Manoj Pandey

Nonlinear normal modes (NNMs) are defined as curved invariant structures (manifolds) in the phase-space of a nonlinear dynamical system, which confine the trajectories initiated on them to themselves and hence provide a reduced order subspace for the system to evolve. In this work, NNMs, obtained through graph style parameterization method, are used to study synchronization dynamics of two non-isochronous, frequency detuned limit cycle oscillators (LCOs) under displacement-based/reactive and velocity-based/dissipative coupling, also considering the effect of delayed interactions. The NNMs contain closed curves corresponding to either the in-phase or out-of-phase synchronized oscillations. The motion initiated on these NNMs is found to evolve strictly on them, unless it encounters a fixed point or closed orbit with an out-of-plane unstable direction, in which case it may even transition to the other manifold. In all other cases, the motion settles onto a stable fixed point or orbit on either manifold if existent, or on to a quasi-periodic orbit lying between the two manifolds in the absence of one. The NNMs are used to uncouple the governing equations of the system, directly giving the in-phase and out-of-phase synchronization frequencies and stability. Parametric study using averaged equations as well as the direct numerical integration-based evolution of the original system on the NNMs is used to identify various bifurcations. The properties of NNMs are tested for qualitatively different motions hence determined. NNMs for the delay coupled oscillators are obtained, by first expanding the delay terms in a Taylor series. It is found that even a small delay (< 1 % of LCOs time period) brings a significant change in the behavior of the system. The quantitative predictions made by NNM in this case are found to be good for finite delay times (≈ 10 % of LCOs time period), while qualitative agreement was achieved for even larger delays. The results obtained here agree with prior work in the literature, at the same time extending them to hitherto unexplored parameter range and provide a new perspective on the same from the point of view of NNMs.

Contact transformation is an operator transformation method in time-independent perturbation theory which is used successfully in molecular spectroscopy to obtain an effective Hamiltonian. FIoquet theory is used to transform the periodic time-dependent Hamiltonian, to a time-independent Floquet Hamiltonian. In this article contact transformation method has been used to get the analytical representation of Floquet Hamiltonian for quadrupolar nuclei with spin I = 1 in the presence of an RF field and first order quadrupolar interaction in magic angle spinning NMR experiments. The eigenvalues of contact transformed Hamiltonian as well as Floquet Hamiltonian have been calculated and a comparison is made between the eigenvalues obtained using the two Hamiltonians. © Indian Academy of Sciences.

In this work, a plasticity based composite interface model is proposed for failure analysis of unreinforced masonry. The hyperbolic composite interface model consists of a single surface yield criterion, which is a direct extension of Mohr-Coulomb criteria with cut in tension region and a cap in compression region. The inelastic behaviour includes potential crack, slip, and crushing of the masonry joints. A micro mechanical based approach is adopted for failure modelling of the masonry. The model is developed by using a fully implicit backward-Euler integration strategy. It is combined with a local/global Newton solver, based on a consistent tangent operator compatible with an adaptive sub stepping strategy. The model is implemented in standard finite element software (ABAQUS) by using user defined subroutine and verification is conducted in all its basic modes. Finally, the model is validated by comparing with experimental results available in the literature.

The effect of combined gradation in area and density on the dynamic response of foams is studied in this work. The cross-sectional area and density are graded such that the product of these quantities results in a tri-linear variation along the foam's length, and hence, the foam type is designated as Type-TL. A theoretical model based on the R-PP-L idealization is developed to study the response of the Type-TL foam struck by a rigid projectile. The Type-TL foam offers force distribution over a considerably larger area than a homogeneous foam, which may benefit structural protection.

Spin dynamics under magic angle spinning has been studied using different theoretical approaches and also by extensive numerical simulation programs. In this article we present a general theoretical approach that leads to analytic forms for effective Hamiltonians for an N-spin dipolar and quadrupolar coupled system under magic angle spinning (MAS) conditions, using a combination of Floquet theory and van Vleck (contact) transformation. The analytic forms presented are shown to be useful for the study of MAS spin dynamics in solids with the help of a number of simulations in two, three, and four coupled, spin-1/2 systems as well as spins in which quadrupolar interactions are also present. © 2010 American Institute of Physics.

