System and source identification from operational vehicle responses: A novel modal model accounting for the track-vehicle interaction1footnote 11footnote 1Manuscript submitted to the Journal of Sound and Vibration, in the present form, on April 1, 2016.

System and source identification from operational vehicle responses: A novel modal model accounting for the track-vehicle interaction111Manuscript submitted to the Journal of Sound and Vibration, in the present form, on April 1, 2016.

Giovanni De Filippis Davide Palmieri Leonardo Soria and Luigi Mangialardi Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Viale Japigia 182, 70126 Bari, Italy
Abstract

Operational Modal Analysis (OMA) is a powerful tool, widely used in the fields of structural identification and health monitoring, and certainly eligible for identifying the real in-operation behaviour of vehicle systems. Several attempts can be found in the literature, for which the usage of algorithms based on the classical OMA formulation has been strained for the identification of passenger cars and industrial trucks. The interest is mainly focused on the assessment of suspension behaviour and, thus, on the identification of the so-called vehicle rigid body modes. But issues arise when the operational identification of a vehicle system is performed, basically related to the nature of the loads induced by the roughness of rolling profiles. The forces exerted on the wheels, in fact, depending on their location, are affected by time and/or spatial correlation, and, more over, do not fit the form of white noise sequences. Thus, the nature of the excitation strongly violate the hypotheses on which the formulation of classical OMA modal model relies, leading to pronounced modelling errors and, in turn, to poorly estimated modal parameters. In this paper, we develop a specialised modal model, that we refer to as the Track-Vehicle Interaction Modal Model, able to incorporate the character of road/rail inputs acting on vehicles during operation. Since in this novel modal model the relationship between vehicle system outputs and modal parameters is given explicitly, the development of new specific curve fitting techniques, in the time-lag or frequency domain, is now possible, making available simple and cost-effective tools for vehicle operational identification. More over, a second, but not less important outcome of the proposed modal model is the usage of the resulting techniques for the indirect characterisation of rolling surface roughness, that can be used to improve comfort and safety.

keywords:
system identification, modal model, vehicle systems, surface roughness, correlated inputs
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1 Introduction

The identification of the real in-operation behaviour is of utmost importance in all the stages of the vehicle design process, from model updating, extensively used as a tool for improving the accuracy of dynamics simulations, to optimisation strategies and techniques, product life-cycle management implementations, and the verification of controllers’ performance Gillespie1992 (); Milliken1995 (); Guiggiani2014 (). In this context, a wheeled vehicle, cruising at constant speed, on a certain road profile, is a common working occurrence for designers, in which the vehicle is mainly subjected to the external loads exerted by the interaction with the rolling rough surface. From an experimental point of view, since unknown steady-state inputs are applied to the system, Operational Modal Analysis (OMA), even referred to as in-Operation or Output-only Modal Analysis, is certainly eligible to perform the system structural identification, moving from output data only Hermans1999 (); Brincker2010 (). Specifically, in the case of linear systems subjected to operational loads, OMA allows for estimating the so-called modal parameters, that is the resonance frequencies, the damping ratios, the mode shapes, and the operational reference vectors, that can be utilised, in turn, to obtain a mathematical model of the system under test, generally referred to as the modal model Heylen2007 ().

Ordinary vehicle systems generally exhibit a very complex dynamics, mainly owing to nonlinearities related to shock absorbers’ operation and to the kinematics imposed by the suspension design Wallaschek1990 (); Surace1992 (). Thus, modal models synthesised under the real operational loadings actually represent an equivalent linearisation of the tested nonlinear systems around interesting and representative working points Caughey1963 (); Hagedorn1988 (). Different speeds, manoeuvres, as cruising or constant radius cornering, the presence of payloads, and of different adopted designs of suspension result in different equivalent linearised models.

A novel, but even challenging application of OMA is being proposed in the framework of vehicle ride dynamics Cossalter2004 (); Soria2012 (). Since modal parameter estimation allows for quantifying the overall damping associated to the vehicle rigid-body modes, OMA of vehicle systems can be employed as a tool for assessing the performance of different suspension systems. More over, in the case of semi-active and active suspensions, the application of OMA is extremely interesting for the validation and optimisation of controllers’ performance Soria2012 ().

