A note on the inverse mode shape problem for bars, beams, and plates

Motivated by the idea of designing a structure for a desired mode shape, intended towards applications such as resonant sensors, actuators and vibration confinement, we present the inverse mode shape problem for bars, beams and plates in this work. The objective is to determine the cross-sectional profile of these structures, given a mode shape, boundary condition and the mass. The contribution of this article is twofold: (i) A numerical method to solve this problem when a valid mode shape is provided in the finite element framework for both linear and nonlinear versions of the problem. (ii) An analytical result to prove the uniqueness and existence of the solution in the case of bars. This article also highlights a very important question of the validity of a mode shape for any structure of given boundary conditions.

A note on new indices for the equatorial Indian Ocean oscillation

It is now well known that there is a strong association of the extremes of the Indian summer monsoon rainfall (ISMR) with the El Nio and southern oscillation (ENSO) and the Equatorial Indian Ocean Oscillation (EQUINOO), later being an east-west oscillation in convection anomaly over the equatorial Indian Ocean. So far, the index used for EQUINOO is EQWIN, which is based on the surface zonal wind over the central equatorial Indian Ocean. Since the most important attribute of EQUINOO is the oscillation in convection/precipitation, we believe that the indices based on convection or precipitation would be more appropriate. Continuous and reliable data on outgoing longwave radiation (OLR), and satellite derived precipitation are now available from 1979 onwards. Hence, in this paper, we introduce new indices for EQUINOO, based on the difference in the anomaly of OLR/precipitation between eastern and western parts of the equatorial Indian Ocean. We show that the strong association of extremes of the Indian summer monsoon with ENSO and EQUINOO is also seen when the new indices are used to represent EQUINOO.

Note: Effect of fluid phase compositions on the formation of substitutionally ordered solid phases from a binary mixture of oppositely charged colloidal suspensions

A binary mixture of oppositely charged colloidal particles can self-assemble into either a substitutionally ordered or substitutionally disordered crystalline phase depending on the nature and strength of interactions among the particles. An earlier study had mapped out favorable inter-particle interactions for the formation of substitutionally ordered crystalline phases from a fluid phase using Monte Carlo molecular simulations along with the Gibbs-Duhem integration technique. In this paper, those studies are extended to determine the effect of fluid phase composition on formation of substitutionally ordered solid phases.

Note: Design and development of an integrated three-dimensional scanner for atomic force microscopy

A compact scanning head for the Atomic Force Microscope (AFM) greatly enhances the portability of AFM and facilitates easy integration with other tools. This paper reports the design and development of a three-dimensional (3D) scanner integrated into an AFM micro-probe. The scanner is realized by means of a novel design for the AFM probe along with a magnetic actuation system. The integrated scanner, the actuation system, and their associated mechanical mounts are fabricated and evaluated. The experimentally calibrated actuation ranges are shown to be over 1 mu m along all the three axes. (c) 2013 AIP Publishing LLC.

Note on high aspect-ratio SU-8 micromechanical structures using mask-less direct laser writing

We report high aspect-ratio micromechanical structures made of SU-8 polymer, which is a negative photoresist. Mask-less direct writing with 405 nm laser is used to pattern spin-cast SU-8 films of thickness of more than 600 um. As compared with X-ray lithography, which helps pattern material to give aspect ratios of 1:50 or higher, laser writing is a less expensive and more accessible alternative. In this work, aspect ratios up to 1:30 were obtained on narrow pillars and cantilever structures. Deep vertical patterning was achieved in multiple exposures of the surface with varying dosages given at periodic intervals of sufficient duration. It was found that a time lag between successive exposures at the same location helps the material recover from the transient changes that occur during exposure to the laser. This gives vertical sidewalls to the resulting structures. The time-lags and dosages were determined by conducting several trials. The micromechanical structures obtained with laser writing are compared with those obtained with traditional UV lithography as well as e-beam lithography. Laser writing gives not only high aspect ratios but also narrow gaps whereas e-beam can only give narrow gaps over very small depths. Unlike traditional UV lithography, laser writing does not need a mask. Furthermore, there is no adjustment for varying the dosage in traditional UV lithography. A drawback of this method compared to UV lithography is that the writing time increases. Some test structures as well as a compliant microgripper are fabricated.

