Surface Elasticity Research Articles

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

Characteristics of wave propagation in pre-stressed viscoelastic Timoshenko nanobeams with surface stress and magnetic field influences

The current investigation explores the behavior of pre-stressed viscoelastic Timoshenko nanobeams under the influence of surface effects and a longitudinal magnetic field. Utilizing a modified version of non-local strain gradient theory through the Kelvin–Voigt viscoelastic model, a closed-form dispersion relation using a suitable analytical approach has been derived. To account for surface stresses, Gurtin–Murdouch surface elasticity theory has been employed. Additionally, the study delves into the impact of a longitudinal magnetic field on a single-walled carbon nanotube, considering Lorentz magnetic forces. The validity of the findings is established by deriving results in the absence of surface effects and magnetic fields, aligning well with existing literature. The investigation indicates that pre-stress has marginal effects on flexural and shear waves, while surface effects, magnetic fields, non-locality, characteristic length, and nanotube diameter significantly influence the phase velocity. Additionally, the threshold velocity and blocking diameter are discussed for the model.

Multiscale Mixed Elastohydrodynamics in Inclined Fixed Pad Thrust Bearing Involving Surface Roughness and Low Clearance

The film pressures, surface separation profiles and carried loads of the hydrodynamic inclined fixed pad thrust bearing with the sandwich films were calculated by the multiscale approach when the bearing clearance was low and the combined effects of the adsorbed layer, the surface roughness and the surface elastic deformation were incorporated; The sandwich film consists of two physically adsorbed layers respectively on the coupled bearing surfaces and the intermediate continuum fluid film. The results show that the surface roughness intrinsically generates the film pressure ripples, but the rough surface is somewhat flattened and the film pressure ripples are a bit reduced by the surface elastic deformation. For a given sliding speed and a given original bearing clearance, the surface elastic deformation results in the significant reductions of both the maximum film pressure and the carried load of the bearing especially for a strong fluid-bearing surface interaction. In spite of this, the slope of the increase of the bearing load with the magnitude () of the surface roughness is nearly the same for the elastic surfaces with that for the rigid surfaces. The presence of the surface roughness does not qualitatively alter the influence of the surface elastic deformation on the performance of the bearing.

On the tractive rolling nanocontact of an exponentially graded coating-substrate structure

This paper studies the tractive rolling nanocontact occurring between an exponentially graded coating-substrate structure and a circular rigid indenter. Employing the framework of Steigmann–Ogden surface elasticity, it models the surface effects inherent in the nanocontact of graded coatings. The contact area is assumed to comprise a central stick zone bounded by two distinct slip zones. Central to the investigation is the utilization of the nonclassical Flamant solution, which serves as the foundational framework for deriving integral equations governing the continuity of both vertical and tangential displacement gradients. Utilizing Gauss–Chebyshev quadratures, the paper discretizes and collocates these integral equations, along with the force equilibrium conditions and shear traction smooth condition at the leading side stick/slip transition point. An iterative algorithm is then developed to tackle the resultant algebraic system, particularly concerning the discretized contact pressure and friction traction. The paper rigorously validates its proposed solution method and numerical algorithm against existing literature results, showcasing their accuracy and reliability. Moreover, it conducts extensive parametric studies to unravel the effects of various parameters, such as surface material properties, coefficient of friction, inhomogeneity index, and thickness of the exponentially graded coating. These analyses uncover the significant role of surface effects in shaping contact pressure, frictional traction, stresses, subsidence distributions, and stick–slip zones. Notably, the inclusion of surface effects is found to reduce maximum stress and subsidence while inducing a shift of the stick region towards the rolling direction. The parametric exploration of graded coating properties also offers insights into tailoring nanocontact responses for gradient nanostructures.

Stochastic process model for interfacial gap of purely normal elastic rough surface contact

In purely normal elastic rough surface contact problems, Persson’s theory of contact shows that the evolution of the probability density function (PDF) of contact pressure with the magnification is governed by a diffusion equation. However, there is no partial differential equation describing the evolution of the PDF of the interfacial gap. In this study, we derive a convection–diffusion equation in terms of the PDF of the interfacial gap based on stochastic process theory, as well as the initial and boundary conditions. A finite difference method is developed to numerically solve the partial differential equation. The predicted PDF of the interfacial gap agrees well with that by Green’s Function Molecular Dynamics (GFMD) and other variants of Persson’s theory of contact at high load ranges. At low load ranges, the obvious deviation between the present work and GFMD is attributed to the overestimated mean interfacial gap and oversimplified magnification-dependent diffusion coefficient used in the present model. As one of its direct application, we show that the present work can effectively solve the adhesive contact problem under the DMT limit. The current study provides an alternative methodology for determining the PDF of the interfacial gap and a unified framework for solving the complementary problem of random contact pressure and random interfacial gap based on stochastic process theory.