The application of solid-state NMR methodology for bio-molecular structure determination requires the measurement of constraints in the form of 13C-13C and 13C-15N distances, torsion angles and, in some cases, correlation of the anisotropic interactions. Since the availability of structurally important constraints in the solid state is limited due to lack of sufficient spectral resolution, the accuracy of the measured constraints become vital in studies relating the three-dimensional structure of proteins to its biological functions. Consequently, the theoretical methods employed to quantify the experimental data become important. To accentuate this aspect, we re-examine analytical two-spin models currently employed in the estimation of 13C-13C distances based on the rotational resonance (R2) phenomenon. Although the error bars for the estimated distances tend to be in the range 0.5-1.0 Å, R2 experiments are routinely employed in a variety of systems ranging from simple peptides to more complex amyloidogenic proteins. In this article we address this aspect by highlighting the systematic errors introduced by analytical models employing phenomenological damping terms to describe multi-spin effects. Specifically, the spin dynamics in R2 experiments is described using Floquet theory employing two different operator formalisms. The systematic errors introduced by the phenomenological damping terms and their limitations are elucidated in two analytical models and analysed by comparing the results with rigorous numerical simulations. © 2010 Taylor & Francis.

Design problems using high fidelity numerical methods such as Finite Element Analysis (FEA) can be computationally intensive, especially if they require multiple runs for different loading conditions or varying parameters. Hence reduced-order models (ROM) which can reproduce the simulation results with high accuracy, while working at a very low computational budget are desirable. Subspace projection-based ROMs are widely used for the analysis of linear systems, using linear eigenmodes as projection basis and can be extended to nonlinear systems using empirical eigenmodes such as Proper Orthogonal modes (POM). However, problems involving moving contact are difficult to handle for such procedure due to moving boundary conditions of the underlying PDE. Here we use the approach proposed by [1] towards reduced-order dynamic analysis of Hyperelastic wheel rolling, incorporating the geometric and material nonlinearities in addition to contact with a view to extend it to tire rolling analysis. The simulations are performed using commercial Finite Element software Abaqus and the ROM based results are verified using Matlab and show a very good match for displacement and contact forces with a model that is orders of magnitude smaller than the full order system.

Computational modeling of failure in quasi-brittle materials at various length scales is important. In this work we present a rate independent cohesive zone model for modeling failure in quasi-brittle materials. The proposed model can simulate cracking, slipping, and crushing of planes through a traction-separation law. A single surface hyperbolic failure criterion, which naturally comes as a direct extension of Coulomb friction criterion with cut-off in tension and cap-off in compression, has been developed. A Euler backward integration scheme together with a global-local Newton solver compatible with a substepping strategy has been used in numerical computations. The proposed model is then used for modeling of shear wall panels. The numerical results obtained are validated by comparing them with experimental results available in literatures.

In this paper, dynamic characteristics of a beam with breathing crack is considered. Breathing crack is modeled as a bilinear oscillator. The stiffness of the cracked beam is estimated by using influence coefficients based on castigliano's theorem and strain energy release rate (SERR). The equation of motion of breathing cracked beam is formulated using finite element method using Hamilton's principle. The equation of motion of breathing cracked beam is converted into Mathieu - Hill type equation to obtain the regions of dynamic instability of beam using Harmonic balance method and it is further solved for Eigen frequency of the cracked beam. The increase in breathing crack depth increases the instability region and it is found that the effect of the crack location near to the fixed end is more for the cantilever beam, and this also increases the instability region. It is found that increase in dynamic instability index increases the instability regions of the cracked structure. In addition to that, the effect of static and dynamic loads are also investigated and discussed. The study has been conducted for the first two instability boundaries of the cracked structure only. It is hence seen that assuming the crack to remain open underestimates the stability boundaries of the system. Permitting the crack to open and close (breath) yields a stability boundaries in between the open and uncracked beam.

In this paper we investigate the static and dynamic behavior of an electrically actuated multilayered circular microplate. The dynamic analogue of Von Karman equation is used to model the governing equation of microplate which accounts for different sources of nonlinearities. We employ a multi-mode reduced order model (ROM) of the governing equations based on Galerkin discretization method to solve for governing and compatibility equations. The eigenvalue problem associated with the governing equation is solved numerically for first few natural frequencies and modeshapes. The microplate is then loaded with direct voltage (DC) load until instability phenomena known as static pull-in is observed. Then, the dynamic forced vibration is investigated by actuating the device with DC voltage superimposed by a small AC voltage. We investigate the transition from hardening to softening behavior near primary resonance for a combination of static and dynamic loads. A MEMS device is fabricated from a polyimide layer coated with metals from top and bottom to validate the theoretical simulation results with the experimental results. The simulated theoretical results are in good agreement with the experiments