During the last decade, several robust methods have been developed for estimating reliable modal parameters from output-only data Parloo2003 (). Basically, these methods rely on the Natural Excitation Technique (NExT) assumptions James1992 (); Shen2003 (); Caicedo2004 (); Peeters2005 (): (i) The unknown loads acting on the system have to fit the form of white noise sequences, (ii) in case of multi-point excitation, the external inputs are required to be strictly uncorrelated. The NExT assumptions are commonly satisfied in civil engineering applications, as is the case of high rise buildings and suspension bridges, where typically the structures are excited by environmental loadings due to traffic, wind and waves Peeters2007 ().

In this paper, we first focus on the issues that arise when an operational structural identification of a vehicle system is performed by using classical OMA based techniques. These issues are related to several aspects: (i) The presence of high modal density and closely-spaced modes; (ii) the high amount of damping owing to the presence of shock-absorbers; (iii) the nature of the loads induced by the rolling surface. Specifically, we show that the latter point implies a strong violation of the NExT assumptions. In fact, the trend of each excitation, transmitted to the vehicle through the wheel stiffness and damping, is quite far from the flat frequency shaping of white noise Turkay2005 (); Andren2006 (). In addition, depending on their location, the inputs acting on the wheels result affected by time and/or spatial correlation Butkunas1966 (); Styles1976 (); Ammon1992 (); Bogsjo2008 (); Song2013 (); DeFilippis2013 (). As a consequence, the application of classical OMA methodologies for post-processing the vehicle output responses may lead to pronounced modelling errors and, therefore, to poorly estimated modal parameters.

Thus, by assuming that the rolling surface is an homogeneous Gaussian random field Dodds1973 (); Newland1993 (), we formulate of a novel OMA modal model, referred to as the Track-Vehicle Interaction Modal Model (TVIMM), specialised and suitable for developing new identification procedures aimed at the estimation of the modal parameters of a vehicle system. More over, we show that the coefficients of the adopted empiric surface model can be even estimated as a further outcome of our OMA processing. Since direct measurements performed by using specific profilometers can result to be expensive, this second application of the model has relevant interest in road and railway health monitoring. In this field, in fact, the indirect characterisation of surface roughness can be employed as a tool for improving safety and comfort Gonzalez2008 ().

The rest of the paper is organised as follows. In Section 2, we obtain the representation of surface-induced forces exerted on a vehicle and discuss the issues arising when identification procedures based on the classical OMA formulation are used to process the output responses of vehicle systems. In Section 3, we propose a novel OMA formulation, specifically designed for the operational identification of vehicle systems, based on a specialised modal model accounting for the track-vehicle interaction. In Section 4, we offer a numerical demonstration of the developed formulation. Concluding remarks are summarised in Section 5. In Appendix A, we provide mathematical proof of some fundamental equations used in Section 3.

2 System and source identification from operational vehicle responses

The nature of loadings exerted on a vehicle by the rolling surface plays a fundamental role in the process of operational identification of the system. We here exploit the properties of homogeneous Gaussian random fields to derive a model of the surface-induced forces. By comparing the resulting surface inputs with those permitted by complying the NExT assumptions, we conclude that a specialised OMA formulation is needed, able to provide a correct modelling of the system and of the main excitation source, that is the surface roughness.

2.1 Classical OMA approach

By processing the output responses of a degrees of freedom (dofs) system, OMA leads to the estimation of the modal parameters, that is the poles , the modal vectors and the operational reference vectors (with ), where the symbol indicates matrix transposition. The computation of system’s poles is of fundamental interest as they contain information about the resonance frequencies and the damping factors

 λn,λ∗n=−ζnωn±iωn√1−ζ2n, (1)

where indicates complex-conjugation. The application of methods of system identification to OMA has given rise to a large variety of estimation techniques Heylen2007 (); Peeters2001 (), comprising, generally, several steps. The most advanced procedures are based on the usage of the modal model, commonly utilised to complete the identification step and for validating the results of the estimation process Peeters2005 (). Recently, a new approach has been proposed, allowing for the computation of modal parameters directly from the modal model El-Kafafy2015 ().

