differentialequation of motion (31). A set of 500 simulations has been carriedout by generating a corresponding vector of random values of thevariable k assuming a Gaussian distribution withmean valueµk =
800 N/cm and a coefficient of variation of 5%.To investigate the efficiency of the proposed approach several
tests, with different numbers of RBFs (Nϕ), have been carried outconsidering 25 s; the results of the proposed approach have beencompared with MCS.
As a first step, the case with 2 RBFs has been considered. Ateach time step a system of Nϕ × N equations has to be solved.Results show that the approximation with 2 RBF match quite wellthe simulation with respect to the mean value of displacementand velocity (Fig. 6). On the contrary, if standard deviation of bothdisplacement and velocity are plotted together with MCS results,it is possible to see that the approximation is not able to reproducethe response process (Fig. 6): the approximation function, with 2RBFs, is not able tomatch the results of the simulationwith respectto the standard deviation of the response process.
The failure of the approximationwas also predicted by the errorfunction that, for the analyzed case, is defined as follows:
e2 = E
m ¨x + gx, ˙x, t
− f (t)
2
where x (t, b) =
Nϕk=1
wk (t) ϕk (b) (39)
and Fig. 7 reports the time varying of the error function. Thedifficulty of the 2 RBFs approximation to predict the systemresponse can be better explained if the system response behaviouris analyzed considering several time-cuts (Fig. 8).
Based in the above results an improved approximation with 5RBFs has been considered.
Preliminarily the error function has been calculated; the timevarying of the error function (Fig. 9) seems to predict good resultsin the case of the adopted approximation. This is confirmed by
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 127
Fig. 4. Mean value (rotation θ ): comparison betweenMCS and Galerkin/neural approach (GNa) results (up) (Nϕ = 2); standard deviation (rotation θ ): comparison betweenMCS and GNa results (down) (Nϕ = 2).
Fig. 5. System response (rotation θ ) at different instants (10, 20, 40 and 50 s): comparison between MCS and Galerkin/neural approach (GNa) approximation (Nϕ = 5).
Fig. 10 where results of the approximation are compared withMCS. The approximation function, with 5 RBFs, is able tomatch theresults of the simulation with respect to the standard deviation ofboth displacement and velocity. The ability of the approximationfunction to match the actual response can be analyzed againconsidering several time-cuts (Fig. 11).
3.3. Cart and pendulum problem
As a third case study a 2 DOFs system subjected to a timedependent load acting on the system mass is considered. Theanalysed mechanical system is sketched in Fig. 12. The equation
of motion can be written as follows:
(M + m) x + m l θ cos θ − m l θ2 sin θ = 0
m l2 θ + m l x cos θ + m l x θ sin θ − mg l sin θ = f (t)(40)
where the forcing process, for illustrative purposes, has beenassumed as a sinusoidal one:
f (t) = a · sin (ω t) . (41)
Uncertainties of the system are assumed to be concentrated inthe length l of the pendulum. The parameters characterizing thesystem behaviour are as follows:m = 10 N s2/m;M = 30 N s2/m;El = 2 m; g = 9.81 m/s2; a = 10 N; ω = 15 rad/s. Based on the
128 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Fig. 6. Mean value (displacement x): comparison betweenMCS and Galerkin/neural approach (GNa) results (up) (Nϕ = 2); standard deviation (displacement x): comparisonbetween MCS and GNa results (down) (Nϕ = 2).
Fig. 7. Error function (Nϕ = 2).
proposed approach nonlinear restoring terms can bewritten in thefollowing form:
gxx, x, x, θ, θ , θ , t; l
= (M + m) x + m l θ cos θ − m l θ2 sin θ
gθ
x, x, x, θ, θ , θ , t; l
= m l2 θ + m l x cos θ + m l x θ sin θ − mg l sin θ.
(42)
So the original equation of motion (40) is written as:
gxx, x, x, θ, θ , θ , t; l
= 0
gθ
x, x, x, θ, θ , θ , t; l
= f (t).
