Active Vibration Suppression of One-dimensional Nonlinear Structures Using Optimal Dynamic Inversion

A flexible robot arm can be modeled as an Euler-Bernoulli beam which are infinite degrees of freedom (DOF) system. Proper control is needed to track the desired motion of a robotic arm. The infinite number of DOF of beams are reduced to finite number for controller implementation, which brings in error (due to their distributed nature). Therefore, to represent reality better distributed parameter systems (DPS) should be controlled using the systems partial differential equation (PDE) directly. In this paper, we propose to use a recently developed optimal dynamic inversion technique to design a controller to suppress nonlinear vibration of a beam. The method used in this paper determines control forces directly from the PDE model of the system. The formulation has better practical significance, because it leads to a closed form solution of the controller (hence avoids computational issues).

Spectral Finite Element Formulation for Nanorods via Nonlocal Continuum Mechanics

In this article, the Eringen's nonlocal elasticity theory has been incorporated into classical/local Bernoulli-Euler rod model to capture unique properties of the nanorods under the umbrella of continuum mechanics theory. The spectral finite element (SFE) formulation of nanorods is performed. SFE formulation is carried out and the exact shape functions (frequency dependent) and dynamic stiffness matrix are obtained as function of nonlocal scale parameter. It has been found that the small scale affects the exact shape functions and the elements of the dynamic stiffness matrix. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave dispersion properties of carbon nanotubes.

Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements

This paper presents a spectral finite element formulation for uniform and tapered rotating CNT embedded polymer composite beams. The exact solution to the governing differential equation of a rotating Euler-Bernoulli beam with maximum centrifugal force is used as an interpolating function for the spectral element formulation. Free vibration and wave propagation analysis is carried out using the formulated spectral element. The present study shows the substantial effect of volume fraction and L/D ratio of CNTs in a beam on the natural frequency, impulse response and wave propagation characteristics of the rotating beam. It is found that the CNTs embedded in the matrix can make the rotating beam non-dispersive in nature at higher rotation speeds. Embedded CNTs can significantly alter the dynamics of polymer-nanocomposite beams. The results are also compared with those obtained for carbon fiber reinforced laminated composite rotating beams. It is observed that CNT reinforced rotating beams are superior in performance compared to laminated composite rotating beams. © 2012 Elsevier Ltd. All rights reserved.

Biomechanics of Substrate Boring by Fig Wasps: Role of Zinc in Insect Cuticle

There are many biomechanical challenges that a female insect must meet to successfully oviposit and ensure her evolutionary success. These begin with selection of a suitable substrate through which the ovipositor must penetrate without itself buckling or fracturing. The second phase corresponds to steering and manipulating the ovipositor to deliver eggs at desired locations. Finally, the insect must retract her ovipositor fast to avoid possible predation and repeat this process multiple times during her lifetime. From a materials perspective, insect oviposition is a fascinating problem and poses many questions. Specifically, are there diverse mechanisms that insects use to drill through hard substrates without itself buckling or fracturing? What are the structure-property relationships in the ovipositor material? These are some of the questions we address with a model system consisting of a parasitoid fig wasp - fig substrate system. To characterize the structure of ovipositors, we use scanning electron microscopy with a detector to quantify the presence of transition elements. Our results show that parasitoid ovipositors have teeth like structures on their tips and contain high amounts of zinc as compared to remote regions. Sensillae are present along the ovipositor to aid detection of chemical species and mechanical deformations. To quantify the material properties of parasitoid ovipositors, we use an atomic force microscope and show that tip regions have higher modulus as compared to remote regions. Finally, we use videography to show that ovipositors buckle during oviposition and estimate the forces needed to cause substrate boring based on Euler buckling analysis. Such methods may be useful for the design of functionally graded surgical tools.

Rotating beams isospectral to axially loaded nonrotating uniform beams

In this paper, the stiffness and mass per unit length distributions of a rotating beam, which is isospectral to a given uniform axially loaded nonrotating beam, are determined analytically. The Barcilon-Gottlieb transformation is extended so that it transforms the governing equation of a rotating beam into the governing equation of a uniform, axially loaded nonrotating beam. Analysis is limited to a certain class of Euler-Bernoulli cantilever beams, where the product between the stiffness and the cube of mass per unit length is a constant. The derived mass and stiffness distributions of the rotating beam are used in a finite element analysis to confirm the frequency equivalence of the given and derived beams. Examples of physically realizable beams that have a rectangular cross section are shown as a practical application of the analysis.

On the Chern number of I-admissible filtrations of ideals

Let I be an m-primary ideal of a Noetherian local ring (R, m) of positive dimension. The coefficient e(1)(I) of the Hilbert polynomial of an I-admissible filtration I is called the Chern number of I. A formula for the Chern number has been derived involving the Euler characteristic of subcomplexes of a Koszul complex. Specific formulas for the Chern number have been given in local rings of dimension at most two. These have been used to provide new and unified proofs of several results about e(1)(I).

Turbulence in the two-dimensional Fourier-truncated Gross-Pitaevskii equation

We undertake a systematic, direct numerical simulation of the twodimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. Firstly, there are transients that depend on the initial conditions. In the second regime, powerlaw scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other.

