Anil K. Bajaj

Amplitude Modulated Dynamics of a Resonantly Excited Autoparametric Two Degree-of-Freedom System

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The “locked pendulum” mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.

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.

Amplitude modulated and chaotic dynamics in resonant motion of strings

Abstract The weakly non-linear, resonant motions of a stretched string to a harmonic, planar excitation are investigated. One mode truncation of the in-plane and out-of-plane equation results in a two-degree-to-freedom system which is examined by using the method of averaging and the method of integral manifolds. It is shown that the averaged equations, for small enough damping, possess non-planar constant solutions which become unstable by a Hopf bifurcation. The resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string. There are two limit cycle branches for the averaged equations, one arising due to the Hopf bifurcation and the other due to a global saddle-node bifurcation. The two branches merge as the damping is further reduced. Also, with variation in detuning, the isolated branch exhibits period-doubling bifurcations, chaotic attractors and merging of attractors, giving rise to Rossler as well as Lorentz-type solutions. The truncated string equations are also directly integrated. It is shown that there are non-planar periodic responses that bifurcate into amplitude modulated motions on a 2-torus. Changes in detuning result in torus-doubling, merging of tori and then destruction of the torus leading to chaotic amplitude modulations. The results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response.

Non-Linear Vibrations and Chaos in Harmonically Excited Rectangular Plates with One-to-One Internal Resonance

Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1 : 1 resonance in the (1, 2) and (3, 1) plate modes.

Nonlinear aerodynamic damping of sharp-edged flexible beams oscillating at low Keulegan-Carpenter numbers

Slender sharp-edged flexible beams such as flapping wings of micro air vehicles (MAVs), piezoelectric fans and insect wings typically oscillate at moderate-to-high values of non-dimensional frequency parameter β with amplitudes as large as their widths resulting in Keulegan–Carpenter (KC) numbers of order one. Their oscillations give rise to aerodynamic damping forces which vary nonlinearly with the oscillation amplitude and frequency; in contrast, at infinitesimal KC numbers the fluid damping coefficient is independent of the oscillation amplitude. In this article, we present experimental results to demonstrate the phenomenon of nonlinear aerodynamic damping in slender sharp-edged beams oscillating in surrounding fluid with amplitudes comparable to their widths. Furthermore, we develop a general theory to predict the amplitude and frequency dependence of aerodynamic damping of these beams by coupling the structural motions to an inviscid incompressible fluid. The fluid–structure interaction model developed here accounts for separation of flow and vortex shedding at sharp edges of the beam, and studies vortex-shedding-induced aerodynamic damping in slender sharp-edged beams for different values of the KC number and the frequency parameter β. The predictions of the theoretical model agree well with the experimental results obtained after performing experiments with piezoelectric fans under vacuum and ambient conditions.

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.

A Microresonator Design Based on Nonlinear 1 : 2 Internal Resonance in Flexural Structural Modes

A unique T-beam microresonator designed to operate on the principle of nonlinear modal interactions due to 1 : 2 internal resonance is introduced. Specifically, the T-structure is designed to have two flexural modes with natural frequencies in a 1 : 2 ratio, and the higher frequency mode autoparametrically excites the lower frequency mode through inertial quadratic nonlinearities. A Lagrangian formulation is used to model the electrostatically actuated T-beam resonator, and it includes inertial quadratic nonlinearities, cubic nonlinearities due to midplane stretching and curvature of the beam, electrostatic potential, and effects of thermal prestress. A nonlinear two-mode reduced-order model is derived using linear structural modes in desired internal resonance. The model is used to estimate static pull-in bias voltages and dynamic responses using asymptotic averaging. Nonlinear frequency responses are developed for the case of resonant actuation of a higher frequency mode. It is shown that the lower frequency flexural mode is excited for actuation levels above a certain threshold and generates response component at half the frequency of resonant actuation. The effects of damping, thermal prestress, and mass and geometric perturbations from nominal design are thoroughly discussed. Finally, experimental results for a macroscale T-beam structure are briefly described and qualitatively confirm the basic analytical predictions. The T-beam resonator shows a high sensitivity to mass perturbations and, thus, holds great potential as a radio frequency filter-mixer and mass sensor.

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.