Exact solution for free vibration analysis of non-uniform thickness functionally graded porous nanosheet with surface effect based on variable nonlocal and length-scale parameters

In this article, the analytical solution for free oscillation of non-uniform thickness functionally graded porous (FGP) rectangular nanosheets resting on a Pasternak foundation with various boundary conditions (BCs) based on the nonlocal strain gradient theory and surface effect are discovered. The nonlocal stress and strain effects are captured on the nonlocal strain gradient hypothesis, while the surface energy effects are based on the surface elasticity hypothesis. The FGP nanosheet has a nonlinear thickness change in both x and y directions and is placed on the Pasternak elastic foundation. The unique point of this study is to investigate the change of nonlocal and length-scale coefficients along the thickness as mechanical properties of the material. Applying classical shear deformation theory (ignore the effect of shear deformation) combined with Hamilton’s principle, the motion balance equation of non-uniform thickness nanosheets is established. The accuracy of the proposed method is verified through reliable publications. A set of results of natural vibration frequencies of variable thickness FGP nanosheets under the influence of factors are discovered and discussed. The results of this study can be used in design calculations of MEM and NEMs systems in mechanics.

Unsteady chemically reactive Maxwell nanofluid flow through a porous elastic surface with Cattaneo–Christov model

Unsteady chemically reactive Maxwell nanofluid flow through a porous elastic surface with Cattaneo–Christov model

Dynamics of acoustically levitated ice impacts on smooth and textured surfaces: Effects of surface roughness, elasticity, and structure

Atmospheric ice collisions on exposed outdoor infrastructure are a rarely explored surface design challenge. A primary obstacle to realistic ice particles in the laboratory setting is freezing a droplet without a solid surface present and then allowing it to impact a substrate without physical intervention. Using acoustically levitated ice formation with subsequent release onto a controlled area, a third class of ice-countering system called “ice-impacting” is introduced. Stainless steel 316 (SS), epoxy resin-coated (ER), and laser-textured (LT) surfaces with known surface roughness, hardness, and structural characteristics were subjected to 40 ice droplet impacts each. The impact effect on surface properties and the effect of these properties on solid-solid interfacial impact dynamics, in turn, were examined using an analysis framework based on fundamental conservation laws. Elasticity was the most significant factor in droplet behaviour as the ER surface exhibited rebounding for 78 % of impacts, nearly double the rates of the stainless steel surfaces. However, surface roughness also played a role, particularly for droplets with rotational motion, as immobilization occurred for 66 % of impacts on the rougher LT surface. Moreover, the nanostructures on that textured surface resulted in droplet redirection perpendicular to the surface directionality. In contrast, the other surfaces experienced no consistent change in rebound angle. Elasticity also affected momentum retention: the ER surface had a translational restitution coefficient of 0.32 compared to 0.17 for the SS/LT surfaces. Surface roughness was a predominant aspect of energy retention, as the LT surface had a translational-to-rotational energy transfer coefficient of 0.07 (0.23 for the smoother surfaces), resulting in an overall energy retention coefficient of 0.09 compared to 0.28 for the SS and ER surfaces on average. The results show that the interplay between elasticity, roughness, and structure at a dynamic solid-solid interface must be considered to optimize their effectiveness when designing ice-countering surfaces.

Upgrade in the elastic surface algorithm for airfoil inverse design with variable attack angle via controlled leading-edge movement

Upgrade in the elastic surface algorithm for airfoil inverse design with variable attack angle via controlled leading-edge movement

Electromechanical responses of a piezoelectric semiconducting nanoplate with the Steigmann-Ogden surface

Abstract Piezoelectric semiconductors (PSCs) find widespread applications in smart electronic devices due to their unique combination of piezoelectric and semiconductive properties. With the increasing demand for smaller and more efficient electronic devices, the performance of their components needs to be carefully optimized, especially when they are scaled down to nanoscale sizes. Pioneering studies have demonstrated that the surface elastic properties play a significant role in determining the mechanical performance of nanoscale materials and structures. Therefore, it is important to comprehensively investigate the effects of surface elasticity, including surface residual stress, surface membrane stiffness and surface bending stiffness, on the electromechanical responses of a PSC nanoplate. Additionally, it is also crucial to examine the influence of flexoelectricity at the nanoscale. Our results demonstrate that surface elastic properties predominantly impact mechanical performance, while the flexoelectric effect plays a more prominent role in electric field and redistribution of charge carriers. In particular, the significance of surface bending rigidity, which was often overlooked in previous literature, becomes pronounced when the thickness of a PSC nanoplateisless than 10 nm. Furthermore, the dependence of natural vibration frequency on surface elastic moduli, flexoelectricity and size is also systematically explored. The redistributions of electric potential and charge carriers across the cross-section are also evidently affected. Our findings provide valuable insights for improving the performance of electronic devices that utilize nanoscale PSCs.