Generalized model predictive static programming and its application to 3D impact angle constrained guidance of air-to-surface missiles

A new `generalized model predictive static programming (G-MPSP)' technique is presented in this paper in the continuous time framework for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. A key feature of the technique is backward propagation of a small-dimensional weight matrix dynamics, using which the control history gets updated. This feature, as well as the fact that it leads to a static optimization problem, are the reasons for its high computational efficiency. It has been shown that under Euler integration, it is equivalent to the existing model predictive static programming technique, which operates on a discrete-time approximation of the problem. Performance of the proposed technique is demonstrated by solving a challenging three-dimensional impact angle constrained missile guidance problem. The problem demands that the missile must meet constraints on both azimuth and elevation angles in addition to achieving near zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Both stationary and maneuvering ground targets are considered in the simulation studies. Effectiveness of the proposed guidance has been verified by considering first order autopilot lag as well as various target maneuvers.

Fractal dimension method for delamination detection in composite beams

A new delaminated composite beam element is formulated for Timoshenko as well as Euler-Bernoulli beam models. Shape functions are derived from Timoshenko functions; this provides a unified formulation for slender to moderately deep beam analyses. The element is simple and easy to implement, results are on par with those from free mode delamination models. Katz fractal dimension method is applied on the mode shapes obtained from finite element models, to detect the delamination in the beam. The effect of finite element size on fractal dimension method of delamination detection is quantified.

Shock-boundary layer interaction and transonic flutter

The transonic flutter dip of an aeroelastic system is primarily caused by compressibility of the flowing fluid. Viscous effects are not dominant in the pre-transonic dip region. In fact, an Euler solver can predict this flutter boundary with considerable accuracy. However with an increase in Mach number the shock moves towards the trailing edge causing shock induced separation. This shock-boundary layer interaction changes the flutter boundary in the transonic and post-transonic dip region significantly. We discuss the effect of viscosity in changing the flutter boundary in the post-transonic dip region using a RANS solver coupled to a two-degree of freedom model of the structural dynamics of a wing.

Transonic flutter, shocks, and parameter effects

There is a drop in the flutter boundary of an aeroelastic system placed in a transonic flow due to compressibility effects and is known as the transonic dip. Viscous effects can shift the lo-cation of the shock and depending on the shock strength the boundary layer may separate leading to changes in the flutter speed. An unsteady Euler flow solver coupled with the structural dynamic equations is used to understand the effect of shock on the transonic dip. The effect of various system parameters such as mass ratio, location of the center of mass, position of the elastic axis, ratio of uncoupled natural frequencies in heave and pitch are also studied. Steady turbulent flow results are presented to demonstrate the effect of viscosity on the location and strength of the shock.

Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference Delta U across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L x +/-infinity. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a ``vortex gas'' comprising N point vortices of the same strength (gamma = L Delta U/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32 000 vortices, with ensemble averages evaluated over up to 10(3) realizations and integration over 10(4)L/Delta U. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/Delta U. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via ``violent'' and ``slow'' relaxations P.-H. Chavanis, Physica A 391, 3657 (2012)], over flow time scales of order 10(2) and 10(4)L/Delta U, respectively (for N = 400). The final state involves a single structure that stochastically samples the domain, possibly constituting a ``relative equilibrium.'' The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter lambda close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166 +/- 0.0002 times Delta U. Regime II, extensively studied in the turbulent shear flow literature as a self-similar ``equilibrium'' state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term ``explosive'' as it lasts less than one L/Delta U. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers J. Sommeria, C. Staquet, and R. Robert, J. Fluid Mech. 233, 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

On Additive Combinatorics of Permutations of Z(n)

Let Z(n) denote the ring of integers modulo n. A permutation of Z(n) is a sequence of n distinct elements of Z(n). Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of Z(n), namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s (n) and t (n) respectively. The case when n is even is trivial in both the cases, with s (n) = 1 and t (n) = n!. For n odd, we prove (n phi(n))/2(k) <= s(n) <= n!.2(-)(n-1)/2/((n-1)/2)! and 2 (n-1)/2 . (n-1/2)! <= t (n) <= 2(k) . (n-1)!/phi(n), where k is the number of distinct prime divisors of n and phi is the Euler's totient function.

Generalized Model Predictive Static Programming and Angle-Constrained Guidance of Air-to-Ground Missiles

A new generalized model predictive static programming technique is presented for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. Two key features for its high computational efficiency include one-time backward integration of a small-dimensional weighting matrix dynamics, followed bya static optimization formulation that requires only a static Lagrange multiplier to update the control history. It turns out that under Euler integration and rectangular approximation of finite integrals it is equivalent to the existing model predictive static programming technique. In addition to the benchmark double integrator problem, usefulness of the proposed technique is demonstrated by solving a three-dimensional angle-constrained guidance problem for an air-to-ground missile, which demands that the missile must meet constraints on both azimuth and elevation angles at the impact point in addition to achieving near-zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Simulation studies include maneuvering ground targets along with a first-order autopilot lag. Comparison studies with classical augmented proportional navigation guidance and modern general explicit guidance lead to the conclusion that the proposed guidance is superior to both and has a larger capture region as well.

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection-diffusion-reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin-Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin-Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.