In the case of classical OMA, the modal model is formulated both in the time-lag and in the angular frequency domains, by referring to correlation functions and power spectral densities (PSDs), respectively. Thus, by considering a system of external forces relying on the NExT assumptions, the input PSD matrix can be written as

 Sf(ω)=⎡⎢ ⎢⎣S10⋱0SN⎤⎥ ⎥⎦. (2)

In fact, since the inputs are required to be strictly uncorrelated white noise sequences, the matrix in Eq. (2) has only constant entries along the main diagonal. With referring to a dofs system subjected to the operational loadings Eq. (2), the formulation of the modal model can be given in terms of output correlation matrix and output PSD matrix , as

 Rq(τ)=2N∑n=1ϕnψTne+λnτh(τ)+ψnϕTne−λnτh(−τ), (3)
 Sq(ω)=2N∑n=1ϕnψTniω−λn+ψnϕTn−iω−λn, (4)

where indicates the Heaviside step function.

2.2 Random fields for surface profiles modelling

Generally, in case of OMA, input loads are uncontrollable and, in addition, remain unmeasured. For this reason, the analyst has to verify that the operational loads are suitable to adequately and effectively excite the system in the frequency range of interest.

A vehicle system in steady-state working conditions is subjected to both external and on-board excitations Gillespie1992 (). On-board forces are related to the operation of rotating parts and are specifically imputable to engine operation and to non-uniformities in assemblies and components of the driveline. These forces introduce harmonics in the spectra, whose frequencies are expected to depend on the engine rotating velocity and, in turn, on the vehicle speed. External loads are mainly related to surface roughness, that represent the only present source able to generate an effective broad band excitation. Specifically, the cut-off frequency of the input spectrum depends on roughness spatial frequency content, vehicle velocity and wheel dynamic behaviour.

Random fields for surface modelling have extensively been studied and the usage of different approaches has emerged Dodds1973 (); Newland1993 (). Basically, the following assumptions are commonly adopted: (i) the surface roughness is an homogeneous random field; (ii) the height of the asperities satisfies to a zero-mean Gaussian distribution; (iii) the pavement unevenness is an ergodic random process. Homogeneity implies that the statistical properties of surface roughness are independent on spatial observations. This assumption has implications similar to stationarity for one-dimensional random processes. The second assumption ensures that the output responses of a linear system subjected to pavement excitations satisfy, as well, to a zero-mean Gaussian distribution. Ergodicity guarantees that average is equal to expectation calculated over the whole ensemble.

In simple track-vehicle interaction models, the unilateral contact point hypothesis is often utilised to describe the forces transmitted to the vehicle through the wheel stiffness and damping. This assumption is made without loss of generality, by considering that the distributed contact in the wheel-pavement interface acts as a low-pass filter, whose bandwidth is governed by the contact interface itself Sun2006 (). In this case, the surface roughness can be supposed to be a single-track random process, describing the longitudinal profile along the wheel path in the travelling direction Turkay2005 (). Generally, an empiric parametric model is adopted to fit the measured PSD, and the captured surface roughness is classified or employed in the simulations. The non-smoothed PSD is often approximated by a simple function, involving the usage of only few parameters. A literature survey on existing different approximations for longitudinal road profiles is presented in Ref. Andren2006 ().

When more than one profile is needed, the ordinary coherence function is introduced to express the relationship between multiple tracks ISO8608 (). Considering two parallel surface profiles and , with the longitudinal spatial variable, separated by a distance , the ordinary coherence function is defined in the angular spatial frequency domain as

 Γp(ν)=∣∣Sdidj(ν)∣∣√Sdi(ν)Sdj(ν), (5)

where and are the auto-PSDs of and , respectively, and is the cross-PSD between the two profiles. By definition, the coherence function is a real even function ranging from 0 to 1. As a consequence, two separated surface profiles perfectly overlap at wavelengths corresponding to amplitude values of the coherence function equal to 1. The expression of cross-PSD is obtained from Eq. (5) as

 Sdidj(ν)=Γp(ν)√Sdi(ν)Sdj(ν)e−iβp(ν), (6)

where is the difference between the phases related to and .

