Analytical study of the natural bending oscillations of a concave beam with parabolic change in thickness

The synthesis of factorization and symmetry methods produced a general analytical solution to the fourth-order differential equation with variable coefficients. The form and structure of the variable coefficients correspond, in this case, to the problem of the oscillations of a concave beam of variable thickness. The solution to this equation makes it possible to study in detail the oscillations of such and similar, for example convex, beams at the different fixation of their ends' sections. A practical confirmation has been obtained that the beam whose thickness changes in line with the concave parabola law H=a2x2+1, where a is the concave factor, demonstrates an increase in the natural frequencies of its free oscillations with an increase in its rigidity. As an example, the object's maximum deflection dependence on the beam rigidity factor has been established. The nature of this dependence confirmed the obvious conclusion that the deflections had decreased while the rigidity had increased. The evidence from the calculation results can be a testament to the correctness of the reported procedure of problem-solving. The considered problem and the analytical solution to it could serve as a practical guide to the optimal design of beam structures. In this case, it is very important to take into consideration the place and nature of the distribution of cyclical extreme operating stresses. The resulting ratios to solve the problem make it possible to simulate the required normal stresses in both the fixation and central zones when the rigidity parameter is changed. Designers could predict such a parabolic profile of the beam, which would ensure the required reduction of maximum stresses in the place of fixing the beam. The considered example of solving the problem of the natural oscillations of the beam with rigid fixation of the ends illustrates the effectiveness of the factoring and symmetry methods used. The developed solution algorithm could be extended to study the natural bending oscillations of the beam at other fixing techniques, not excluding a variant of a completely free beam

The integrated MHD thermal performance with slip conditions on metachronal propulsion: the engineering material for biomedical applications

PurposeParticular attention is given to the viscous damping force parameter, stiffness parameter, rigidity parameter, and Brinkman number and plotted their graph for thermal distribution, momentum profile and concentration profile.Design/methodology/approachIn the field of engineering, biologically inspired propulsion systems are getting the utmost importance. Keeping in view their developmental progress, the present study was made. The theoretical analysis explores the effect of heat and mass transfer on non-Newtonian Sisko fluid with slip effects and transverse magnetic field in symmetric compliant channel. Using low Reynolds number, so that the authors neglect inertial forces and for keeping the pressure constant during the flow, channel height is used largely as compared to the ratio of wavelength. The governing equations of fluid flow problem are solved using the perturbation analysis.FindingsResults are considered for thickening, thinning and viscous nature of fluid models. It is found that the velocity distribution profile is boosted for increasing values of the Sisko fluid parameter and porous effect, while thermal profile is reducing for Brinkman number (viscous dissipation effects) for all cases. Moreover, shear-thicken and shear-thinning behavior of non-Newtonian Sisko fluid is also explained through the graphs.Originality/valueHear-thicken and shear-thinning behavior of non-Newtonian Sisko fluid is also explained through the graphs.

Non-Rigid Rotating Motion Effect on Creep Behavior for Infinite Cylinders under Thermomechanical Loading

The effect of non-rigid motion on creep analysis for cylinders have been investigated. inner surface of the cylinder is exposed to a uniform heat flux and for cooling the outer surface of the cylinder an air steam is applied to it. The cylinder is subjected to a body force. That produced by rotating cylinder about its own axis. It is assumed the cylinder is not rigid. Therefore, using equations of equilibrium, stress–strain and kinematic equation, governed equation which includes redial displacement and creep strains, is obtained. The material creep constitutive model is defined by the Bailey-Norton time-dependent creep law. From a solution consisting of analytical solution and iteration method has been used to obtain history of stresses and deformations during creep evolution of rotating cylinders. Therefore, Prandtle-Ruess equation substituted in equilibrium equation in the rate form, the radial displacement rate is obtained for plane strain condition. To investigate the effect of non-rigidity, the non-rigidity parameter (ζ) is defined and its effect on the creep behavior of the structure is investigated. It was revealed that Considering the structure as rigid does neglect the amount of stresses and radial displacement. History of stresses and displacement during 20 years are studied and it is observed that the changes have not accrued in recent years. The effect of velocity is another parameter that is investigated its effect on structural behavior during the time. it was revealed that, velocity has significant effect on structural behavior which cause trend of variation behavior change from linear to polynomial curve.

Mesophase behaviour of a cyanobiphenyl molecule in polar aprotic solvent: Rigidity effect

In this paper, a theoretical study has been carried out on a liquid crystal compound named p-n-propyl cyanobiphenyl (3CB). The different modes of interaction energy values in a polar aprotic solvent (ethyl acetate) for small amount of translation and rotation are calculated. The corresponding probabilities have been calculated at both room temperature (300 K) and transition temperature (303.3 K). The rigidity parameter for stacking and in-plane interactions has been estimated and then the stability of molecule according to probability and rigidity at definite translation and rotation has been concluded. The change in the characteristics and stability of the compound at transition temperature has been observed. The dependence of mesophase behaviour with change in the certain configurations and orientation of the molecules have been discussed. These observed results provide an insight about the process of mesophase structure and its formation. The present compound may guide in establishing the other molecular models with transition temperature nearer to room temperature.

