Additional Stiffness Research Articles

A unified analytical expression of the tangent stiffness matrix of holonomic constraints

While an accurate computation of the tangent stiffness matrix of multibody systems is usually of critical importance in various numerical analyses, one contribution often neglected in dynamics is represented by the tangent stiffness matrix of constraints. This term, resulting from the change of joints reactions with respect to the coordinates of the connected bodies, is indeed easily discarded for the sake of performance due to its slight effect on dynamic simulation results. On the contrary, it plays a critical role when static or eigenvalue analyses are required, especially for those cases featuring free motions. While the topic of holonomic constraint equations and their respective Jacobian matrix is not new in literature, this article aims to provide a general and unified formulation based on quaternion parametrization together with a consistent analytical expression of the tangent stiffness matrix derived through linearization. The article includes also the rheonomic contribution to holonomic constraints. The formulations presented in this work are built on a mixed-basis formulation, in use in many engineering applications, and allow to easily derive specialized versions (e.g. revolute, cylindrical, prismatic joints, etc.) from the same equation set. Examples demonstrate the doubly-positive effect of this additional stiffness term: first, static analyses are able to converge to the actual equilibrium position and second, the eigenvalues analyses proved to be more consistent. For this latter set of tests the non-linear dynamic results, observed in the frequency domain, are compared against those coming from eigenvalue analysis, in order to prove the augmented accuracy of the results.

Transient state analysis of a rub-impact rotor system during maneuvering flight

Maneuvering flight substantially affects the dynamic behavior of rotors; particularly, such flight may cause rubbing between a rotor and stator, which is one of the most serious damages in aircraft engines. In this paper, a nonlinear dynamic model for describing the dynamic characteristics of a rub-impact rotor system during maneuvering flight is established based on the Lagrange equations. Subsequently, numerical simulations employing the Newmark method are performed, delving into the detailed discussion of the influence of parameters such as rotational speed and maneuvering flight on the transient and steady-state responses of the rotor system. The effect mechanism of maneuver load and its coupling with rub impact is revealed. The results show that the impact response induced by maneuvering flight is more obvious in the subcritical state than in the supercritical state. The additional stiffness and damping are also induced; in particular, the additional damping has a coupling effect. Moreover, the rub impact imposes an additional constraint on the rotor system, thereby weakening the influence of the maneuver load and becoming the major factor that determines the dynamic behavior of the rotor system at high speeds.

Development of modular cryogenic indentation apparatus for investigating micro-region mechanical properties of materials

High-precision cryogenic micromechanical testing apparatuses are of significant importance in investigating the micro-region mechanical properties and deformation behaviors of materials at cryogenic temperatures. In this study, a modular cryogenic indentation apparatus was designed and developed utilizing a contact-atmosphere hybrid refrigeration technique with a minimum temperature of −150 °C. The temperature of both the indenter tip and the specimen was independently controlled through a dual-channel proportional–integral–derivative (PID) feedback loop to eliminate contact thermal drift to within 0.1 nm/s. The application of cryogenic nitrogen (N2) atmosphere refrigeration eliminated additional stiffness at the indenter tip completely. The feasibility of the apparatus was demonstrated through calibrations and tests conducted on standard fused silica and single crystal nickel. Highly accurate indentation curves and micro-region mechanical properties of single crystal nickel were obtained at variable cryogenic temperatures. With decreasing temperature, both slip bands and pile-up around residual indentation imprints were weakened. Notably, at −150 °C, the slip bands disappeared, and the phenomenon of pile-up was replaced by sink-in. This developed cryogenic indentation apparatus will be a powerful tool to reveal the effect of cryogenic temperatures on the evolution mechanisms of materials’ micro-region mechanical properties.