(43)
Based on the proposed approach, again the first step of the solutionis represented by the derivation of the equation ofmotion (43)withrespect to time:
mtXθ
x, x, θ, θ , k
...θ + mtXX
x, x, θ, θ , k
...x
+ ctXθ
x, x, θ, θ , k
θ + ctXX
x, x, θ, θ , k
x
+ ktXθ
x, x, θ, θ , k
θ + ktXX
x, x, θ, θ , k
x = 0
mtθθ
x, x, θ, θ , k
...θ + mtθX
x, x, θ, θ , k
...x
+ ctθθ
x, x, θ, θ , k
θ + ctθX
x, x, θ, θ , k
x
+ ktθθ
x, x, θ, θ , k
θ + ktθX
x, x, θ, θ , k
x = f (t)
(44)
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 129
Fig. 8. System response (displacement x) at different instants (5, 10, 15 and 20 s): comparison between MCS and Galerkin/neural approach (GNa) approximation (Nϕ = 2).
Fig. 9. Error function (Nϕ = 5).
where the following notations have been introduced:
The second step of the solution procedure is the Radial BasisFunction approximation of the system response x = (θ, x)t :whose
derivative is the following:
x (t, b) =
Nϕk=1
wkx (t) ϕk (b)
x (t, b) =
Nϕk=1
wkx (t) ϕk (b)
...x (t, b) =
Nϕk=1
...wkx (t) ϕk (b) .
(46)
The last step is the evaluation of the weightw functions by meansof the training of the net. Several tests, with different numbers of
130 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Fig. 10. Mean value (displacement x): comparison between MCS and Galerkin/neural approach (GNa) results (up) (Nϕ = 5); standard deviation (displacement x):comparison between MCS and GNa results (down) (Nϕ = 5).
Fig. 11. System response (displacement x) at different instants (5, 10, 15 and 20 s): comparison betweenMCS and Galerkin/neural approach (GNa) approximation (Nϕ = 5).
RBFs (Nϕ), have been carried out and 15-s analysis time has beenconsidered. For each time step a system of Nϕ × N equations hasto be solved.
The problem has been preliminarily solved, for comparativepurposes, by means of MCS and the response of the system hasbeen obtained by direct numerical integration of the differentialequation of motion (40). A set of 500 simulations has been carriedout by generating a vector of random values of the variable l (forthis task a Gaussian distribution has been assumed with meanvalue µl = 2 m and a coefficient of variation of 5%).
As a first step, the case with 2 RBFs has been considered.Fig. 13 reports the mean value and the standard deviation ofthe system displacement; Fig. 14 reports the mean value andthe standard deviation of the system rotation. Results of theapproximation are compared with those obtained with the MCSand show that the approximation with 2 RBFs is not able to matchthe physical behaviour of the mechanical system. The inefficiencyof the degree of RBFs approximation to predict the system responsecan be better, also in this case, analyzing the time-cuts (rotation)(Fig. 15).
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 131
Fig. 12. The mechanical 2 DOFs nonlinear system (cart and pendulum problem).
This failure was also predicted by the a posteriori error functionthat for the present case is defined as follows:
x (t, b) =
Nϕk=1
wk (t) ϕk (b) (47)
gxx, ˙x, ¨x, θ ,
˙θ,
¨θ, t
= 0
gθ
x, ˙x, ¨x, θ ,
˙θ,
¨θ, t
= f (t)
(48)
e2 = E(G − F (t))2
. (49)
As a second step, the improvement of the approximation assuming5 RBFs has been considered. In this case the approximation is ableto match the system response as the error function for both thedisplacement (Fig. 16) and the rotation (Fig. 17) degree of freedomoffers acceptable results. This is confirmed by the comparisonbetween the proposed approach and the MCS results (Figs. 18 and19). The efficiency of the approximation to predict the mechanical
system response is shown again by the analysis of the time-cutsresponse (Fig. 20).
