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Concepts, mathematical modelling, experimental verification and implementation

Final Report Summary - PARM-2 (VIBRO-IMPACT MACHINES BASED ON PARAMETRIC RESONANCE:Concepts, mathematical modelling, experimental verification and implementation)

Description of the project PARM2:

The aim of this project was to enhance the performance and output of vibrating machines and decrease their ecological footprint by employing parametric resonance (PR) in their operation. Compared to regular resonance, PR is characterized by much higher intensities within a wider range of frequencies. The advantage of such a PR-based machine was demonstrated with a prototype parametric resonance screener (PRS), developed and produced by the project partner LPMC, Loginov and Partners Mining Company (Kiev, Ukraine). The PRS demonstrated large amplitude, high-frequency lateral oscillations and self-vibro-insulation, and could process a naturally wet fine granular material. Since PR is unstable, its use as an effective operating mode assumes, among other things, the establishment of a stabilized regime. This inspiring and challenging high-tech task for the combined efforts of applied mathematicians and engineers, the extraordinary features of PR, and its relevance to existing and open non-trivial theoretical and engineering problems, provided a compelling motivation to undertake this interdisciplinary research.

The PARM2 network involved eight partners. Five of them represented Academic Institutions: AU, Aberystwyth University (UK), specialized in the mathematical modelling of dynamic fracture in lattice and discrete structures; UoL, the University of Liverpool (UK), specialized in Mathematical and Numerical Modelling of complex structures and metamaterials; LU, Loughborough University (UK), specialised in numerical modelling of various vibro-impact mechanics and experimentation; RUT, Rzeszow University of Technology (Poland), specialised in Mechanical Engineering and Prototyping; TAU, Tel-Aviv University (Israel), held and shared expertise in areas of MEMS and NEMS. The remaining three were Industrial Partners: the aforementioned LPMC; EUROTECH (Poland), SME, working on creating targeted machinery; ISOTOP (Israel), a company specialising in advanced sophisticated monitoring technologies.

A research consortium was created to achieve the following main research objectives:
(a) development of failure theory due to resonant and localized waves in continuous and discrete structures;
(b) construction of mathematical models for the operational regimes of material separation with a PRS;
(c) development of a technically sound control of PR amplitude;
(d) conducting of theoretical and experimental investigations into the possibility of PR stabilization and amplitude control based on MEMS/NEMS;
(e) design of PRS-related grids with the required stiffness and minimal bending stresses, to provide the desired high-amplitude vibration resistance, and design of other types of PR-based separators and crushers;
(f) development of fracture theory under impact loading, where the enhancement of vibro-cutting/drilling tools, via development of the underpinning theory and application of the PR principle, was also an objective;
(g) conducting of experiments to verify the theoretically predicted parametric-resonance regimes of the vibrating screen;

The exchange of fundamental and technical concepts between the macro-scale research and micro/nano PR studies was one of the project themes, and project activities were based in close cooperation and targeted secondments between academia and industry. These opened opportunities for ESRs to be trained through research, working on all interdisciplinary aspects, beginning with conceptual design and modelling, and finishing with prototypes and demonstrations of new PR-based machines and tools.

The research targets were achieved via the following Work Packages:
WP1 - Mathematical and numerical analysis of nonlinear dynamic models for both PR-based and other dynamic systems;
WP2 - Methods for control and stabilization of PR vibro-impact mechanisms;
WP3 - Conceptual design of vibro-impact mechanisms;
WP4 - Theoretical and experimental research on fatigue and fracture under vibro-impact conditions.

Main results and their potential impact:
- Mathematical models for the propagation of resonant waves, and their localisation in structured plates and lattices, have been developed. Additionally, a novel phenomenon in Elasto-Dynamically Inhibited Transmission (EDIT) was discovered for the specifically structured plates, and it was shown that this phenomenon was related to the occurrence of simultaneously resonant trapped modes with even and odd symmetries.

- A semi-analytical model for the dynamic response of a lattice was developed to study dynamic anisotropy and primitive waveforms in discrete elastic systems. Special attention was given to localised vibrations in periodic lattices containing defects of finite size. Various aspects of wave cloaking, such as possible efficient designs of a cloak and different ways of measuring the quality of the cloaking, were investigated.

- Various novel dynamic problems for discrete and discrete-continuous periodic systems were solved. A ‘classical’ fracture-related and dynamic moving-contact problem for an elastic half-plane, which was formulated in the middle of the last century but remained debatable, finally received its complete solution. A new condition concerning energy fluxes through singular points was proposed, which guarantees the uniqueness of that problem's solution. The condition incorporates for the first time all speed ranges, including the super-Rayleigh subsonic and intersonic speed regimes. Using this approach, the driving forces of dynamic fracture caused by the main underlying factors were determined and identified as the stress-field singular points, wave radiation and fracture resistance. The regimes for which the failure wave may exist and propagate uniformly or accelerate, were identified. Such phenomena have been observed in several technological disasters (see for example

- The theory of quasi-static and dynamic brittle fracture has been developed, taking into account the internal potential energy. The crack speed and the criterion for crack initiation due to internal energy were determined. Different scenarios for the crack growth and non-trivial associated phenomena were revealed. The analytical technique relevant to this and related problems was developed, where a selective discrete transform was introduced. Spontaneous failure waves, which can propagate under micro-level stresses, were investigated, and new opportunities to carefully explain the famous Prince Rupert’s drop paradox were opened.

- A mathematical model of a vibrating screen operating in PR mode has been developed, and an exact analytical solution obtained. This allows assessment of the effects of the structural parameters (the masses, stiffness, damping, the excitation frequency and level) on the vibration amplitude and stability, enabling implementation of passive control of the screen.

- Computer codes based on the above-mentioned model were developed to account for the dynamic interaction between the screen and the processed material.

- The influence of the moisture of the processed material on the dissipation and material-separation rate was determined theoretically and experimentally. These, in particular, were related to the interaction between the vibrating mesh and the material, where the collisions were shown to be almost inelastic. The dissipation level was then theoretically estimated using this interaction.

- Theoretical and experimental investigations into the possibility of PR stabilization and amplitude control based on MEMS/NEMS were successfully conducted.

- Fracture mechanisms for crack growth under cyclic impact loads were identified and the related criteria were formulated.

- Different types of grid pattern were suggested and investigated. The corresponding new design of the PRS was developed.

All project research findings were published in scientific journals and disseminated during presentations at numerous international conferences. A prototype of a new PR screener was constructed and thoroughly investigated, allowing identification of the most effective working regimes. Many of the fundamental results for waves and fracture waves (or any transition waves) propagating in complex lattice structures and metamaterials were obtained, allowing the creation of effective clocking effects in the structures and estimation of its workability. New fundamental results on PR and MEMS will allow researchers and industrial partners to create new machines and tools that operate using the PR phenomenon. Some of the vibrating tools, regimes and principles were proposed in publications, while many of the developed ground-breaking ideas go beyond the specific research topics planned within the project.