A. K. Bajaj

Purdue University, West Lafayette, IN, United States

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Publications (86)111.91 Total impact

  • [Show abstract] [Hide abstract]
    ABSTRACT: Nonlinear viscoelastic behavior is a characteristic of many engineering materials including flexible polyurethane foam, yet it is difficult to develop dynamic models of systems that include these materials and are able to predict system behavior over a wide range of excitations. This research is focused on a specific example system in the form of a pinned-pinned beam interacting with a viscoelastic foundation. Two cases are considered: (1) the beam and the foundation are glued so that they are always in contact and the foundation can undergo both tension and compression, and (2) the beam is not glued to the foundation and the foundation reacts only in compression so that the contact region changes with beam motion. Static as well as dynamic transverse and axial forces act on the beam, and the Galerkin method is used to derive modal amplitude equations for the beam-foundation system. In the second case of the beam on tensionless foundation, loss of contact between the beam and the foundation can arise and determination of the loss-of-contact points is integrated into the solution procedure through a constraint equation. The static responses for both cases are examined as a function of the foundation nonlinearity and loading conditions. The steady-state response of the system subject to static and harmonic loads is studied by using numerical direct time-integration. Numerical challenges and the accuracy of this approach are discussed, and predictions of solutions by the three-mode and five-mode approximate models are compared to establish convergence of solutions. Frequency responses are studied for a range of foam nonlinearities and loading conditions.
    Journal of vibration and acoustics 02/2014; 136(3). · 1.27 Impact Factor
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    ABSTRACT: Many engineering materials and foundations such as soils demonstrate nonlinear and viscoelastic behaviour. Yet, it is challenging to develop static and dynamic models of systems that include these materials and are able to predict the behaviour over a wide range of loading conditions. This research is focused on a specific example: a pinned–pinned beam interacting with polyurethane foam foundation. Two cases, when the foundation can react in tension and compression as well as only in compression, are considered. The model developed here is capable of predicting the response to static as well as dynamic forces, whether concentrated or distributed. Galerkin’s method is used to derive modal amplitude equations. In the tensionless foundation case, the contact region changes with beam motion and the estimation of the co-ordinates of the lift-off points is embedded into the solution procedure. An efficient solution technique is proposed that is capable of handling cases where there are multiple contact and non-contact regions. Depending on the loading profiles a high number of modes may need to be included in the solution and to speed up computation time, a convolution method is used to evaluate the integral terms in the model. The adaptability of the solution scheme to complicated loading patterns is demonstrated via examples. The solution approach proposed is applicable to dynamic loadings as well and in these cases the automated treatment of complicated response patterns makes the convolution approach particularly attractive. The influence of various parameters on the static response is discussed.
    International Journal of Solids and Structures 07/2013; 50(s 14–15):2328–2339. · 1.87 Impact Factor
  • F Wang, A K Bajaj
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    ABSTRACT: This work discusses model reduction for non-linear structural systems under harmonic excitations. Model reduction can be achieved by different techniques, one of the recent techniques being the non-linear normal modes (NNMs) of the system. The dimension reduction achievable depends on the possibilities of internal resonances. A master–slave separation of degrees of freedom is used, and a non-linear relation between the slave and master coordinates is constructed based on the method of multiple timescales. More specifically, three cases involving external resonance of a mode without any internal resonance, and subharmonic as well as superharmonic resonances for systems with 1:2 internal resonances are considered. Reduced-order models based on the ‘Conservative NNMs’ as well as ‘Damped NNMs’ are constructed. The steady-state periodic responses of the reduced models determined by the method of multiple timescales are compared to exact solutions of the system models computed by the bifurcation analysis and parameter continuation software AUTO. The analysis is specifically applied to a spring-mass-pendulum system with external excitation, and to an elastic three-beam-tip-mass structure, which is first reduced to a high-fidelity non-linear discretized model through a Galerkin approximation. Both systems exhibit essential quadratic non-linearities and couplings between the various generalized coordinates. The NNMs of the two systems are used to perform model reductions when excited by harmonic excitations. It is seen that for systems with essential inertial quadratic non-linearities, the technique for model-order reduction through multiple timescales approximation based on NNMs over-predicts the softening non-linear response in each of the cases studied.
    ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 1989-1996 (vols 203-210) 10/2011; 225(10):2422-2435. · 0.63 Impact Factor
  • Fengxia Wang, Anil K. Bajaj
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    ABSTRACT: Nonlinear dynamics of elastic structures with two-mode interactions have been extensively studied in the literature. In this work, nonlinear forced response of elastic structures with essential inertial nonlinearities undergoing three-mode interactions is studied. More specifically, a three-beam structural system with attached mass is considered, and its multidegree-of-freedom discretized model for the structure undergoing planar motions is carefully studied. Linear modal characteristics of the structure with uniform beams depend on the length ratios of the three beams, the mass of the particle relative to that of the structure, and the location of the mass particle along the beams. The discretized model is studied for both external and parametric resonances for parameter combinations resulting in three-mode interactions. For the external excitation case, focus is on the system with 1:2:3 internal resonances with the external excitation frequency near the middle natural frequency. For the case of the structure with 1:2:5 internal resonances, the problem involving simultaneous principal parametric resonance of the middle mode and a combination resonance between the lowest and the highest modal frequencies is investigated. This case requires a higher-order approximation in the method of multiple time scales. For both cases, equilibrium and bifurcating solutions of the slow-flow equations are studied in detail. Many pitchfork, saddle-node, and Hopf bifurcations appear in the amplitude response of the three-beam structure, thus resulting in complex multimode responses in different parameter regions. KeywordsNonlinear elastic structures-Inertial nonlinearities-Multiple internal resonances-Substructure synthesis-Method of multiple scales-Bifurcation analysis
    Nonlinear Dynamics 01/2010; 62(1):461-484. · 3.01 Impact Factor
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    Gauri Joshi, Anil K Bajaj, Patricia Davies
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    ABSTRACT: Vehicle occupants are exposed to low frequency vibrations with possible harmful effects such as mild discomfort, lower back pain, and even injury to the spine. Occupational drivers and operators of heavy machinery are exposed to significantly longer duration and higher levels of vibration. Thus, the modeling and prediction of biodynamic response of seated occupants to such vibrations is very important. Since the properties of seating foam affect the response of the occupant, there is need for good models of seat-occupant systems through which the effects of foam properties on the dynamic response can be directly evaluated. A nonlinear planar seat-occupant model which incorporates the nonlinear viscoelastic behavior of seating foam has been developed. This model is used to study response of the occupant to harmonic excitation applied at the seat base, in terms of the frequency response in vertical and fore-and-aft directions, the deflection shapes at resonance, as well as the seat-to-head-transmissibility. In addition, to better understand the role of flexible polyurethane foam in characterizing the system behavior, the response of a single-degree-of-freedom foam-block system is also studied. The effects of different masses riding on the foam block and undergoing vertical vibrations at different acceleration levels are also investigated.
    Industrial Health 01/2010; 48(5):663-74. · 0.87 Impact Factor
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    Michael G. Snow, Anil K. Bajaj
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    ABSTRACT: This work presents an uncertainty analysis of a comprehensive model for an electrostatic MEMS switch. The goal is to elucidate the effects of parameter variations on certain performance characteristics. A sufficiently detailed model of an electrostatically actuated beam is developed. This model accounts for various physical effects, including the electrostatic fringing field, finite length of electrodes, squeeze film damping, and contact between the beam and the dielectric layer. The performance characteristics of immediate interest are the static and dynamic pull-in voltages for switch. Using Latin Hypercube and other sampling methods, the model is evaluated to find these performances characteristics when variability in the model’s geometric and physical parameters is specified. Response surfaces of these results were constructed via Multivariate Adaptive Regression Splines (MARS). Using a Direct Simulation Monte Carlo (DSMC) technique on these response surfaces gives smooth PDF’s of the outputs. The relative variation in output due to each input is used to determine the critical parameters.
    PRISM: NNSA Center for Prediction of Reliability, Integrity and Survivability of Microsystems. 01/2010;
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    ABSTRACT: 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. [2008-0107]
    Journal of Microelectromechanical Systems 07/2009; · 2.13 Impact Factor
  • Fengxia Wang, Anil K. Bajaj
    Journal of Computational and Nonlinear Dynamics - J COMPUT NONLINEAR DYN. 01/2009; 4(2).
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    ABSTRACT: 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 beta with amplitude as large as their widths resulting in Keulegan-Carpenter (KC) numbers or 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 beta. The predictions of the theoretical model agree well with the experimental results obtained after performing experiments with piezoelectric fans under vacuum and ambient conditions.
    Journal of Fluid Mechanics 01/2009; · 2.18 Impact Factor
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    ABSTRACT: Automotive brake noise has been a major concern in automotive industry and structural modification is a practical and effective way to reduce it. Here, a new method to estimate the critical friction coefficient is summarised, and then a sensitivity analysis is developed. The key is a reduced-order characteristic equation for computing the elastically coupled system's eigenvalues and their derivatives. Based on this reduced-order equation, a sensitivity analysis of the steady-sliding stability with respect to lining stiffness and lining geometry is presented for a drum brake system. This analysis eliminates need for new system models or a full complex eigenvalue analysis.