Optimization of tuned mass dampers for multiple mode vortex-induced vibration mitigation in flexible structures: An application to multi-span continuous bridge

Tuned mass dampers (TMDs) have proven effective in suppressing structural vibration. In existing research on vortex-induced vibration (VIV) control of flexible structures, only one mode is commonly considered at the time, and only basic aerodynamic force models, which can struggle to model amplitude-dependent damping, are considered. However, flexible structures, such as multi-span bridges, exhibit multiple modes with closely spaced frequencies. Neglecting secondary modes may introduce calculation inaccuracies, as these modes can impose additional stiffness, thereby influencing the performance of TMD. This paper presents a framework to optimize the performance of multiple TMDs in the context of multi-mode VIV control for a flexible structure. A multi-span bridge is used as a case study, and the secondary modes and nonlinear aerodynamic effects are considered. The governing equations for the force-structure-MTMDs system are derived in modal coordinates using a polynomial model to model the aerodynamic force. A novel method to calculate the contribution of each mode to determine the minimum number of secondary modes to be considered is proposed. The optimal parameters of MTMDs are obtained using a global optimization method and compared with mode-by-mode design results. Robustness analysis is conducted to verify the effectiveness of the optimization framework. The results show that the polynomial model accurately predicts the amplitude after TMD installation. Secondary modes significantly affect the optimal parameters of TMDs, and the optimized results exhibit better control performance than the mode-by-mode design results.

Web Stiffening of Additive Manufactured Polylactide (PLA) Railroad Tracks Using Fiberglass Wrapping

Three-dimensional (3D) printed railroad tracks can properly address the rendering of a variety of topographic profiles of naturally occurring landscapes and reduce construction costs. Load tests of PLA railroad tracks found that web failure is a dominant failure mode. This paper evaluates the web stiffening strategies of additively fabricated PLA railroad tracks for rail-guided, micro-people movers (MPMs). To strengthen the track web, web-widening with additional PLA material and stiffening using composite reinforcement using glass fiber reinforced polymer (GFRP) wrap have been investigated. The composite wrapping technique was conducted by bonding fiberglass cloth to both sides of the web using a polymeric resin. Flexural load tests were conducted on the specimens with different infill volume percentages. A linear trend was observed on the peak bending capacities of specimens with PLA infill fiber contents ranging between 20% to 100%. Different failure modes were observed for the different percentage specimens during the tests. When compared to their unreinforced counterparts, the 60% infill specimen had the largest increase in strength about 1,500 N in bending with the addition of GFRP wrap. The tangent stiffnesses of all samples were calculated which showed the 100% infill specimen had the highest increase of approximately 50%. The wrapping reinforcement was found to significantly modify the failure mechanisms of the 3D-printed tracks with different infill volume percentages. The strengthening of the 3D-printed tracks using GFRP wrapping is also shown to be similar to the widening of web sections of the additively printed railroad tracks.

DEVELOPMENT OF COMPUTATIONAL SCHEMES OF GENERALIZED KINEMATIC DEVICES THAT PRECISELY REGULATE THE NATURAL FREQUENCY SPECTRUM OF ELASTIC SYSTEMS WITH A FINITE NUMBER OF DEGREES OF MASS FREEDOM, IN WHICH THE DIRECTIONS OF MOTION ARE PARALLEL, BUT DO NOT LIE IN THE SAME PLANE

To date, for some elastic systems with a finite number of degrees of mass freedom, in which the directions of mass movement are parallel and lie in the same plane, methods have been developed for creating additional generalized targeted constraints and generalized targeted kinematic devices. Each generalized targeted constraint increases, and each generalized targeted kinematic device reduces the value of only one selected natural frequency to a predetermined value, without changing the remaining natural frequencies and natural modes. Earlier, for elastic systems with a finite number of degrees of mass freedom, in which the directions of mass motion are parallel, but do not lie in the same plane (for example, plates), an approach for the computing of a matrix of additional stiffness and a method for the development of computational schemes of additional generalized targeted constraints were developed. Also earlier, for such systems, an approach was proposed for the computing of a special matrix with allowance for additional inertial forces that determine a generalized targeted kinematic device. At the same time, the method of development of computational schemes of kinematic devices was not proposed. The distinctive paper is devoted to approach, that makes it possible to develop computational schemes of generalized targeted kinematic devices for such systems as well. A variant of the computational scheme of constraint for the rod system with one degree of activity, is considered. Some special properties of such targeted kinematic devices are revealed.