An Analytical Adhesion Model for Elastic Contact Electrification

Abstract Contact electrification is a universal phenomenon that commonly occurs in almost every solid–solid contact pair. The tribo-charges deposited on two surfaces by contact electrification can significantly affect adhesion; however, contact electrification is often overlooked in the study of adhesive contact. Here, we develop an analytical model to investigate electroadhesion during the contact phase between two initially uncharged dielectric surfaces, namely, an elastic parabolic surface and a rigid flat. A system of nonlinear equations is derived to describe the relationship between the indentation, normal load, radius of contact area, and radius of the charged zone using the Barthel–Maugis–Dugdale model (Barthel, 1999, “Modelling the Adhesion of Spheres: When the Form of the Interaction Is Complex, Colloids. Surf., A., 149, pp. 99105.). The analytical results show good agreement with the numerical results of the full self-consistent contact model. When contact electrification leads to a higher tribo-charge density and a larger charged zone, it has a greater impact on the normal traction, interfacial gap, force-approach curves, jump-out, and dissipated energy. The analytical model developed in this study serves as the foundation for advances in rough surface electroadhesive contact and electroadhesion testing, and it sheds light on the usage of adhesive joints in ultra-high vacuum environments and outer space, where contact electrification has a significant impact.

Substrate deformability and applied normal force are coupled to change nanoscale friction.

Amonton's law of friction states that the friction force is proportional to the normal force in magnitude, and the slope gives a constant friction coefficient. In this work, with molecular dynamics simulation, we study how the kinetic friction at the nanoscale deviates qualitatively from the relation. Our simulation demonstrates that the friction behavior between a nanoscale AFM tip and an elastic graphene surface is regulated by the coupling of the applied normal force and the substrate deformability. First, it is found that the normal load-induced substrate deformation could lower friction at low load while increasing it at high load. In addition, when the applied force exceeds a certain threshold another abrupt change in friction behavior is observed, i.e., the stick-slip friction changes to the paired stick-slip friction. The unexpected change in friction behavior is then ascribed to the change of the microscopic contact states between the two surfaces: the increase in normal force and the substrate deformability together lead to a change in the energy landscape experienced by the tip. Finally, the Prandtl-Tomlinson model also validates that the change in friction behavior can be interpreted in terms of the energy landscape.

Scattering of SH wave in an elastic half-space by a semi-elliptical crater with surface elasticity

Cratering occurs inevitably on the surface of materials at all scales due to comprehensive factors. The presence of craters may have a significant influence on the mechanical performance of the surface of an elastic body. In this paper, we study the SH wave propagation in an elastic semi-infinite medium whose surface has a semi-elliptical crater. We explore the scattering of the SH wave by the crater when surface elasticity is incorporated in the model of deformation to account for possible small-scale effects of the crater. We obtain a series solution for the scattered wave function by employing the Mathieu functions in the corresponding elliptical coordinate system. We discuss in some numerical examples the dynamic stress concentration around the crater relative to the wave frequency and few geometric parameters.

Technical-functional and surface properties of white common bean proteins (Phaseolus vulgaris L.): Effect of pH, protein concentration, and guar gum presence

Legumes are abundant sources of proteins, and white common bean proteins play an important role in air–water interface properties. This study aims to investigate the technical-functional properties of white common bean protein isolate (BPI) as a function of pH, protein concentration, and guar gum (GG) presence. BPI physicochemical properties were analyzed in terms of solubility, zeta potential, and mean particle diameter at pH ranging from 2 to 9, in addition to water-holding capacity (WHC), oil-holding capacity (OHC), and thermogravimetric analysis. Protein dispersions were evaluated in terms of dynamic, interfacial, and foam-forming properties. BPI showed higher solubility (>80 %) at pH 2 and above 7. Zeta potential and mean diameter ranged from 15.43 to −34.08 mV and from 129.55 to 139.90 nm, respectively. BPI exhibited WHC and OHC of 1.37 and 4.97 g/g, respectively. Thermograms indicated decomposition temperature (295.81 °C) and mass loss (64.73 %). Flow curves indicated pseudoplastic behavior, with higher η100 values observed in treatments containing guar gum. The behavior was predominantly viscous (tg δ > 1) at lower frequencies, at all pH levels, shifting to predominantly elastic at higher frequencies. Equilibrium surface tension (γeq) ranged from 43.87 to 41.95 mN.m−1 and did not decrease with increasing protein concentration under all pH conditions. All treatments exhibited ϕ < 15°, indicating predominantly elastic surface films. Foaming properties were influenced by higher protein concentration and guar gum addition, and the potential formation of protein-polysaccharide complexes favored the kinetic stability of the system.