    International Journal of Vehicle Design - INT J VEH DES. 01/2009; 51.
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    ABSTRACT: Hip joint location (H-point) is an important design specification used by car seat manufacturers. Since most modern car seats are full-foam, the H-point location is primarily dependent on quasi-static behavior of foam which is a highly nonlinear and viscoelastic material. In this work, the seat–occupant dynamic model is developed in the form of a planar multi-degree of freedom system that also incorporates nonlinear viscoelastic behavior of flexible polyurethane foam. The foam force is modeled as an additive sum of nonlinear elastic and linear viscoelastic effects. The viscoelastic force is modeled as a convolution integral with a sum of exponentials kernel. The interface between the occupant and the seat at the seat-back and at the seat-bottom is modeled using Coulomb friction type laws. A system identification procedure, based on linear least squares fitting and ARMA modeling, is developed to identify the parameters from data collected in quasi-static foam experiments. The resulting nonlinear integro-differential algebraic model of the seat–occupant system is used to simulate the response of the system and determine the H-point for the occupant. A few parametric studies related to the effect on H-point of different foam types and inertia properties of the occupant are also reported. Such a model can be very useful for investigating the effects on H-point location of design changes in the seat foam. It can also be used to determine the forces at the occupant–seat interface and thus in studies related to occupant's comfort.
    International Journal of Industrial Ergonomics. 01/2008;
  • R WIDDLEJR, A. K. Bajaj, P. Davies
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    ABSTRACT: Previous attempts to model the uniaxial compression response of flexible polyurethane foam using a hyperelastic model for rubber-like materials assumed a zero Poisson’s ratio in the model development. The consequences of relaxing this assumption are explored in this investigation. First, measurements are made of the material’s Poisson’s ratio at high compression levels. The measured values range between 0.5 at 5% compression and −0.05 at 66% compression. The experimental results are combined with those of other investigators and incorporated into a nonlinear viscoelastic model for modeling uniaxial compression behavior. It is found that relaxing the zero Poisson’s ratio assumption decreases the accuracy of the model predictions, indicating that an alternative model structure may be required.
    International Journal of Engineering Science - INT J ENG SCI. 01/2008; 46(1):31-49.
  • Rahul A. Bidkar, Arvind Raman, Anil K. Bajaj
    Journal of Applied Mechanics. 01/2008; 75(4).
  • Fengxia Wang, A. K. Bajaj
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    ABSTRACT: In this work, the nonlinear normal modes (NNMs) of a conservative spring—mass—pendulum system with quadratic inertial nonlinearity are determined by applying the Adomian decomposition method. The technique is applied to the governing boundary value problem that determines the nonlinear normal modes of the two degrees-of-freedom system in configuration space. The NNMs of this system are non-similar and such that the modal curves do not pass through the origin in the configuration space. Both the low energy and the high energy cases are considered, and nonlinear normal modes near 1:2 and 1:1 internal resonances in the two linear modes of vibration of the system are developed. It is seen that the ADM polynomial approximation approach, where the model nonlinearities are approximated by polynomials in the neighborhood of center—center equilibrium, greatly reduces algebraic calculations, and improves the convergence speed and the accuracy of the results obtained. The method also captures the possible multiple nonlinear normal modes that exist near internal resonance conditions. The advantages and disadvantages in using the Adomian decomposition method to construct the nonlinear normal modes of the system are discussed.
    Journal of Vibration and Control 01/2008; 14:107-134. · 1.97 Impact Factor
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    ABSTRACT: A novel microresonator operating on the principle of nonlinear modal interactions due to autoparametric 1:2 internal resonance is introduced. Specifically, an electrostatically actuated pedal-microresonator design, utilizing internal resonance between an out-of-plane torsional mode and a flexural in-plane vibrating mode is considered. The two modes have their natural frequencies in 1:2 ratio, and the design ensures that the higher frequency flexural mode excites the lower frequency torsional mode in an autoparametric way. A Lagrangian formulation is used to develop the dynamic model of the system. The dynamics of the system is modeled by a two degrees of freedom reduced-order model that retains the essential quadratic inertial nonlinearities coupling the two modes. Retention of higher-order model for electrostatic forces allows for the study of static equilibrium positions and static pull-in phenomenon as a function of the bias voltages. Then for the case when the higher frequency flexural mode is resonantly actuated by a harmonically varying AC voltage, a comprehensive study of the response of the microresonator is presented and the effects of damping, and mass and structural perturbations from nominal design specifications are considered. Results show that for excitation levels above a threshold, the torsional mode is activated and it oscillates at half the frequency of excitation. This unique feature of the microresonator makes it an excellent candidate for a filter as well as a mixer in RF MEMS devices.