Investigation of statistical energy analysis coupling loss factors of vibro-acoustic systems under thermal environment

Vibro-acoustic systems in aviation engineering could operate in a thermal environment and suffer from high-frequency dynamic excitations. The high-frequency vibro-acoustic problems are commonly solved by the Statistical Energy Analysis method, in which the Coupling Loss Factor is the key parameter for modeling vibro-acoustic systems. This work investigates the CLFs of vibro-acoustic systems under thermal environment. First, based on the modal data considering different thermal effects, the modal CLFs between structural modes and acoustic modes are predicted by the Statistical modal Energy distribution Analysis. Then, the CLFs between the structural subsystem and the acoustic subsystem are calculated. The simulation models employed in this research include the coupled plate-cavity system and the cylindrical shell-cavity system. The influence of thermal effects on the CLFs is discussed. For the investigated cases, in which the boundaries of structural susbsystem are set to simply supported, the results demonstrate that the CLFs decrease with increasing temperature when only the temperature-dependent material properties are considered. However, when only the additional stress stiffness is considered, the CLFs experience a more dramatic decrease compared to when only the temperature-dependent material properties are taken into account. Moreover, when both effects are considered, the influence of the additional stress stiffness has a leading effect on the CLFs. The proposed method can be conveniently applied to various vibro-acoustic systems operating in the thermal environment.

Theoretical and experimental investigation on nonlinear dynamic of grain-beam system

We experimentally investigated the system of a simply supported beam immersed in a granular medium, which displays jump and new peaks in its amplitude-frequency curve under harmonic excitation. The relationship between the macroscopic nonlinear response and microscopic state of the granular medium was studied using the discrete element method. Using the dynamic modulus and vibrational density of states as indicators, we show that the granular medium changes from a solid-like to liquid-like state when the frequency increases to approach the jumping point. Considering the similarity between this system and acoustic metamaterial beams, an equivalent theoretical model was established for the former. Numerical results of the theoretical model were consistent with the experimental ones. It was revealed that the new peak and jump phenomena were caused by negative equivalent mass, negative equivalent stiffness of the granular medium and abrupt change in the additional stiffness cause by the jamming transition. This theoretical model is useful for studying the nonlinear dynamic characteristics of this coupled grain-beam system.

A Novel Design for Improving the Control on the Stainless-Steel Vessel Welding Process for Superconducting Magnets

Stainless steel vessels see widespread use in superconducting magnets for particle accelerator applications. Their function varies in different magnet designs: they always provide the necessary liquid helium containment, but in some cases are also used to provide azimuthal prestress and can also be welded to the magnet end plate to provide additional longitudinal stiffness. A magnet designed with the bladder and key technology does not rely on the structural role of the vessel. They are structurally supported using azimuthally prestressed aluminum shells, and the longitudinal constraint by rods. In this case, the magnet designer would generally like to minimize the interaction between the magnet and the stainless-steel vessel and to minimize the coil stress variation due to the vessel. The stress state in the vessel and in the coil is a function of the circumferential interference, defined as the vessel azimuthal length minus the magnet circumference. The vessel and the magnet azimuthal length machining tolerances are relatively large resulting in significant stress variations in the superconducting coils. In this paper we introduce an interference-control shim, which can significantly limit the stress variation of the coils for a given variation of the interference. The effectiveness of the interference-control shim is evaluated numerically on the MQXF, the low- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> quadrupole for the High Luminosity LHC.

Experimental and numerical study on the seismic behavior of an external prestressed self-centering frame structure

A kind of external prestressed self-centering frame (EPSCF) structure is proposed, and the numerical analytical modeling is given. In such a structure, the pure hinged beam-column joints and column base joints are adopted, the externally prestressed steel strands are utilized to provide lateral stiffness and restoring forces, and the inter-story metal dampers are installed to control displacement response. Based on the quasi-static cyclic tests of the EPSCF test frame, the prestressing forces and the hysteretic behavior of the EPSCF structure are investigated. The test results show that the load-displacement hysteretic curves of the EPSCF without damper exhibit a linear elastic characteristic with small residual displacement, and the load-displacement hysteretic curves of the EPSCF with damper show a full S-shape and significant two-stage characteristics. Installation of the X-type metal damper provides additional stiffness and improves the energy dissipation capacity for the EPSCF. FE model of EPSCF is established, and the hysteretic curves obtained by tests are in good agreement with the numerical simulation results. Parametric analysis indicates that with the increase of the relative stiffness ratio of the lateral stiffness, the seismic response of EPSCF shows a trend of decreasing structural displacement, increasing structural acceleration and base shear. Moreover, the simulation results suggest that the optimum range for the relative stiffness ratio is roughly 0.02–0.10.

