Anil K. Bajaj

Purdue University, West Lafayette, IN, United States

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Publications (115)128.36 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
  • Astitva Tripathi, Anil K. Bajaj
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    ABSTRACT: Nonlinear phenomenon such as internal resonance have significant potential applications in Micro Electro Mechanical Systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoretical systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance as well as are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer’s prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal frequencies in certain ratios) computationally. In this work, plate structures which are candidates for internal resonances are obtained using a Finite Element Method (FEM) formulation implemented in Matlab to iteratively modify a base structure to get its first two natural frequencies close to the desired ratio (1:2 or 1:3). Once a structure with desired topology is achieved, the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the Hyperelastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes, and the nonlinear response can be obtained by application of perturbation methods such as averaging on the two-mode model.
    ASME 2013 International Mechanical Engineering Congress and Exposition; 11/2013
  • Astitva Tripathi, Anil K. Bajaj
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    ABSTRACT: With advances in technology, hyperelastic materials are seeing increased use in varied applications ranging from microfluidic pumps, artificial muscles to deformable robots. They have also been proposed as materials of choice in the construction of components undergoing dynamic excitation such as the wings of a micro-unmanned aerial vehicle or the body of a serpentine robot. Since the strain energy potentials of various hyperelastic materials are more complex than quadratic, exploration of their nonlinear dynamic response lends itself to some interesting consequences. In this work, a structure made of a Mooney-Rivlin hyperelastic material and undergoing planar vibrations is considered. Since the stresses developed in a Mooney-Rivlin material are at least a quadratic function of strain, a possibility of 1:2 internal resonance is explored. A Finite Element Method (FEM) formulation implemented in Matlab is used to iteratively modify a base structure to get its first two natural frequencies close to the ratio 1:2. Once a topology of the structure is achieved, the linear modes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the hyperelastic structure can be used to develop a nonlinear two-mode model of the structure. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. It is shown that the strain energy potential for the Mooney-Rivlin material makes it possible for internal resonance to occur in such structures.
    ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 08/2013
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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. · 2.04 Impact Factor
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    ABSTRACT: We present simulations of the dynamic response of radio frequency micro-electro-mechanical-systems (RF-MEMS) switches undergoing creep deformation. The model includes a microscale-informed Coble creep formulation incorporated in a beam model of an electrostatically actuated RF-MEMS switch, and it is solved using a Ritz-Galerkin based modal expansion. The resulting effects on the long-term device behavior as well as the implications of uncertainty in the device geometry and material parameters are studied. We find that the addition of creep to the beam model results in an undesired degradation of the device performance, as evidenced by decreases in the closing and release voltages.
    ASME 2012 International Mechanical Engineering Congress and Exposition; 11/2012
  • Yousof Azizi, Patricia Davies, Anil K. Bajaj
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    ABSTRACT: Flexible polyethylene foam, which is used in many engineering applications, exhibits nonlinear and viscoelastic behavior. To date, several models have been proposed to characterize the complex behavior of foams from the computationally intensive microstructural models to continuum models that capture the macroscale behavior of the foam materials. A nonlinear viscoelastic model, which is an extension of previously developed models, is proposed and its ability to capture foam response in uniaxial compression is investigated. It is assumed in the model that total stress is decomposed into the sum of a nonlinear elastic component, which is modeled by a higher order polynomial, and a nonlinear hereditary type viscoelastic component. System identification procedures are developed to estimate the model parameters using uniaxial compression data from experiments conducted at different rates. The performance of this model is compared to that of other nonlinear viscoelastic models.
    ASME 2012 International Mechanical Engineering Congress and Exposition; 11/2012
  • Yousof Azizi, Patricia Davies, Anil K. Bajaj
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    ABSTRACT: Vehicle occupants are exposed to low frequency vibration that can cause fatigue, lower back pain, spine injuries. Therefore, understanding the behavior of a seat-occupant system is important in order to minimize these undesirable vibrations. The properties of seating foam affect the response of the occupant, so there is a need for good models of seat-occupant systems through which the effects of foam properties on the dynamic response can be directly evaluated. In order to understand the role of flexible polyurethane foam in characterizing the complex seat-occupant system behavior better, the response of a single-degree-of-freedom foam-mass system, which is the simplest model representing a seat-occupant system, is studied. The incremental harmonic balance method is used to determine the steady-state behavior of the foam-mass system subjected to sinusoidal base excitation. This method is used to reduce the time required to generate the steady-state response at the driving frequency and at harmonics of the driving frequency from that required when using direct time-integration of the governing equations to determine the steady state response. Using this method, the effects of different viscoelastic models, riding masses, base excitation levels and damping coefficients on the response are investigated.
    ASME 2012 International Mechanical Engineering Congress and Exposition; 11/2012
  • Astitva Tripathi, Anil K. Bajaj
    [Show abstract] [Hide abstract]
    ABSTRACT: Nonlinear phenomena such as internal resonances have significant potential applications in Micro Electro Mechanical Systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoretical systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance as well as are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer’s prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal interactions) computationally. In this work, a preliminary method for computer based synthesis of structures consisting of beams for desired internal resonance is presented. The linear structural design is accompalished by a Finite Element Method (FEM) formulation implemented in Matlab to start with a base structure and iteratively modify it to obtain a structure with the desired properties. Possible design criteria are having the first two natural frequencies of the structure in some required ratio (such as 1:2 or 1:3). Once a topology of the structure is achieved that meets the desired criterion (i.e., the program converges to a definite structure), the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the nonlinear elastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. Many candidate structures are synthesized that meet the desired modal frequency criterion and their nonlinear responses are compared.
    ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 08/2012
  • Rajat Goyal, Anil K. Bajaj
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    ABSTRACT: This work is focused on Uncertainty Quantification (UQ) in a nonlinear MEMS T-beam structure exhibiting 1:2 autoparametric internal resonance. The study is presented by elaborating on the sources of uncertainty in fabrication and operation of the device and their quantification. Nonlinear response of the system is formulated by using a two-mode model constructed with lowest two linear modes of the structure in conjunction with the nonlinear Lagrangian representing the dynamics of the beam structure as well as the excitation mechanism. Thus UQ for the nonlinear resonant MEMS is carried out in two steps. First, propagation and quantification in linear analysis outputs namely mode shapes, natural frequencies and tuning, and secondly, UQ on nonlinear response obtained from averaged equations determined by the averaged Lagrangian. In this paper, applications of UQ techniques on linear part are presented and effects of various parameteric uncertainties on model output are brought out. Sensitivity analysis is performed to reduce the number of parameters and a comparison of sensitivity analysis with sampling is done to establish the accuracy of the method. Response surface analysis is performed using generalized polynomial chaos (gPC) to generate an analytical expression for multi-dimensional uncertainty. A detailed description of the gPC collocation method is also presented. A comparison of response surface method with direct sampling is done to illustrate the efficiency and accuracy of gPC collocation technique for up to 5 uncertain parameters.
    ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 08/2012
  • Fengxia Wang, Anil 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
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    ABSTRACT: Nonlinear viscoelastic behavior is a characteristic of many engineering materials and also biological tissue, 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 polyurethane foam. Two cases are considered: (1) the beam and foam are glued so that they are always in contact and the foam can undergo both stretching and compression, and (2) the beam and foam are not glued so that the contact region changes with beam motion, and the foam only reacts in compression. Static as well as dynamic forces act on the beam and the Galerkin method is used to derive modal amplitude equations for the beam on polyurethane foundation. In the second case, determination of the loss of contact points is integrated into the solution procedure through a constraint relation. The static responses for both cases are examined as a function of the foam nonlinearity and loading conditions, and three and five mode solutions are compared. The steady state response of the system subject to static and harmonic loads is studied by using numerical integration techniques. Numerical challenges and the accuracy of this approach are discussed. Frequency responses are generated for a range of foam nonlinearities and loading conditions.
    ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 01/2011
  • Source
    Michael G. Snow, Anil K. Bajaj
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    ABSTRACT: As MEMS technology develops it is becoming better understood that MEMS designers must account for the large uncertainties characteristic of the relevant manufacturing processes. Uncertainty quantification tasks the designer with evaluating many different possible outcomes from the manufacturing process which creates a demand for models that are accurate and comprehensive, yet fast to evaluate. This work presents a comprehensive reduced-order model of electrostatically actuated switches incorporating a range of effects that are typically included only in FE modeling codes. Specifically, the model accounts for variable electrode geometry, stretching of centerline or large displacement effects, fringing field, squeeze film and rarefied gas damping, and allows for elastic contact with the dielectric substrate. Individual compact models for each of these effects are taken from literature and included in the model for the system. The dielectric substrate is modeled as an elastic foundation. The resulting partial differential equation for the switch modeled as a beam is discritized via a Galerkin method into ordinary differential equations for modal amplitudes. The Galerkin method uses the linear un-damped mode shapes of the beam to approximate the solution. Both cantilever and fixed-fixed type switches are analyzed. Static equilibrium solutions as a function of the applied voltage are developed along with their stability. Static pull-in voltages, first time of switch closure, and voltage for lift-off are studied with the model. To capture the contact dynamics, the contact condition is evaluated with the substrate divided into a large number of elements and the contact force is projected on to the beam basis functions. In the case of cantilever geometry and slow voltage variations, three stable regimes of contact configuration and hysteresis between them are demonstrated.
    PRISM: NNSA Center for Prediction of Reliability, Integrity and Survivability of Microsystems. 01/2011;
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    ABSTRACT: Vehicle occupants are sensitive to low frequency vibrations, and these can affect ride-quality and dynamic comfort. Static comfort, a function of the support provided by the seat, is also important. The transmission of vibration to seated occupants and the support provided by the seat can be controlled by appropriately designing the seats. Optimization of seat design requires accurate models of seat-occupant systems can be used to predict both static settling points and the low frequency dynamic behavior of the occupant around those points. A key element in the seat, which is a challenge to model, is the flexible polyurethane foam in the seat cushion. It is a nonlinear, viscoelastic material exhibiting multiple time-scale behavior. In this work, the static and the low-frequency dynamic response of the occupant is examined through a planar multi-body seat-occupant model, which also incorporates a model of flexible polyurethane foam developed from relatively slow cyclic compression tests. This model also incorporates profiles of the seat and the occupant, and includes relatively simple friction models at the various occupant-seat interfaces. The settling point, the natural frequencies, the deflection shapes of the occupant at particular frequencies, and the dynamic force distribution between the seat and the occupant are examined. The effects of seat foam properties on the responses as well as those of including a flexible seat-back frame are also investigated.
    ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 01/2011
  • Source
    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;
  • 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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    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 09/2009; · 2.29 Impact Factor
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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
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    ABSTRACT: Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.
    Journal of Computational and Nonlinear Dynamics - J COMPUT NONLINEAR DYN. 01/2009; 4(2).
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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.
    Journal of Applied Mechanics 01/2009; 51. · 1.04 Impact Factor

Publication Stats

586 Citations
128.36 Total Impact Points

Institutions

  • 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