3.4. Nonlinear restoring spring
Another example is presented in order to demonstrate thefeasibility of the outlined method. The case study is a 2 DOFsnonlinear restoring spring system, as shown in Fig. 21, subjectedto a time–space dependent load acting on the system mass.
The mechanical and geometric parameters characterizing themechanical system are: the Young modulus E; the truss crosssectional area A; the truss length l and themassm. In the followingapplications the uncertainties of the system are considered tobe on the length l of the trusses. Denoting by x1(t) the verticaldisplacement and x2(t) the horizontal displacement, it is possibleto derive the equation of motion of this nonlinear system by usingthe Lagrange equations:
ddt
∂L∂ xi
−
∂L∂xi
= Qhi. (50)
The corresponding Lagrangian can be expressed as follows:
L =12m(x21 + x22) +
EAl
(∆21 + ∆2
2)
where∆1 =
(l + x2)2 + x21 − l
∆2 =
(l − x2)2 + x21 − l.
(51)
The first term is the well knows kinetic energy, and the second oneis the potential energy where ∆1 and ∆2 represents the trusseselongations.
By substituting Eq. (51) into the Lagrange equations (50), theequations of motion of the nonlinear system can be expressed inthe following form:
m xi (t) + gixj, l; t
= Qi
xj; t
i, j = 1, 2
with gi =∂
∂ xi(L) . (52)
Fig. 13. Mean value (displacement x): comparison between MCS and Galerkin/neural approach (GNa) results (up) (Nϕ = 2); standard deviation (displacement x):comparison between MCS and GNa results (down) (Nϕ = 2).
132 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Fig. 14. Mean value (rotation θ ): comparison betweenMCS andGalerkin/neural approach (GNa) results (up) (Nϕ = 2); standard deviation (rotation θ ): comparison betweenMCS and GNa results (down) (Nϕ = 2).
Fig. 15. System response (rotation θ ) at different instants (2, 5, 10 and 15 s): comparison between MCS and Galerkin/neural approach (GNa) approximation (Nϕ = 2).
In Eq. (52) term l, the trusses length, account for the parameteruncertainties. The system response will depend on the time t andon the length l. In this application the vector b that collects theuncertainties is a one-dimensional vector.
Preliminarily the problem has been solved by MSC (a set of500 simulations has been carried out by generating a vector ofrandom values of the variable lwith assigned mean and variance).A Gaussian distribution has been assumed considering a meanvalue µl = 5 m and a coefficient of variation of 5%. The responseof the system has been obtained by direct numerical integration ofthe differential equations (52) governing the problem in which the
forces acting on the mass were of the type:
Qi (xi; t) = a · sin (2π t) · cos (xi) . (53)
The analyzed time history had a duration of 10 s; the value of theparameter a for the time dependent load is assumed 200 N forthe x1 DOF, and 20 N for the x2 DOF in the calculations. Fig. 22reports the time histories of the two DOFs when the differentialequations (52) are solvedwith respect to the parameter l fixedwitha value equal to its mean. It is possible to observe the differencesbetween the behaviour of the responses of the system along thetwo degrees of freedom. The system is more stiff along the x2
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 133
Fig. 16. Displacement x error function (Nϕ = 5).
Fig. 17. Rotation θ error function (Nϕ = 5).
direction (along the truss direction) and then it shows a lesssensitive behaviour with respect to the applied load (and hence amore regular displacement function), if compared with the otherresponse.
In order to estimate the quality of the approximation offered bythe use of the RBF, several cases are investigated where differentvalues of the radial basis function number in Eq. (2) have beentaken into account. The output of the outlined procedure, thecoefficients wi(t) (and respectively wi(t) and wi(t)), are obtainedby integration of Eq. (22). Indicating with Nϕ the number ofneurons adopted, for each time step a system of Nϕ × N equationshas to be solved.