Based on the homogeneity assumption, the following properties can be exploited for simplifying the description of multiple tracks Dodds1973 (); Newland1993 ():
(i) The auto-PSDs related to parallel surface profiles are coinciding and, thus, equal to the same function

 Sdi(ν)=Sdj(ν)=Sd(ν); (7)

(ii) the cross-PSD between two parallel surface profiles is a real and even function depending on auto-PSD and coherence function

 Sdidj(ν)=Γp(ν)Sd(ν), (8)

in which it is specifically useful to notice the cancellation of the phase difference in Eq. (8).

2.3 Problem statement

We here consider the generic -wheel vehicle system represented in Fig. 1, with , even if the following considerations can be easily extended to systems equipped with wheels. The considered geometry includes different trackwidths (with ) and different wheelbases (with ). We hypothesise that the vehicle travels at constant velocity on an homogeneous Gaussian surface. By writing Eqs. from (5) to (8) in the angular frequency domain , we obtain the following PSD matrix of surface-induced displacements

 Sr(ω) (9)

where indicate the time-delays between the inputs acting on different axles.

To describe the forces transmitted to the vehicle through the wheel stiffness and damping, we first introduce the following static gain matrix

 Gfr=[{0}(N−Nt)×Nt{0}(N−Nt)×NtKtCt], (10)

where and are diagonal matrices in which stiffness and damping terms are, respectively, collected. We, second, account for the PSDs of the derived processes, in addition to Eq. (2.3), in case tyre damping is included in the wheel model. We, finally, obtain the following representation for the input PSD matrix related to surface-induced forces

 Sf(ω)=Gfr(D(ω)⊗Sr(ω))GTfrwithD(ω)=[1iω−iωω2], (11)

where the matrix allows for taking into account the contributions related to road-induced velocities and the symbol denotes the Kronecker product. We stress that the sign of the imaginary part of Eq. (11) depends on the adopted definition of correlation function, where a change of definition in the time-lag domain leads to a complex-conjugate expression in the frequency domain.

By comparing Eqs. (2) and (11), we conclude that for a vehicle system subjected to operational loadings, the modal model resulting from the classical OMA approach (Eqs. (3) and (4)) is no more valid. The arisen issues are mainly related to the nature of the loads induced by the rolling surface, that violates the NExT assumptions. First, we notice that forces applied to each wheel are basically coloured excitations, with magnitude inversely proportional to the frequency raised to a certain power, and shaping depending on the auto-PSD of the road profile ( Eq. (7)) and on the wheel parameters ( Eq. (10)). In addition, being systems equipped with more than one wheel, forces are affected by time and/or spatial correlation: (i) wheels mounted on the same axle are subjected to spatially correlated inputs ( in Eq. (2.3)); (ii) wheels travelling on the same path and located on different axles are subjected to time correlated inputs ( in Eq. (2.3)); (iii) wheels travelling on separated paths and located on different axles are subjected to time and spatially correlated inputs ( in Eq. (2.3)).

Owing to the aforementioned effects, the application of classical OMA methodologies to vehicle responses may lead to pronounced modelling errors and, therefore, to poorly estimated modal parameters. The formulation of a specialised modal model for correctly describing the track-vehicle interaction is required and, in turn, a specific OMA formulation is needed to understand how Eqs. (3) and (4) are modified. We stress that a modal model providing the relationship between the generic output of the system and its modal parameters in an explicit form is needed for the formulation of a whatever procedure for modal parameter estimation based on suitable curve fitting algorithms. We more over comment that since operational vehicle responses incorporate information about the surface roughness, the identification procedures based on this novel OMA formulation would allow for estimating, in addition to modal parameters, even the coefficients of the adopted empiric surface models.