Dynamic modeling of spur gear system under marine ship heaving-pitching motion

Sea waves often result in multi-freedom motions of ships, and affect the stability of marine gear systems. This study presents a novel dynamic modeling method for spur gear system considering the influence of marine ship heaving-pitching motion and investigates the spur gear dynamics. An excitation transmission path from wave to gear is illustrated and incorporated into the gear dynamic model. The hydrodynamic force is analyzed and the hull heaving-pitching motion is solved using the New Strip Method (NSM). Considering the time-varying meshing stiffness, transmission error, and tooth side clearance, the motion differential equation of the spur gear system impacted by the heaving-pitching coupling motion of the hull is established. The influences of key wave and navigation parameters on gear dynamics are analyzed. The results show that ship heaving-pitching motion causes additional stiffness and damping excitation, and leads to typical nonlinear phenomena of the spur gear system in a marine ship, such as period, quasi-period, and chaos, and causes a sharp increase in the meshing impact amplitude up to a maximum of 36.81%. The changes in wave frequency, the hull's heading angle, and hull's speed have a significant impact on gear dynamics, whereas the wave amplitude has less of an impact.

Multi-Physics Simulation and Experimental Verification of Magnetorheological Damper with Additional Stiffness

Single-rod magneto-rheological dampers (MRD) have the advantages of a simple mechanism, high reliability, and broad application range. They are widely used in various semi-active vibration control fields. However, their working mode requires a compensating mechanism to perform volume compensation on the rod, leading to additional stiffness for the system. Ignoring this point makes it tough to establish an accurate mechanical model to describe its performance in the design stage, affecting its application. To address this issue, this study proposes a multi-physics simulation model based on gas compensation for single-rod MRD to characterize their mechanical performance accurately. Firstly, the mechanism and mechanical model of the single-rod gas compensation MRD are introduced. Secondly, considering that its performance is affected by the coupling effect of multiple physical fields, including magnetic, flow, and solid mechanics fields, the control equations and boundary conditions of each field are analyzed separately, and a multi-physics coupling simulation model is established by COMSOL. In particular, the gas compensation unit is considered in the multi-physics simulation model. The effect of the compensating mechanism on the mechanical performance of the damper under different excitation speeds, currents, and initial pressures is analyzed. Finally, the accuracy of the proposed method is verified through the demonstration power test. The results show that the simulation can describe the additional stiffness in the damper. The average error between experimental value and simulation value is 7%. This demonstrates the degree of agreement between the experiment and simulation.

Analysis of Track Bending Stiffness and Loading Distribution Effect in Rail Support by Application of Bending Reinforcement Methods

Railway track is a linearly inhomogeneous object that consists of geometrical and elastic discontinuities such as bridges, transition zones, rail joints and crossings. The zones are subjected to the development of local instabilities due to quicker deterioration than the other tracks. Until now, there have been no efficient approaches that could fully exclude the problem of accelerated differential settlements in the problem zones. Many structural countermeasures are directed at controlling the sleeper/ballast loading with the help of fastenings/under-sleeper pad elasticities, sleeper forms and additional bending stiffness reinforcements. However, the efficiency of the methods is difficult to compare. The current paper presents a systematic approach in which the loading distribution effect in the rail support by application of two bending reinforcement methods is compared: auxiliary rail and under-sleeper beam. The study considers only the static effects to reach a clear understanding the influence of the main factors. The track equivalent bending stiffness criterion is proposed for comparing reinforcement solutions. The analysis shows that the activation of the bending stiffness of the reinforcement beams depends on the relative ratio of the rail fastenings stiffness and track support stiffness under sleepers (or under the under-sleeper beam). The comparison demonstrates that conventional auxiliary rail reinforcement solutions are ineffective due to their weak bending because of the high elasticity of fastening clips and the main rail fastenings. The share of an auxiliary rail is maximally 20% in the track bending stiffness and cannot be significantly improved by additional rails. The under-sleeper beam-based reinforcement solutions show noticeably higher efficiency. The highest effect can be achieved by the activation of the horizontal shear interaction between the under-sleeper beam and the rail. The additional track bending stiffness of the under-sleeper-based solutions is about 3.5 times more of the rail one and could be potentially increased to 6–10 times.