As a first test, a case with 5 RBFs has been considered. Fig. 23shows the expected value of the displacement for the degree offreedom x1 comparing simulation and the RBF approximation. Theapproximation function matches very well the simulation until3.5 s, after that it is possible to observe a difference betweenthe two responses. A similar behaviour is possible to observe inthe standard deviation of the displacement in the x1 direction:until 3.5 s the approximation with RBF is able to match thefrequencies of the mechanical response; after divergence appearson the absolute values even frequency seems conserved. Fig. 24shows the mean value and the standard deviation of the velocityfor the degree of freedom x1 comparing results obtained with
134 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
Fig. 18. Mean value of (displacement x): comparison between MCS and Galerkin/neural approach (GNa) results (up) (Nϕ = 5); standard deviation (displacement x):comparison between MCS and GNa results (down) (Nϕ = 5).
Fig. 19. Mean value (rotation θ ): comparison betweenMCS andGalerkin/neural approach (GNa) results (up) (Nϕ = 5); standard deviation (rotation θ ): comparison betweenMCS and GNa results (down) (Nϕ = 5).
simulation and with the RBF approximation. Again the methodworks well until 3.5 s, after that a difference appears (analysingthe system response with respect to the second degree offreedom x2, that seems to be less sensitive, a good agreementbetween simulation and approximation is observed for the entireanalysis).
Increasing the number Nϕ of radial basis functions up to 10 norelevant differences appear with respect to the case with 5 RBFs. Abetter approximation of the PDF is, of course, obtained but againthe approximation is able to reproduce the simulation only fora limited range. The approximation is quite good until 5 s, afterthat a difference appears in the absolute values of the response.
A similar behaviour is observed for the standard deviation of thedisplacement x1. Generally the approximation is able to reproducethe response’s frequencies with small errors, but errors appear onthe absolute value of the structural response.
If the number Nϕ of radial basis functions is increased to 20, theagreement between the expected value of the response and thesimulation improves considerably both in terms of displacementsand velocities. With respect to the previous cases it is possibleto observe that the procedure succeeds to follow the simulationpattern: only a slight difference is observable. The approximationis able to reproduce peaks of the response with small errors, for allthe duration of the analysis.
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 135
Fig. 20. System response (rotation θ ) at different instants (2, 5, 10 and 15 s): comparison between MCS and Galerkin/neural approach (GNa) approximation (Nϕ = 5).
Fig. 21. The mechanical 2 DOFs system.
Fig. 22. Response (in terms of displacements) with respect to the mean value ofthe parameter l.
A sensible improvement of results can be obtainedby increasingthe value Nϕ of the RBF functions. Taking into account 50 RBFfunctions the mean value and the standard deviation of theresponse for the two degrees of freedom are shown in Figs. 25
and 26. No significant difference appears between simulation andthe RBF approximation showing that the proposed method is ableto reproduce peaks of the response with small errors for all theduration of the analysis.
Finally, a run with 200 RBF functions has been carried out,in order to analyse the behaviour of the proposed theoreticalapproach but no relevant differences appear with respect to thecase with 50 RBFs, so to use such a high Nϕ number of functionsseems unjustified. Analysing the time history of both displacementad velocity it is possible to observe that the approximation with 50RBFs followswith a good agreementwith the results obtainedwithMCS. The error parameter can provide a criterion in order to acceptor reject the obtained solution.
Previous examples demonstrate the ability of the proposedapproach to approximate the dependence of the nonlinearstructural system on the uncertain parameters of the system.The error function, defined via a Galerkin projection scheme,additionally allows us to have control of the degree of the adoptedapproximation. In its actual form, the adopted solution strategymanages to approximate the system response process only byincreasing the numbers of RBFs. It has been verified that anincreasing degree of uncertainties demands increasing degree ofRBF approximation, leading to an increasing of computationalcosts. Uncertainties have been analysed assuming a coefficient ofvariation variable in a range between 5% and 20% that has beenassumed as the target of the considered problems. Improvements
136 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
0 1 2 3 4 5 6 7 8 9 10-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Mean value displacement (x1)
time [s]
E[x
1] [c
m]
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4Standard deviation displacement (x1)
time [s]
x1
MCS
GNa
MCS
GNa
Fig. 23. Mean value of the response: x1 displacement (5 RBFs) (up); standard deviation of the response: x1 displacement (5 RBFs) (down).