SURFACE WAVE SCATTERING BY AN ELASTIC PLATE SUBMERGED IN WATER WITH UNEVEN BOTTOM

Wave interaction with a vertical elastic plate in presence of undulating bottom topography is considered, assuming linear theory and utilizing simple perturbation analysis. First order correction to the velocity potential corresponding to the problem of scattering by a vertical elastic plate submerged in a fluid with a uniform bottom is obtained by invoking the Green’s integral theorem in a suitable manner. With sinusoidal undulation at the bottom, the first-order transmission coefficient (T1) vanishes identically. Behaviour of the first order reflection coefficient (R1) depending on the plate length, ripple number, ripple amplitude and flexural rigidity of the plate is depicted graphically. Also, the resonant nature of the first order reflection is observed at a particular value of the ratio of surface wavelength to that of the bottom undulations. The net reflection coefficient due to the joint effect of the plate and the bottom undulation is also presented for different flexural rigidity of the plate. When the rigidity parameter is made sufficiently large, the results for R1 reduce to the known results for a surface piercing rigid plate in water with bottom undulation.

Couple Stress Impacton S-Wavesin an Anisotropic Pre-Stressed Medium under Gravity and Magnetic Field

The gravity fieldand magnetic field impact with the inclusion of couple stress s r onshear waves in a transversely isotropic elastic solid medium is studied. Thedispersion equations for the particular problem have been obtained. Severalparticular cases are also examined. The numerical calculations are carriedout for specific rigidity parameter A, density parameter C, anisotropic factor N , initial stress parameter P , gravity parameter G, magnetic fieldparameter H , couple stress parameter  as well as wave number κ and theoutcomes are visually represented by graph.

Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

New models are investigated in this paper, that describe equilibrium states of plates with Signorini type nonpenetration conditions. In these models, it is assumed that under appropriate loading, plates have special deformations with already known configurations of edges. For this case, we deal with new non-penetration conditions that allow us to describe more precisely the possibility of contact interaction of plate edges. Using the method of fictitious domains, it is proved that an original contact problem for a plate can be obtained by passing to the limit when a rigidity parameter tends to infinity from a family of auxiliary problems formulated in a wider domain. The mentioned family of problems model an equilibrium state of plates with a crack and depend on the positive rigidity parameter. For these problems, to prevent a mutual penetration of the opposite crack faces boundary conditions of inequality type are imposed on the inner boundary corresponding to the crack. For the problem, describing a plate with a crack that intersects the external boundary at zero angle (a case of a boundary having one cusp), the unique solvability is proved.

Investigation of Free Vibrations of Three-Layered Circular Shell Supported by Annular Ribs of Rigidity

The construction of a mathematical model and the development of an algorithm of free vibrations investigation in the three-layered circular shell with a light-weight aggregate supported by annular rigidity ribs are considered in paper. The hypotheses of Kirchoff-Lyav are accepted for external bearing layers of shell and for aggregate there is accepted the linear law of tangential displacements change by thickness. The boundary conditions of a shell region closed between the ribs are established. Using the boundary transition, conditions along the lines of the ribs, taking into account and without deformations of displacement in the ribs, but without taking into account the torsional rigidity in the ribs are determined. The equation of motion of supported three-layered shell is obtained. The frequencies of free vibrations were investigated and values of parameter of the first frequency of free vibrations for a shell, supported by one and three rigidity ribs, were calculated. There are given values depending on the physical and mechanical properties of materials and geometric dimensions of the shell, the curvature parameter, and the rigidity parameter of an aggregate.

Effects Wall Properties on Peristaltic Transport of Rabinowitsch Fluid through an Inclined Non-Uniform Slippery Tube

The effects of slip and wall properties on the peristaltic mechanism of Rabinowitsch fluid flowing through a non-uniform inclined tube is investigated under the assumptions of long wavelength and small Reynold’s number. The governing equations of motion, momentum, and energy are rendered dimensionless by using suitable similarity transformations. The effects of the velocity slip parameter , thermal slip parameter, wall rigidity parameter, wall stiffness parameter and the viscous damping force parameter on velocity, temperature and streamlines are analyzed for shear thinning, viscous, and shear thickening fluid models. From the results, it is found that an increase in the value of velocity and thermal slip parameter enhances the velocity and temperature profiles for viscous and shear thinning fluids. Also, the volume of trapped bolus improves for an increase in the value of rigidity and stiffness parameter for all the three liquids, whereas it decreases for an increase in the value of the viscous damping force parameter.