    Nonlinear Dynamics 01/2008; · 3.01 Impact Factor
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    Khalid El Rifai, George Haller, Anil K. Bajaj
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    ABSTRACT: In this paper, a study of the global dynamics of an autoparametric four degree-of-freedom (DOF) spring–mass–pendulum system with a rigid body mode is presented. Following a modal decoupling procedure, typical approximate periodic solutions are obtained for the autoparametrically coupled modes in 1:2 internal resonance. A novel technique based on forward-time solutions for finite-time Lyapunov exponent is used to establish global convergence and domains of attraction of different solutions. The results are compared to numerically constructed domains of attraction in the plane of initial position and initial velocity for the pendulum. Simulations are also provided for a few interesting cases of interest near critical values of parameters. Results also shed some light on the role played by other modes present in a multi-DOF system in shaping the overall system response.
    Nonlinear Dynamics 01/2007; 49(1):105-116. · 3.01 Impact Factor
  • Fengxia Wang, Anil K. Bajaj
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    ABSTRACT: Nonlinear normal modes for elastic structures have been studied extensively in the literature. Most studies have been limited to small nonlinear motions and to structures with geometric nonlinearities. This work investigates the nonlinear normal modes in elastic structures that contain essential inertial nonlinearities. For such structures, based on the works of Crespo da Silva and Meirovitch, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion. The motion of each substructure is represented by a finite number of substructure admissible functions in a way that the geometric compatibility conditions are automatically assured. The multi degree-of-freedom reduced-order models capture the essential dynamics of the system and also retain explicit dependence on important physical parameters such that parametric studies can be conducted. The specific structure considered is a 3-beam elastic structure with a tip mass. Internal resonance conditions between different linear modes of the structure are identified. For the case of 1:2 internal resonance between two global modes of the structure, a two-mode nonlinear model is then developed and nonlinear normal modes for the structure are studied by the method of multiple time scales as well as by a numerical shooting technique. Bifurcations in the nonlinear normal modes are shown to arise as a function of the internal mistuning that represents variations in the tip mass in the structure. The results of the two techniques are also compared.
    Nonlinear Dynamics 01/2007; 47(1):25-47. · 3.01 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: Most modern car seats are full-foam and thus the H-point location depends on quasi-static behavior of foam. In this work, planar multi-degree-of-freedom models of seat-occupant systems that incorporate nonlinear viscoelastic behavior of polyurethane foam are developed. The foam force is modeled as an additive sum of nonlinear elastic and linear viscoelastic effects and the model parameters are identified using a parameter identification technique. A possible model for seat-occupant interface is introduced. The resulting nonlinear integro-differential-algebraic model is used to determine the H-point for the system.
    12/2006: pages 1-10;
  • A. Vyas, A.K. Bajaj, A. Raman, D. Peroulis
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    ABSTRACT: A microresonator concept based on 1:2 internal resonance between the modes of the resonator is explored in this study. The response of the structure under electrostatic actuation is computed and the simulated currents at the input and output ports are presented. Results show that the output current for the T-beam is non-zero for a very small band of frequencies. Unlike linear filters, the proposed non-linear resonator provides filtering and mixing since the output signal is at half the input signal frequency. Furthermore, the proposed device has significantly higher out-of-band rejection as compared to an equivalent linear micromechanical filter. Because of these unique characteristics these microresonators hold a great potential for use in RF filtering and mixing applications
    Silicon Monolithic Integrated Circuits in RF Systems, 2006. Digest of Papers. 2006 Topical Meeting on; 02/2006
  • R. Deng, P. Davies, A.K. Bajaj
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    ABSTRACT: 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.
    Signal Processing. 01/2006;

Publication Stats

554 Citations
111.91 Total Impact Points


  • 1986–2013
    • Purdue University
      • • Ray W. Herrick Laboratories
      • • School of Mechanical Engineering
      West Lafayette, IN, United States
  • 1997
    • University of Wisconsin - Whitewater
      • Department of Mathematical and Computer Sciences
      Whitewater, WI, United States
  • 1991
    • California Institute of Technology
      Pasadena, California, United States
  • 1988
    • Pennsylvania State University
      University Park, Maryland, United States