3 The Track-Vehicle Interaction Modal Model

We here utilise a 7 dofs system (), generally referred to as the full-car model, to introduce the theoretical background of the proposed Track-Vehicle Interaction Modal Model. To obtain the analytical expression of the generic system output, we solve a Duhamel integral in modal coordinates. Moving from Eq. (11), we compute the correlation matrix of surface-induced forces and achieve the TVIMM formulation by using properties of convolution integrals and Fourier transform.

3.1 Full-car model

The full-car model (Fig. 2) is the simplest mathematical description of a four-wheel vehicle (), whose predicted output responses incorporate all the effects of the issues discussed in Section 2, which make no longer possible the usage of classical OMA methodologies for post-processing. Since full-car model offers a good trade-off between model complexity and accuracy, this linear lumped-parameters system is commonly utilised for simulating the ride dynamics of passenger and race cars Milliken1995 (); Guiggiani2014 (). The model preserves the multi-input nature of road excitation and, different from half-car model, allows for evaluating the contributions to vehicle responses due to roll disturbance produced by two parallel tracks. The geometry comprises one single trackwidth and one single wheelbase . In Fig. 2, we indicate the dofs of the model by using the following notation: , , and represent the heave, the roll and the pitch rigid body motions of the sprung mass (the body); , , , denote the vertical displacements at the four corners of the sprung mass (strut mounts to body); , , , are the rattle displacements of the unsprung masses (the wheels). More over, with regards to system parameters, , , represent the mass and the moments of inertia associated with the body; , , , denote the unsprung masses; , , , are the stiffness and damping coefficients of the absorbers; , , , indicate the stiffness and damping coefficients of the tyres; , are the distances of the front and rear axle, respectively, from the center of gravity of the unsprung mass, the sum of which equals the wheelbase.

3.2 Duhamel integral in modal coordinates

Since vehicles are generally non-proportional damping systems, we adopt a representation in state-space form. Thus, we recast the motion equations of a dofs system into an equivalent set of 2 first-order differential equations. Specifically, by denoting with the vector of Lagrangian coordinates and with the vector of external loads, we write the well-known set of system dynamics equations as

 M¨q(t)+C˙q(t)+Kq(t)=f(t), (12)

where is the time variable, and , , and are the mass, damping, and stiffness matrices, respectively. By adding the following further set of differential equations

 M˙q(t)−M˙q(t)={{0}}N×N, (13)

and by making the substitutions

 x(t)=[q(t)˙q(t)]˙x(t)=[˙q(t)¨q(t)]u(t)=[f(t){{0}}N×1], (14)

we obtain the set of system dynamics equations in state-space form

 P˙x(t)+Qx(t)=u(t), (15)

where and are partitioned as

 P=[CMM{0}N×N]Q=[K{0}N×N{0}N×N−M]. (16)

Here, the product leads to the so-called state matrix . The eigenvalue decomposition of the state matrix is

 A=VΛV−1, (17)

where is the eigenvalue matrix, containing the system poles

 Λ=diag(λ1,…,λN,λ1∗,…,λN∗), (18)

with diag denoting a diagonal matrix, while is the eigenvector matrix, having the following structure

 V=[ψ1⋯ψNψ∗1⋯ψ∗Nλ1ψ1⋯λNψNλ∗1ψ∗1⋯λ∗Nψ∗N]. (19)

We decouple the equations of motion by using the following coordinate transformation from the physical to the modal space

 x(t)=Vp(t)⇒[q(t)˙q(t)]=2N∑n=1[ψnλnψn]pn(t), (20)

where is the modal state vector. By substituting Eq. (20) in Eq. (15) and pre-multiplying both sides by , we obtain 2 independent differential equations, that we collect in the following compact form

 Ma˙p(t)+Mbp(t)=VTu(t), (21)

where and are two diagonal matrices generally referred to as modal a and modal b. Specifically,

 Ma=VTPV=diag(ma1,…,maN,m∗a1,…,m∗aN) (22)

and

 Mb=VTQV=diag(mb1,…,mbN,m∗b1,…,m∗bN), (23)

while

 VTu(t)=[ψ1⋯ψNψ∗1⋯ψ∗N]Tf(t). (24)