Analysis of Biomechanical Response After Corneal Crosslinking with Different Fluence Levels in Porcine Corneas

Purpose To evaluate corneal stiffening of porcine corneas induced by corneal crosslinking (CXL) with constant irradiance as a function of total fluence. Methods Ninety corneas from freshly enucleated porcine eyes were divided into five groups of 18 eyes. Groups 1–4 underwent epi-off CXL using a dextran-based riboflavin solution and an irradiance of 18 mW/cm2, group 5 served as the control group. Groups 1 to 4 were treated with a total fluence of 20, 15, 10.8, and 5.4 J/cm2, respectively. Thereafter, biomechanical measurements were performed on 5 mm wide and 6 mm long strips using an uniaxial material tester. Pachymetry measurements were performed on each cornea. Results At 10% strain, the stress was 76, 56, 52, and 31% higher in groups 1–4, respectively compared to the control group. The Young’s modulus was 2.85 MPa for group 1, 2.53 MPa for group 2, 2.46 MPa for group 3, 2.12 MPa for group 4, and 1.62 MPa for the control group. The difference between groups 1 to 4 and the control group 5 were statistically significant (p = <0.001; p = <0.001; p = <0.001; p = 0.021). In addition, group 1 showed significantly more stiffening than group 4 (p = <0.001), no other significant differences were found. Pachymetry measurements revealed no statistically significant differences among the five groups. Conclusion Additional mechanical stiffening can be achieved by increasing the fluence of the CXL. There was no threshold detected up to 20 J/cm2. A higher fluence could compensate the weaker effect of accelerated or epi-on CXL procedures.

A semi-analytical method for vibro-acoustic characteristics of orthogonal stiffened laminated cylindrical shells

This paper presents a semi-analytical model for vibro-acoustic characteristics of submerged orthogonal stiffened laminated cylindrical shells strengthened with orthogonal stiffeners. The first-order shear deformation theory (FSDT) combined with the Lekhnitsky smeared stiffener technique is adopted to describe the displacement field, and the spectral boundary element method is utilized to model the sound field. For modeling of a complex stiffened shell, the shell is divided into several main structures with the additional stiffness and mass of the smeared stiffeners. Then, the main structure is separated into sub-segments that match the boundary elements to ensure high accuracy of acoustic calculations. The continuous boundary conditions between sub-segments are simulated by using the artificial spring technique. The Fourier series and Legendre orthogonal polynomials are selected as admissible functions to describe the displacement field and the sound field. The vibration governing equations and Helmholtz integral equation are coupled with each other and solved simultaneously. High accuracy is demonstrated by comparing the results of the developed model with the literature. In numerical investigations, the influences of different external fluids, layup schemes, quantities and shapes of stiffeners (width and height) on vibro-acoustic characteristics of orthogonal stiffened laminated cylindrical shells (OSLCS) are investigated.

Fracture mechanical stability of DCB and 4ENF tests complemented with linear springs

In this work the stability of the DCB and 4ENF fracture specimens is analyzed using some beam theories and the compliance calibration method. The main goal is that the systems are complemented with linear springs and it is investigated how the springs influence the stability of crack propagation from the theoretical standpoint. Large number of experiments were carried out including crack initiation tests providing the experimental load displacement responses, the compliance datasets and the qualitative observation of crack initiation. Apart from the analytical method the compliance calibration method was employed to evaluate the experimental data and to determine the fit curves of the compliance datasets, the R-curves and finally the bifurcation curves or in other words the critical displacement and crack length relationships including several additional spring stiffnesses for both test configurations. The results indicate that - as expected - the smaller the spring stiffness is the worst the stability of the DCB specimen will be, leading to the possibility of destabilization. On the contrary the analysis and experiments confirm that the 4ENF test cannot be destabilized independently of the stiffness of additional springs.

