A new aspect of generalized integral operator and an estimation in a generalized function theory

In this paper we investigate certain integral operator involving Jacobi–Dunkl functions in a class of generalized functions. We utilize convolution products, approximating identities, and several axioms to allocate the desired spaces of generalized functions. The existing theory of the Jacobi–Dunkl integral operator (Ben Salem and Ahmed Salem in Ramanujan J. 12(3):359–378, 2006) is extended and applied to a new addressed set of Boehmians. Various embeddings and characteristics of the extended Jacobi–Dunkl operator are discussed. An inversion formula and certain convergence with respect to δ and Δ convergences are also introduced.

Dynamics of thermoelastic half-plane by action of periodic loads and heat flows at its boundary

The problem of the dynamics of a thermoelastic half-space under periodic surface forces and heat flows is solved using the model of coupled thermoelasticity. The Green’s tensor for one boundary value problem is constructed utilizing Fourier transformation. Analytical solutions for arbitrary surface forces and heat flow using the theory of generalized functions are constructed. To solve this boundary value problem, generalized function theory, tensor and differential algebra, the operator method, and integral transformations were used. The solutions obtained make it possible to investigate the thermal stress–strain state of an array with natural and artificial thermal sources and mass power forces acting at its surface.

ACTIVE CONTROL OF AN FGM BEAM UNDER FOLLOWER FORCE WITH PIEZOELECTRIC SENSORS/ACTUATORS

In this paper, an active controller is used to suppress the flutter vibration of a beam. The beam is made of Functionally Graded Material (FGM) and subjected to a follower force and arbitrary lumped mass. The properties of the FGM layer are functionally graded in the thickness direction according to the volume fraction power law distribution. The piezoelectric layers, which are attached to both sides of the beam, are used as sensors and actuators. The beam is fixed from one end and elastically restrained by a spring at the other end. To investigate the effect of the controller on the vibration response of the beam, parameters such as the follower force, spring stiffness, mass ratio and attachment location are included in the analysis based on the generalized function theory and Lagrange–Rayleigh–Ritz technique. The vibration responses of the system are presented in the simulation results, where excellent agreement of the controller scheme is observed.

Bending-Torsional Flutter of a Composite Rectangular Cantilever Wing Subjected to Engine Thrust

The bending-torsional flutter characteristics of a cantilever composite wing subjected to engine thrust are presented. The engine thrust modeled as a follower force and the wing modeled as a two degree of freedom beam. In order to consider the spanwise and chordwise location and the properties of the engine location, the generalized function theory is used. Unsteady Theodorsen aerodynamic theory in time domain, is used. The general laminate composite theory is used for modeling the effects of the ply angles on the flutter boundary. The Ritz method is subsequently applied to convert the partial differential equations into a set of ordinary differential equations. In order to precisely consider the location of the engine Dirac delta function is used. Moreover, the numerical results are compared with the published results and excellent agreement is observed. Numerical results highlighting the effect of engine thrust on the wing instability, is presented.

Nonlinear Aeroelastic Analysis of Bending-Torsion Wings Subjected to a Transverse Follower Force

This paper deals with the nonlinear aeroelastic behaviors of bending-torsion wings subjected to a transverse follower force. The nonlinear structural wing formulation is based on von Karman large deformation theory. In order to accurately consider the spanwise location of the follower force, the generalized function theory is used. Also, Peter’s finite-state unsteady aerodynamic model is considered. The governing equations are obtained using Hamilton’s principle. Furthermore, the Galerkin method is applied to convert the partial differential equations into a set of nonlinear ordinary differential equations, which will be solved through the numerical integration scheme. Wing dynamic behaviors are investigated through frequency spectra and the bifurcation diagrams of Poincaré maps. In addition, the postcritical region, which includes all periodic, quasiperiodic, and chaotic pockets, is indeed found to exist. Furthermore, the results indicate noticeable effects of the follower force magnitude and location as well as the air stream velocity on critical and postcritical behaviors of a wing.

BUCKLING AND FLUTTER OF A COLUMN ENHANCED BY PIEZOELECTRIC LAYERS AND LUMPED MASS UNDER A FOLLOWER FORCE

In this paper, enhancement of the buckling and flutter capacities of a column by the attachment of an arbitrary lumped mass and a pair of piezoelectric layers on the column sides are studied. The column is subjected to a follower force, with one end fixed and the other end elastically restrained by a spring. To investigate the effects of the piezoelectric layers on the column instability capacity and frequency, parameters such as the follower force, spring stiffness, and mass ratio and attachment location are included in the analysis based on the generalized function theory and Lagrange–Rayleigh–Ritz technique. The numerical results reveal that if each of the design parameters of the piezoelectric layers, follower force, spring stiffness, or external mass exceeds a certain critical value, the column will undergo buckling and flutter instabilities. Besides, comparisons are made with available results in the literature and excellent agreement is observed.