By considering the generic system motion equation of set Eq. (21) referred to the -th mode of vibration,

 man˙pn(t)+mbnpn(t)=ψnTf(t)withmbn=−λnman, (25)

and by assuming zero initial conditions, we retrieve the solution in the form

 pn(t)=ψnTmant∫−∞f(ϵ)eλn(t−ε)dε. (26)

Eq. (26), usually written in physical coordinates, is the well-known Duhamel integral. By combining Eqs. (20) and (26), we obtain the system response in terms of the physical Lagrangian coordinates

 q(t)=2N∑n=1ψnpn(t)=2N∑n=1ψnψnTmant∫−∞f(ρ)eλn(t−ρ)dρ. (27)

We comment that Eq. (26) can be interpreted as a convolution integral encompassing the impulse response matrix of the system under study

 g(t)=2N∑n=1ψnψnTmaneλnt. (28)

3.3 Input and output correlation functions

By considering the generic outputs and in Eq. (27), evaluated at the separated time instants and , respectively,

 qi(t)=2N∑n=1ψinψnTmant∫−∞f(ρ)eλn(t−ρ)dρ, (29)

and

 qj(t+τ)=2N∑m=1ψjmψmTmamt+τ∫−∞f(σ)eλm(t+τ−σ)dσ, (30)

where the following relations hold between the dummy integration variables

 ρ→tandσ→t+τ⇔σ=ρ+τandτ=σ−ρ, (31)

we derive the resulting output cross-correlation function

 Rqiqj(τ) =E[qi(t)qj(t+τ)] =2N∑n=12N∑m=1ψinψjmmanmamt+τ∫−∞t∫−∞ψnTE[f(ρ)fT(ρ+τ)]ψmeλn(t−ρ)eλm(t+τ−σ)dρdσ =2N∑n=12N∑m=1ψinψjmmanmamt+τ∫−∞t∫−∞ψnTRf(τ)ψmeλn(t−ρ)eλm(t+τ−σ)dρdσ, (32)

where the symbol indicates the expectation computed over the ensemble and is the input correlation matrix related to the external forces, which, by definition, is the inverse Fourier transform of the input PSD matrix Eq. (11).

By particularising to the case of full-car model (Fig. 2) the stiffness and damping entries of the static gain matrix Eq. (10)

 Kt=diag(kft,kft,krt,krt)Ct=diag(cft,cft,crt,crt), (33)

we decompose the PSD matrix of surface-induced displacements through the following summation

 Sr(ω)=Sd(ω)Nw∑p=0Γp(ω)Δp(ω)withΓ0(ω)=1, (34)

where in the case of full car model and the matrices and are defined as

 Δ0(ω)=⎡⎢ ⎢ ⎢ ⎢⎣10e−iωτ10010e−iωτ1e+iωτ10100e+iωτ101⎤⎥ ⎥ ⎥ ⎥⎦, (35)
 Δ1(ω)=⎡⎢ ⎢ ⎢ ⎢⎣010e−iωτ110e−iωτ100e+iωτ101e+iωτ1010⎤⎥ ⎥ ⎥ ⎥⎦. (36)

By substituting Eq. (34) in Eq. (11), we obtain the Fourier pair and in the case of interest

 Sf(ω)=Sd(ω)Nw∑p=0Γp(ω)Gfr(D(ω)⊗Δp(ω))GTfrwithΓ0(ω)=1, (37)

and

 Rf(τ)=Rd(τ)∗Nw∑p=0Hp(τ)∗(Gfr(d(τ)⊗δp(τ))GTfr)withH0(τ)=1, (38)

where is the auto-correlation function associated to surface profiles and (with ) represents the inverse Fourier transform (IFT) of the ordinary coherence function referred to two parallel tracks. The terms , and correspond to a representation in the time-lag domain of the matrices included in Eq. (37), that is

 (39)
 (40)
 (41)

where indicates the Dirac delta function, and the symbols and denote the first and second order derivative operators, respectively. By combining Eqs. (38) and (3.3), we achieve the final expression of the cross-correlation function