Note on modelling directionality in piezoresistivity

When computing the change in electrical resistivity of a piezoresistive cubic material embedded in a deforming structure, the piezoresistive and the stress tensors should be in the same coordinate system. While the stress tensor is usually calculated in a coordinate system aligned with the principal axes of a regular structure, the specified piezoresistive coefficients may not be in that coordinate system. For instance, piezoresistive coefficients are usually given in an orthogonal cartesian coordinate system aligned with the <100> crystallographic directions and designers sometimes deliberately orient a crystallographic direction other than <100> along the principal directions of the structure to increase the gauge factor. In such structures, it is advantageous to calculate the piezoresistivity tensor in the coordinate system along which the stress tensors are known rather than the other way around. This is because the transformation of stress will have to be done at every point in the structure but piezoresistivity tensor needs to be transformed only once. Here, using tensor transformation relations, we show how to calculate the piezoresistive tensor along any arbitrary Cartesian coordinate system from the piezoresistive coefficients for the <100> coordinate system. Some of the software packages that simulate the piezoresistive effect do not have interfaces for calculation of the entire piezoresistive tensor for arbitrary directions. This warrants additional work for the user because not considering the complete piezoresisitive tensor can lead to large errors. This is illustrated with an example where the error is as high as 33%. Additionally, for elastic analysis, we used hybrid finite element formulation that estimates stresses more accurately than displacement-based formulation. Therefore, as shown in an example where the change in resistance can be calculated analytically, the percentage error of our piezoresistive program is an order of magnitude lower relative to displacement-based finite element method.

A note on D1-D5 entropy and geometric quantization

We quantize the space of 2-charge fuzzballs in IIB supergravity on K3. The resulting entropy precisely matches the D1-D5 black hole entropy, including a specific numerical coefficient. A partial match (ie., a smaller coefficient) was found by Rychkov a decade ago using the Lunin-Mathur subclass of solutions - we use a simple observation to generalize his approach to the full moduli space of K3 fuzzballs, filling a small gap in the literature.

Note: A resonating reflector-based optical system for motion measurement in micro-cantilever arrays

A robust, compact optical measurement unit for motion measurement in micro-cantilever arrays enables development of portable micro-cantilever sensors. This paper reports on an optical beam deflection-based system to measure the deflection of micro-cantilevers in an array that employs a single laser source, a single detector, and a resonating reflector to scan the measurement laser across the array. A strategy is also proposed to extract the deflection of individual cantilevers from the acquired data. The proposed system and measurement strategy are experimentally evaluated and demonstrated to measure motion of multiple cantilevers in an array. (C) 2015 AIP Publishing LLC.

A note on tetrablock contractions

A commuting triple of operators (A, B, P) on a Hilbert space H is called a tetrablock contraction if the closure of the set E = {(a(11),a(22),detA) : A = GRAPHICS] with parallel to A parallel to <1} is a spectral set. In this paper, we construct a functional model and produce a set of complete unitary invariants for a pure tetrablock contraction. In this construction, the fundamental operators, which are the unique solutions of the operator equations A - B* P = DPX1DP and B - A* P = DPX2DP where X-1, X-2 is an element of B(D-P) play a pivotal role. As a result of the functional model, we show that every pure tetrablock isometry (A, B, P) on an abstract Hilbert space H is unitarily equivalent to the tetrablock contraction (MG1*+G2z, MG2*+G1z, M-z) on H-DP*(2). (D), where G(1) and G(2) are the fundamental operators of (A*, B*, P*). We prove a Beurling Lax Halmos type theorem for a triple of operators (MF1*+F2z, MF2*+F1z, M-z), where epsilon is a Hilbert space and F-1, F-2 is an element of B(epsilon). We also deal with a natural example of tetrablock contraction on a functions space to find out its fundamental operators.

A NOTE ON SMALL GAPS BETWEEN NONZERO FOURIER COEFFICIENTS OF CUSP FORMS

It is shown that there are infinitely many primitive cusp forms f of weight 2 with the property that for all X large enough, every interval (X, X + cX(1/4)), where c > 0 depends only on the form, contains an integer n such that the n-th Fourier coefficient of f is nonzero.

A Note On The Numerical Simulations Of Flow Past A Wavy Square-Section Cylinder

The flow past a square-section cylinder with a geometric disturbance is investigated by numerical simulations. The extra terms, due to the introduction of mapping transformation simulating the effect of disturbance into the transformed Navier-Stokes equations, are correctly derived, and the incorrect ones in the previous literature are pointed out and analyzed. Furthermore, the relationship between the vorticity, especially on the cylinder surface, and the disturbance is derived and explained theoretically. The computations are performed at two Reynolds numbers of 100 and 180 and three amplitudes of waviness of 0.006, 0.025 and 0.167 with another aim to explore the effects of different Reynolds numbers and disturbance on the vortex dynamics in the wake and forces on the body. Numerical results have shown that, at the mild waviness of 0.025, the Karman vortex shedding is suppressed completely for Re = 100, while the forced vortex dislocation is appeared in the near wake at the Reynolds number of 180. The drag reduction is up to 21.6% at Re = 100 and 25.7% at Re = 180 for the high waviness of 0.167 compared with the non-wavy cylinder. The lift and the Strouhal number varied with different Reynolds numbers and the wave steepness are also obtained.