0 1 2 3 4 5 6 7 8 9 10-30
-20
-10
0
10
20
30Mean value velocity (x1)
time [s]
E[x
1] [c
m/s
]
MCS
GNa
0 1 2 3 4 5 6 7 8 9 100
5
10
15
20Standard deviation velocity (x1)
time [s]
x1
MCS
GNa
Fig. 24. Mean value of the response: x1 velocity (5 RBFs) (up); standard deviation of the response: x1 velocity (5 RBFs) (down).
are expected by an optimization of the RBF parameters (centreand decay parameters) and of their distribution over the sampledomain by means of an enhanced training.
4. Concluding remarks and perspective
The paper has presented an approximate approach for the dy-namics analysis of mechanical systems with randomness in theirparameters. The proposed methodology is based on a neural net-work approximation via Radial Basis Functions (RBF) of the systemmechanical response; a Galerkin approach is used in order to eval-uate the coefficients of the expansion (with the aim of minimizing
an appropriate error function). After recalling the background, theapproach has been firstly discussed and secondly applied to theanalysis of SDOF and MDOF mechanical systems. The numericalsolutions for the considered systems has been analysed and testedwith the aim to evaluate the dependency of the response on thedegree of the approximation and to evaluate the ability of the er-ror function (a posteriori analysis) to predict the efficiency of thedegree of approximation. Results have been compared with thoseobtained by means of Monte Carlo Simulation (MCS) so as to pro-vide the validation of the proposed approach. Moreover to betterexplain the novelty of the approach the first case study has beencompared with a classical perturbative approach. The comparison
M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138 137
0 1 2 3 4 5 6 7 8 9 10-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Mean value displacement (x1)
time [s]
E[x
1] [c
m]
MCS
GNa
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1Standard deviation displacement (x1)
time [s]
x1
MCS
GNa
Fig. 25. Mean value of the response: x1 displacement (50 RBFs) (up); standard deviation of the response: x1 displacement (50 RBFs) (down).
0 1 2 3 4 5 6 7 8 9 10-30
-20
-10
0
10
20
30Mean value velocity (x1)
time [s]
E[x
1] [c
m/s
]
MCS
GNa
0 1 2 3 4 5 6 7 8 9 100
5
10
15Standard deviation velocity (x1)
time [s]
x1
MCS
GNa
Fig. 26. Mean value of the response: x1 velocity (50 RBFs) (up); standard deviation of the response: x1 velocity (50 RBFs) (down).
with the results offered by MCS shows that the proposed methodis able to provide a good approximation of the relevant character-istic of the system response (mainly the first two moments of theresponse)with a relatively lownumber of RBFs (and, consequently,with reduced computational costs). The resulting approach is quitegeneral and the proposed expansion for the solution contains themain necessary probabilistic information (mean value and stan-dard deviation of the system response) that allows us to charac-terize the response process; moreover the control parameter (the
a posteriori error function) is very useful to verify the correctness ofthe number of terms in the series approximation. The obtained re-sults make this method attractive and demonstrate that, althoughlimited for the time being, it is a promising candidate to approachuncertain dynamic problems; further investigations are needed toassess its efficiency with respect to other applications. Improve-ments are expected by the optimization of the RBF parameters(centre and decay parameters) and of their distribution over thesample domain by means of an enhanced training.
138 M. Betti et al. / Probabilistic Engineering Mechanics 29 (2012) 121–138
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