 Rqiqj(τ)=Rd(τ)∗Nw∑p=0Hp(τ)∗2N∑n=12N∑m=1ψinψjmmanmamt+τ∫−∞t∫−∞ψTnGfr(d(τ)⊗δp(τ))GTfrψmeλn(t−ρ)eλm(t+τ−σ)dρdσ. (42)

3.4 Modal model in the time-lag and frequency domains

Moving from Eq. (42), by means of calculations detailed in Appendix A, we obtain the relationship between the operational modal parameters (, , , , ) and the generic output cross-correlation function . We here rephrase the expression of the output correlation matrix , in the following compact matrix notation

 Rq(τ)=2N∑n=1¯φn(τ)ψTne+λnτ+ψn¯φTn(−τ)e−λnτ, (43)

where the terms are the lag-dependent operational reference vectors, defined as

 ¯φn(τ)=Rd(τ)∗NW∑p=0Hp(τ)∗(αpnh(τ)+NL∑l=1βplne+λnτlh(τ+τl)+χplne−λnτlh(τ−τl))withH0(τ)=1. (44)

By Fourier transforming Eq. (43), we obtain the expression of the output PSD matrix

 Sq(ω)=2N∑n=1φn(ω)ψTniω−λn+ψnφTn(−ω)−iω−λn, (45)

where the terms are the frequency-dependent operational reference vectors, defined as

 φn(ω)=Sd(ω)NW∑p=0Γp(ω)(αpn+NL∑l=1βplne+iωτl+χplne−iωτl)withΓ0(ω)=1. (46)

The modal model that we introduce through Eqs. (43) and (45), namely the TVIMM, incorporates a combination of several contributions: (i) and describe the statistical properties of the single surface profiles; (ii) and account for the time correlation between the inputs applied to wheels travelling on the same path and mounted on different axles; (iii) and encompass the spatial correlation due to loads acting on wheels travelling on separated paths. Thus, the vehicle output responses contain information on the dynamics of the system and on its interaction with the surface profiles. Not considering this fundamental last contribution from the resulting modal model (Eqs. (43) and (45)) increase the modelling errors of the identification procedure. As a consequence, the unknown parameters result poorly estimated.

We consider useful to stress that although Eqs. (43) and (45) have been obtained by only considering the rigid-body modes of the full-car model, this novel OMA formulation is of general validity and can be utilised for real-world vehicle systems. Specifically, it is able to include and describe, without loss of generality, even the presence of deformable-body modes of vibration.

3.5 Comments on the usage of half spectra

Several robust OMA algorithms in the frequency domain rely on the estimation of the so called half spectra Peeters2007 (), defined as

 S(+)qiqj(ω)=2N∑n=1ϕinψjniω−λn. (47)

The main advantages of this representation consist in: (i) The usage of lower model orders without affecting the quality of the fitting; (ii) the adoption of well-known windowing functions for reducing the effect of leakage and the influence of samples at the higher time-lags, which are the most affected by noise.

In the case of the classical OMA formulation, half spectra are computed by Fourier transforming the correlation functions corresponding to positive time-lags

 R(+)qiqj(τ)=2N∑n=1ϕinψjne+λnτh(τ). (48)

We here prove that in presence of time correlation this property does no longer hold. In fact, the lag-dependent operational reference vectors do not vanish at the positive time-lags

 ¯φTn(−τ)=Rd(τ)∗NW∑p=0Hp(τ)∗NL∑l=1βplnTe−λnτlh(−τ+τl)forτ>0, (49)

implying that the Fourier transform of Eq. 43 for does not lead to the first fraction summation in Eq. 45.

