Patricia Davies

Experimental Techniques and Identification of Nonlinear and Viscoelastic Properties of Flexible Polyurethane Foam

Identification of the vibrational behavior of polyurethanefoams used in automotive seats is described. The dynamic system consistsof a rigid block mounted on a 3″ cube of foam material, which serves asthe only flexible component. When constrained to undergo linearunidirectional motion, the dynamic system is modeled as a single degreeof freedom system, governed by an integro-differential equation. Inaddition to a relaxation kernel representing the linear viscoelasticbehavior of the foam, the model includes a polynomial type stiffness toaccount for the foams strain-based nonlinearities. The relaxationkernel is assumed to be of an exponential type. Experimentalmethodologies for obtaining repeatable, accurate measurements of thesystems response to an impulse and to single frequency harmonic baseexcitations are described. Analysis methods are then investigated forextracting the relevant linear, nonlinear, and viscoelastic parameters.Characterization of these foam properties as functions of compressionlevel is also presented.

Flexible polyurethane foam modelling and identification of viscoelastic parameters for automotive seating applications

Abstract A hereditary model and a fractional derivative model for the dynamic properties of flexible polyurethane foams used in automotive seat cushions are presented. Non-linear elastic and linear viscoelastic properties are incorporated into these two models. A polynomial function of compression is used to represent the non-linear elastic behavior. The viscoelastic property is modelled by a hereditary integral with a relaxation kernel consisting of two exponential terms in the hereditary model and by a fractional derivative term in the fractional derivative model. The foam is used as the only viscoelastic component in a foam–mass system undergoing uniaxial compression. One-term harmonic balance solutions are developed to approximate the steady state response of the foam–mass system to the harmonic base excitation. System identification procedures based on the direct non-linear optimization and a sub-optimal method are formulated to estimate the material parameters. The effects of the choice of the cost function, frequency resolution of data and imperfections in experiments are discussed. The system identification procedures are also applied to experimental data from a foam–mass system. The performances of the two models for data at different compression and input excitation levels are compared, and modifications to the structure of the fractional derivative model are briefly explored. The role of the viscous damping term in both types of model is discussed.

Resonant dynamics of an autoparametric system : A study using higher-order averaging

Abstract The autoparametric system consisting of a pendulum attached to a primary spring-mass is known to exhibit 1:2 internal resonance, and amplitude-modulated chaos under harmonic forcing conditions. First-order averaging studies and an analysis of the amplitude dynamics predicts that the response curves of the system exhibit saturation. The period-doubling route to chaos is observed following a Hopf bifurcation to limit cycles. However, to answer questions about the range of the small parameter e (a function of the forcing amplitude) for which the solutions are valid, and about the persistence of some unstable dynamical behavior, like saturation, higher-order non-linear effects need to be taken into account. Second-order averaging of the system is undertaken to address these issues. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of e appearing in the averaged equations. For larger e, second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response. For the response between the two Hopf bifurcation points from the coupled-mode solution branch, the period-doubling as well as the Silnikov mechanism for chaos are observed. The predictions of the averaged equations are verified qualitatively for the original equations.

Identification of Nonlinear and Viscoelastic Properties of Flexible Polyurethane Foam

Analysis of the steady-state response of a polyurethane foam and masssystem to harmonic excitation is presented. The foams uni-directionaldynamic behavior is modeled by using nonlinear stiffness, linearviscoelastic and velocity proportional damping components. Therelaxation kernel for the viscoelastic model is assumed to be a sum ofexponentials. The harmonic balance method is used to develop one- andtwo-term approximations to periodic solutions, and the equationsdeveloped are utilized for system identification. The identificationprocess is based on least-squares minimization of a sub-optimal costfunction that uses response data at various excitation frequencies andamplitudes. The effects of frequency range, spacing and amplitudes ofthe harmonic input on the results of the model parameter estimation arediscussed. The identification procedure is applied to measurements ofthe steady-state response of a base-excited foam-mass system. Estimatesof the system parameters at different levels of compression and inputamplitudes are thus determined. The choice of model-order and thefeasibility of describing the system behavior at several inputamplitudes with a single set of parameters are also addressed.

Vocal affect in three‐year‐olds: A quantitative acoustic analysis of child laughter

Recordings were obtained of the laughter vocalizations of four 3-year-old children during three sessions of spontaneous free-play between mother and child in a laboratory playroom. Acoustic analysis was used to determine laughter durations, laughter events, F0, and harmonic characteristics, and to suggest a taxonomy of laughter types. Melodic contours were assessed from patterns of F0 change during laughter. Mean duration of laughs ranged from 200 ms to 2.0 s, but events within a laugh were usually about 200-ms duration. Laughs were intuitively classified into four major types, and, following the acoustic analyses, were further defined and classified into types and subtypes of exclamatory and dull comment; chuckle; basic, variable, and classical rhythmical; and squeal. Melodic contours included more rising contours than previously reported for cry, but there was great variability in the types of contours produced especially for rhythmical laughs. The results of the acoustic analyses are discussed in relation to (a) the development of a taxonomy of laughter and (b) different features of the vocal affect characteristics of high-intensity emotion.

