A general-purpose approach to computer-aided dynamic analysis of a flexible helicopter

A general purpose mathematical formulation is described for dynamic analysis of a helicopter consisting of flexible and/or rigid bodies that undergo large translations and rotations. Rigid body and elastic sets of generalized coordinates are used. The rigid body coordinates define the location and the orientation of a body coordinate frame (global frame) with respect to an inertial frame. The elastic coordinates are introduced using a finite element approach in order to model flexible components. The compatibility conditions between two adjacent elements in a flexible body are imposed using a Boolean matrix, whereas the compatibility conditions between two adjacent bodies are imposed using the Lagrange multiplier approach. Since the form of the constraint equations depends upon the type of kinematic joint and involves only the generalized coordinates of the two participating elements, then a library of constraint elements can be developed to impose the kinematic constraint in an automated fashion. For the body constraints, the Lagrange multipliers yield the reaction forces and torques of the bodies at the joints. The virtual work approach is used to derive the equations of motion, which are a system of differential and algebraic equations that are highly nonlinear. The formulation presented is general and is compared with hard-wired formulations commonly used in helicopter analysis

Sturm\u27s theorems for generalized derivative and generalized Sturm--Liouville problem

The present paper defines the Sturm separation and Sturm comparison theorems for the generalized derivative. The generalized derivative is defined with respect to weight function and another function. Further, we define the generalized Sturm--Liouville problem (GSLP) and analyze the properties of GSLP such that the eigenvalues of GSLP are real, and for distinct eigenvalues, the associated eigenfunctions are orthogonal. Moreover, using variational approach, we show that GSLP has an infinite eigenvalues

Sturm\u27s theorems for generalized derivative and generalized Sturm--Liouville problem

The present paper defines the Sturm separation and Sturm comparison theorems for the generalized derivative. The generalized derivative is defined with respect to weight function and another function. Further, we define the generalized Sturm--Liouville problem (GSLP) and analyze the properties of GSLP such that the eigenvalues of GSLP are real, and for distinct eigenvalues, the associated eigenfunctions are orthogonal. Moreover, using variational approach, we show that GSLP has an infinite eigenvalues

Quantitative evaluation of essential oils for the identification of chemical constituents by gas chromatography/mass spectrometry

Essential oils are greatly strenuous aromatic materials having various constituents. They are used in the preparation of various precious substances like making perfumes, medicines, cleaning agent, and aromatic treatment etc. The purpose of the present investigation was to identify the major and minor chemical constituent in eighteen essential oils viz., amyris, basil, black pepper, camphor, catnip, chamomile, cinnamon, citronella, dill, frankincense, galbanum, jasmine, juniper, lavender, peppermint, rosemary, tagetes and thyme with the help of gas chromatography /mass spectrometry (GC/MS). In eighteen essential oils the identified compounds studied by GC-MS contain various types of high and low molecular weights of chemical ingredients. Therefore, GC/MS efficiently and speedily screened all the volatile elements present in the essential oils for the quantitative use of these identified chemical constituents for various reasons

Fractional Hamilton formalism within Caputo's derivative

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange formulations lead to the same set of equations.Comment: 8 page

A Scaling Method and its Applications to Problems in Fractional Dimensional Space

A scaling method is proposed to find (1) the volume and the surface area of a generalized hypersphere in a fractional dimensional space and (2) the solid angle at a point for the same space. It is demonstrated that the total dimension of the fractional space can be obtained by summing the dimension of the fractional line element along each axis. The regularization condition is defined for functions depending on more than one variable. This condition is applied (1) to find a closed form expression for the fractional Gaussian integral, (2) to establish a relationship between a fractional dimensional space and a fractional integral, (3) to develop the Bochner theorem, and (4) to obtain an expression for the fractional integral of the Mittag–Leffler function. Some possible extensions of this work are also discussed