4 Numerical demonstration

In this section, we offer a numerical demonstration of the effectiveness of Eqs. (43) and (45). Specifically, we first compute the responses of the full-car model to surface-induced excitations by using the following implicit analytical expression holding in the frequency domain

 Sq(ω)=G∗qf(ω)Sf(ω)GTqf(ω), (50)

representing the well-known input-output formula Newland1993 (). Since the full-car model utilised in this study is a linear mutliple-input mutliple-output system, the output PSD matrix can be calculated at each frequency through Eq. (50). We derive the frequency response function matrix , referred to the lumped-parameters system of Fig. 2, by means of the following relation

 Gqf(ω)=(−ω2M+iωC+K)−1. (51)

Then, we compare the obtained curves with those resulting from the TVIMM. We stress that the comparison, here provided for demonstration purposes, is made between quantities computed by following two different routes: The former is based on Eq. (50), in which the relationship between the generic output of the system and its modal parameters in not explicitly given; the latter is based on the proposed OMA for vehicles’ modal model Eqs. (43) and (45), in which the relationship is instead provided in an explicit form. Only this second formulation allows for developing specialised curve fitting techniques, needed, in turn, for modal parameter estimation. We more over comment that, based on the different approximations proposed in the literature, semi-analytical expressions of and have to be provided, in order to form the input PSD matrix of surface-induced forces (Eq. (11)).

4.1 Matrix formulation of the simulated vehicle model

For the full-car model, two alternative sets of degrees of freedom can be used to describe the dynamics of the sprung mass: (i) The heave , the roll , and the pitch or (ii) the vertical translations at the right-front , left-front , and right-rear corners. In what follows, we analyse both the two cases by introducing two different Lagrangian coordinate vectors, respectively

 (52)

The vector of surface-induced forces is

 f(t)=[000cft˙dA(t)+kftdA(t)cft˙dB(t)+kftdB(t)crt˙dC(t)+krtdC(t)crt˙dD(t)+krtdD(t)]T. (53)

The mass, damping and stiffness matrices of the considered vehicle model, related to the first set of Lagrangian coordinates , are given by

 M1=diag(ms,jsx,jsy,mu1,mu2,mu3,mu4), (54)
 C1=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣2(cf+cr)02(crL1r−cfL1f)−cf−cf−cr−cr00.5(cf+cr)W2100.5cfW1−0.5cfW10.5crW1−0.5crW12(crL1r−cfL1f)02(cfL21f+crL21r)cfL1fcfL1f−crL1r−crL1r−cf0.5cfW1cfL1fcf+cft000−cf−0.5cfW1cfL1f0cf+cft00−cr0.5crW1−crL1r00cr+crt0−cr−0.5crW1−crL1r000cr+crt⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (55)
 K1=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣2(kf+kr)02(krL1r−kfL1f)−kf−kf−kr−kr00.5(kf+kr)W2100.5kfW1−0.5kfW10.5krW1−0.5krW12(krL1r−kfL1f)02(kfL21f+krL21r)kfL1fkfL1f−krL1r−krL1r−kf0.5kfW1kfL1fkf+kft000−kf−0.5kfW1kfL1f0kf+kft00−kr0.5krW1−krL1r00kr+krt0−kr−0.5krW1−krL1r000kr+krt⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (56)

while those related to the second set , can be expressed by

 M2=M1T−1C2=C1T−1K2=K1T−1, (57)

where is the transformation matrix

 T=⎡⎢ ⎢ ⎢ ⎢⎣⎡⎢⎣1−0.5W1−L1f10.5W1−L1f1−0.5W1L1r⎤⎥⎦{{0}}4×4{{0}}4×4{{I}}4×4⎤⎥ ⎥ ⎥ ⎥⎦. (58)

The parameters of the full-car model employed in the simulation Guiggiani2014 () are listed in Tab. 1.

4.2 Single-profile and coherence-based models for parallel road paths

We obtain the input PSD matrix of surface-induced forces by imposing that the auto-PSD of single surface profiles be equal to the ISO 8608 class C approximation. According to ISO 8608 ISO8608 (), in fact, roughness level is classified from A to H, and the form of the fitted one-sided auto-PSD is given as follows

 Sd(ν)=S0(νν0)−eforνa≤ν≤νb, (59)

where rad/m denotes the reference angular spatial frequency and (m3) is the amplitude of the PSD for . A constant velocity auto-PSD can be obtained by imposing the undulation exponent equal to 2. When this occurrence is satisfied, the auto-PSD of surface-induced displacements is calculated through a simple integration of a flat-spectrum velocity signal. Rearranging Eq. (59), the expression of the PSD in the angular frequency domain becomes

 Sd(ω)=Sd