Simplified models of the vibration of mannequins in car seats

Abstract A simplified two-dimensional modelling approach to predict the vibration response of mannequin occupied car seats about a static settling point is demonstrated to be feasible. The goal of the research is to develop tools for car seat designers. The two-dimensional model, consisting of interconnected masses, springs and dampers is non-linear due to geometric effects but, under the excitations considered, the model behaviour is linear. In this approach to modelling, the full system is initially broken down into subsystems, and experiments are conducted with subsystems to determine approximate values for the stiffness and damping parameters. This approach is necessary because of the highly non-linear behaviour of foam where stiffness changes with compression level, and because the simplified model contains more structure than is necessary to model the relatively simple measured frequency response behaviour, thus requiring a good initial starting point from which to vary parameters. A detailed study of the effects of changing model parameters on the natural frequencies, the mode shapes and resonance locations in frequency response functions is given, highlighting the influence of particular model parameters on features in the seat–mannequin systems vibration response. Reasonable qualitative as well as good quantitative agreement between experimental and simulation frequency response estimates is obtained. In particular, the two-dimensional motions at the peaks in the frequency response, a combination of up and down and rotational behaviour is predicted well by the model. Currently research is underway to develop a similar model with non-linear springs, surface friction effects and viscoelastic elements, that predicts the static settling point, a necessary step to aid in the subsystem modelling stage in this dynamic modelling approach.

Estimation of the dynamical properties of polyurethane foam through use of Prony series

A system identification procedure is formulated for estimation of parameters associated with a dynamic model of a single-degree-of-freedom foam-mass system. The foam is modelled as a linear viscoelastic material, whose constitutive law is expressed by an exponential hereditary relaxation kernel. The identification procedure is based on modelling the free response of the system as a Prony series (sum of exponentials terms) and fitting this Prony series to the data. This estimated response model is then utilized to estimate the parameters in the system model based on an explicit solution of the model. The procedure is analyzed for its reliability under different sources of error and uncertainties, such as the presence of weak components and experimental noise, and some modifications are evaluated to improve the robustness of the procedure. Finally, the procedure is applied to experimental data to obtain relevant stiffness, viscous and viscoelastic parameters associated with the system. Variations in values of these parameters as a function of static compression are also investigated.

A nonlinear fractional derivative model for large uni-axial deformation behavior of polyurethane foam

In uni-axial quasi-static tests, flexible polyurethane foam undergoing large compressive loading and unloading deformation exhibits highly nonlinear and viscoelastic behavior. In particular, the response in the first cycle is observed to be significantly different from the response in subsequent cycles. In addition, the stresses in the loading paths are higher than those in unloading paths. This quasi-static response is modeled as an additive sum of a nonlinear elastic and a linear viscoelastic response. The nonlinear elastic behavior is modeled as a polynomial function of compressive strain, while the viscoelastic behavior is modeled as a parallel combination of five-parameter fractional derivative models. The focus of this paper is to develop a multi-element fractional derivative model that can capture the multi-cycle behavior. A parameter estimation procedure based on separating the contributions of the viscoelastic elements and the nonlinear elastic component is developed. This approach is applied to both simulation and experimental data. The combination of a nonlinear elastic component and a two-element fractional derivative model is found to predict the observed responses reasonably well. The results for two distinct foams from tests with different compression rates are compared and discussed.

Measurement of the attributes of complex tonal components commonly found in product sound

All rotating components in machinery produce sounds that contain tonal components, and the presence of these tones can significantly affect the quality of the product sound. Tone corrections for metrics based on weighted, average sound pressure level have been used since the late 1960s to assess annoyance due to aircraft noise and to rate climate control machinery. Much research has also been focused on measuring the strength of well separated tones in noise. Metrics such as the Prominence Ratio, the Tone-to-Noise Ratio, and variants, as well as more complex models such as the Joint Nordic Method, Aures Tonalness, and Virtual Pitch, produce values that often correlate well with subjects judgments on the level of the tonal features that they perceive when listening to the sound. However, when sounds are more complex, these metrics do not always work well. Models for the tonalness of two types of tonal sounds are considered here: narrow-band random noise and tones with random frequency fluctuations. The influence of bandwidth and roll-off rate on perceived tonalness are explored for the narrow-band sounds, and the effect of the range and the rate of change of the frequency variation on perceived tonalness is explored for frequency modulated sounds. It was found that roll-off rate affected the perception of tonalness, and that when frequency variations could be tracked or were very small, metrics derived from averaged spectra produced inaccurate predictions of tonalness. Based on the results of these two investigations modifications to tonal metrics are proposed.

Tonal strength of harmonic complex tones in machinery noise

The sounds of machines often contain families of harmonically related sine waves that are referred to as harmonic complex tones. The perceived tonal strength of these types of sounds can adversely influence people’s impressions of the sound. While a complex tone is comprised of many sine waves, usually only one prominent pitch sensation is produced. It can be argued that harmonic complex tones are perceived as a single entities, not as a sum of individual tones. A series of psychoacoustics tests was conducted to evaluate tonal prominence of harmonic complex tones. Two sounds of equal loudness were played to subjects. One was a harmonic complex tone in noise and the other was a single tone in noise. Subjects were asked to equalize the perceived tonalness of the two sounds by adjusting the tone to noise ratio of the single tone in noise. Tonalness and Terhardt’s pitch perception models were applied to the pairs of sounds used in each test. The feasibility of replacing harmonic complex tones with a tonally e...