Fixed point and a Cantilever beam problem in a partial b-metric space

We determine the common fixed point of two maps satisfying Hardy-Roger type contraction in a complete partial b-metric space without exploiting any variant of continuity or commutativity, which is indispensable in analogous results. Towards the end, we give examples and an application to solve a Cantilever beam problem employed in the distortion of an elastic beam in equilibrium to substantiate the utility of these improvements.

Enhanced Energy Harvesting of Flexural Waves in Elastic Beams by Bending Mode of Graded Resonators

We show efficient elastic energy transfer and wave confinement through a graded array of resonators attached to an elastic beam. Experiments demonstrate that flexural resonators of increasing lengths allow to reduce wave scattering and to achieve the rainbow effect with local wavefield amplifications. We show that the definition of a monotonically decreasing distribution of the natural frequencies of the resonators along the wave propagation direction, is the preferable choice to increase the energy efficiency of the system. The proposed configuration is suitable for micro-fabrication, envisaging practical applications for micro-scale vibration energy harvesting.

Analysis of Double Elastic Steel Wind Driven Magneto-Electric Vibration Energy Harvesting System

This research proposes an energy harvesting system that collects the downward airflow from a helicopter or a multi-axis unmanned rotary-wing aircraft and uses this wind force to drive the magnet to rotate, generating repulsive force, which causes the double elastic steel system to slap each other and vibrate periodically in order to generate more electricity than the traditional energy harvesting system. The design concept of the vibration mechanism in this study is to allow the elastic steel carrying the magnet to slap another elastic steel carrying the piezoelectric patch to form a set of double elastic steel vibration energy harvesting (DES VEH) systems. The theoretical DES VEH mechanism of this research is composed of a pair of cantilever beams, with magnets attached to the free end of one beam, and PZT attached to the other beam. This study analyzes the single beam system first. The MOMS method is applied to analyze the frequency response of this nonlinear system theoretically, then combines the piezoelectric patch and the magneto-electric coupling device with this nonlinear elastic beam to analyze the benefits of the system’s converted electrical energy. In the theoretical study of the DES VEH system, the slapping force between the two elastic beams was considered as a concentrated load on each of the beams. Furthermore, both SES and DES VEH systems are studied and correlated. Finally, the experimental data and theoretical results are compared to verify the feasibility and correctness of the theory. It is proven that this DES VEH system can not only obtain the electric energy from the traditional SES VEH system but also obtain the extra electric energy of the steel vibration subjected to the slapping force, which generates optimal power to the greatest extent.

Main Roof Breakage and Vibration Induced Coal Burst Occurring in Longwall Roadways

Rock burst is one major threat to mining safety and economy. Rock burst occurring in the longwall mining roadway accounts for 85% of the total amount of burst events. This paper investigates the causality mechanism of rock burst in longwall roadways by establishing a finite elastic beam model in the working face based on the elastic foundation theory. The breakage process of the main roof and related dynamic effects are analysed. The result shows that the movement of the main roof shows free vibration under certain damping resistance. It is also found that the roof dominant vibration frequency increases with the increase in the thickness and elastic modulus of the roof. During roof vibration, the vertical stress applied on the coal mass is unloaded. The destressing of the roof-coal interface causes the coal mass in the roadway rib to slip into the roadway under the horizontal ground stress, resulting in rock burst. The possibility of rock burst increases with increase in the strength and thickness of the roof and horizontal ground stress within the coal mass. This mechanism explains the occurrence of rock burst in the mining roadway; it provides the fundamental theory for the prevention and controlling technologies of longwall roadway rock burst.

Waves in Flexural Beams with Nonlinear Adhesive Interaction

The paper studies the initial boundary value problem related to the dynamic evolution of an elastic beam interacting with a substrate through an elastic-breakable forcing term. This discontinuous interaction is aimed to model the phenomenon of attachment-detachment of the beam occurring in adhesion phenomena. We prove existence of solutions in energy space and exhibit various counterexamples to uniqueness. Furthermore we characterize some relevant features of the solutions, ruling the main effects of the nonlinearity due to the elastic-breakable term on the dynamical evolution, by proving the linearization property according to Gérard (J Funct Anal 141(1):60–98, 1996) and an asymptotic result pertaining the long time behavior.

Hydroelastic waves propagating along a frozen channel with non-uniform thickness of ice

Hydroelastic waves propagating along a channel covered with ice of non-uniform thickness are considered. The channel has a rectangular cross section. The fluid in the channel is inviscid and incompressible. The ice is modeled as a thin elastic plate. The ice thickness changes linearly. The problem is reduced to the problem of the wave profile across the channel, which is solved using the normal modes of an elastic beam with non-uniform thickness. It is shown that with the decrease in the change in the ice thickness, the modes approach the normal modes of an elastic beam with a constant thickness. The behavior of the dispersion relations of the hydroelastic waves depending on the parameter describing the change in the ice thickness is studied.

Energetics of collapsible channel flow with a nonlinear fluid-beam model

We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.

Application of some special operators on the analysis of a new generalized fractional Navier problem in the context of q-calculus

The key objective of this study is determining several existence criteria for the sequential generalized fractional models of an elastic beam, fourth-order Navier equation in the context of quantum calculus (q-calculus). The required way to accomplish the desired goal is that we first explore an integral equation of fractional order w.r.t. q-RL-integrals. Then, for the existence of solutions, we utilize some fixed point and endpoint conditions with the aid of some new special operators belonging to operator subclasses, orbital α-admissible and α-ψ-contractive operators and multivalued operators involving approximate endpoint criteria, which are constructed by using aforementioned integral equation. Furthermore, we design two examples to numerically analyze our results.

The Reduced-Degree-of-Freedom Model for Seismic Analysis of Predominantly Plan-Symmetric Reinforced Concrete Wall–Frame Building

A reduced-degree-of-freedom (RDOF) model for seismic analysis of predominantly plan-symmetric reinforced concrete (RC) wall–frame buildings is introduced. The RDOF model of the wall–frame building consists of elastic beam–column elements with concentrated plasticity used for simulating cantilever walls and predominantly plan-symmetric RC frame buildings that are represented by the improved fish-bone (IFB) model. In this paper, the capability of the RDOF model is demonstrated for two frame buildings and two wall–frame buildings. The RDOF models were defined directly from the building information model. This is an advantage of RDOF models with respect to single-degree-of-freedom (SDOF) models, while the computational robustness of the RDOF models also makes them attractive for the seismic analysis of building stock. The imposed cyclic displacement analyses conducted for the investigated buildings proved that the condensation of the degrees of freedom for RDOF models was appropriate. Consequently, only minor differences were observed for maximum storey drift IDA curves, maximum storey acceleration IDA curves, and seismic fragility functions for different limit states. However, development is needed to make RDOF models appropriate for preliminary seismic performance assessment of plan-irregular buildings.