Analytically optimal parameters of supplemental brace-damper system in single-degree-of-freedom structure based on stability maximization criterion

In this study, optimal parameters (i.e., optimal additional stiffness ratio α and damping ratio λ) for supplemental brace-damper system in a single-degree-of-freedom (SDOF) structure are analytically derived. Different from the design formulations for supplemental brace-damper system proposed by Londoño, which are essentially fitting formulae through extensive parametric study, analytically optimal parameters for supplemental brace-damper system are derived by combining the research work by Londoño and the stability maximization criterion (SMC). Similar to Optimum Brace Stiffness (OBS) criterion and Optimum Damper Size (ODS) criterion proposed by Londoño, SMC-based OBS criterion and SMC-based ODS criterion are respectively proposed to assure a desired structural performance represented by the targeted overall damping ratio of the considered system. Through numerical study, it is demonstrated that the seismic response of the proposed SMC-based OBS criterion and SMC-based ODS criterion excellently agree with those of the existing OBS criterion and ODS criterion, respectively.

A Method for Solving the Additional Stiffness Introduced by Flexible Joints in Stewart Platform Based on FEM Modal Analysis

The usage of flexible joints has greatly facilitated the widespread application of the Stewart construction mechanism in many advanced fields. This research focuses on the problem that the flexible joints in the Stewart mechanism cause additional stiffness effects on the whole system. A method for solving the additional stiffness of flexible joints combining the finite element method (FEM) is proposed, which avoids the complex theoretical derivation process and allows the advantages of high versatility and accuracy. Three aspects make up the main content. Firstly, based on dynamics theory, the theoretical analysis and demonstration of the method are conducted according to the symmetry characteristics of the Stewart platform. Next, the additional stiffness of a designed Stewart platform with flexible joints was solved following the method proposed, and the obtained results were verified through FEM simulation under the given conditions, which give a maximum natural frequency deviation against the theory of 2.26%. Thirdly, dynamics tests associated with the Stewart platform were conducted, and the deviation between the Stewart Platform’s natural frequencies separately obtained from the tests and the theory does not exceed 5%, which demonstrates the effectiveness and accuracy of the method in engineering applications. This study aims to provide technical support for the development of Stewart mechanisms with flexible joints.

Development of a reversible top-hat forming approach for reducing flange wrinkling in flexible roll forming

Flexible roll forming (FRF) addresses the need for flexible manufacturing and light-weighting in electric vehicle production, but its widespread implementation is prevented by severe wrinkling in the flange. Wrinkling in sheet metal forming depends on the tool support and material deformation. In FRF the tool effects on wrinkling are complex, given that the sheet undergoes unsupported deformation when it enters and exits the forming roll and there is no fundamental understanding of material deformation in the critical forming regions where wrinkling occurs. This prevents an effective tool and process design. This work develops a new model to analyse the stability of the flange by combining plastic bifurcation analysis with thin-shell assumptions. The model defines a stress ratio parameter that can be used to determine when in the process, deformation becomes unstable considering the tool support conditions and material deformation. Based on this a new approach for wrinkling reduction is developed that includes the forming and bending back of a top-hat shape to provide additional flange stiffness when wrinkling is likely to initiate. The top-hat shape is optimised using a series of linear perturbation ‘buckle’ analyses to provide the highest possible stiffness increase in the flange. This is followed by experimental manufacturing trials to test and prove the new concept for wrinkling reduction and to validate the model. It is shown that the new top-hat forming approach can eliminate wrinkling in the flange even when flexible roll forming Ultra High Strength Steels. The analytical model results suggest that this is due to the stiffening effect of the top-hat and the reduction of the effective flange length which decreases the stress ratio. The new model in combination with the new forming approach to eliminate flange wrinkling represents a vital step towards a more effective process and tool design that will enable the widespread application of FRF in automotive manufacturing.

Modelling and testing of a pressure-differential wave energy converter with flexible membranes

We investigate a wave energy converter composed of two submerged, bottom-fixed air-filled cylindrical chambers, separated by a distance of ideally half a wavelength, and closed on top by flexible membranes. A passing wave creates a pressure differential between the two chambers which drives the membranes and causes air to flow through a turbine. Using linear analysis and physical model testing, we show that the device exhibits broad-banded power absorption. Our linear model successfully predicts the pressure differential and absorbed power measured in the tests, except at low frequencies, where the differences are attributed to unstable membrane conditions due to a negative total stiffness. Using a simple model, we show that the flat-membrane equilibrium, in the absence of additional stiffness, is unstable. By introducing an additional stiffness to the membrane, this instability can be avoided and the natural period of the device can theoretically be tuned to any value. With a matching flow resistance, it is possible to attain absorbed power values close to the theoretical limit.