Investigation of the Bending Behavior in Silicon Nanowires: A Nanomechanical Modeling Perspective

Nanowires (NWs) play a crucial role across a wide range of disciplines such as nanoelectromechanical systems, nanoelectronics and energy applications. As NWs continue to reduce in dimensions, their mechanical properties are increasingly affected by surface attributes. This study conducts a comprehensive examination of nanomechanical models utilized for interpreting large deformations in the bending response of silicon NWs. Specifically, the Heidelberg, Hudson, Zhan, SimpZP and ExtZP nanomechanical models are explored regarding their capability to predict the elastic properties of silicon NWs with varying critical dimensions and crystal orientations. Molecular dynamics simulations are employed to model silicon NWs with unreconstructed surface states. The calculation of intrinsic stresses and the methodology for quantifying surface properties, including surface stresses and surface elasticity constants, are carried out using atomistic modeling. The findings reveal significant disparities of up to 100 GPa among nanomechanical models in interpreting a singular force-deflection response obtained for a silicon NW. Inadequate consideration of surface and intrinsic effects in nanomechanical modeling of NWs leads to substantial variability in their mechanical properties. This investigation yields valuable insights into the surface characteristics of silicon NWs, thereby enhancing our understanding of the essential role played by nanomechanical models in the intricate interpretation of mechanical properties at the nanoscale.

Effect of slip velocity on Newtonian fluid flow induced by a stretching surface within a porous medium

This article aims to examine an unsteady 2-D laminar flow of magnetohydrodynamic fluid caused by an elastic surface immersed in a permeable medium under the influence of thermal radiation and extended heat flux. Thermal conductivity and viscosity both are supposed to vary with temperature. This flow model also includes velocity slip, heat source, and joule heating. The governing equations of the fluid, including momentum and energy equations, of the proposed problem are transfigured into a system of interconnected non-linear ordinary differential equations through similarity transformations. The resultant equations are solved efficiently by employing the shooting technique in combination with the fourth-order Runge-Kutta method. Numerical values and the effect of numerous governing factors on the flow field, temperature distribution, local skin friction coefficient, and Nusselt number are showcased via graphs and tables. The investigation reveals that velocity slip, heat source, and porosity parameters enhance the temperature field while diminishing the velocity field. Furthermore, the velocity slip parameter notably reduces both the coefficient of skin friction and the Nusselt number.

P-775 What is the potential of cervicovaginal mucus as a biomarker for identifying the fertile window and the effectiveness of AI-enhanced prediction for scalability and accessibility?

Abstract Study question What is the potential of cervicovaginal mucus as a biomarker for identifying the fertile window and the effectiveness of AI-enhanced prediction for scalability and accessibility? Summary answer Cervicovaginal mucus is a reliable biomarker for the fertile window. AI Foundation models show promise in enhancing this method’s scalability and predictability. What is known already The timing of ovulation corrosponds to the fertile window. Although modern methods of ovulation detections yield high accuracies, they often require the purchase of sensors/equipment which creates a barrier to those in low-resource settings. Natural fertility awareness based methods, such as CVM monitoring, eliminate this problem, but there are limited recent large-scale studies validating cyclical CVM changes. Digitising CVM monitoring could remove the subjectivity of the interpretation of symptoms. Despite the advances of AI in other FABM areas, limited work has been done on computer vision models for CVM evaluation, which could revolutionise accessibility and scalability, particularly in low-resource settings. Study design, size, duration The study involved analysis of self-reported CVM characteristics data from over 57,000 individuals; largest sample size used in any study investigating the use of CVM as a biomarker. Furthermore, a dataset of CVM images were collected from the public survey and labeled manually (n = 13). The study included retrospective and pilot components, employing statistical analyses like ANOVA, chi-squared, and Spearman correlation, and AI techniques like zero-shot learning for binary segmentation of CVM from smartphone-captured images Participants/materials, setting, methods Self-reported CVM data was obtained from a fertility tracking app and analysed using ANOVA and chi-squared tests to identify and validate findings of cyclical variations, and correlation with lifestyle data. Zero-shot learning was employed using foundation models. An ethically-reviewed survey was generated to collect images for subsequent model training Images from the survey were segmented using the Grounding DINO and the SAM. The surface area and elasticity extracted are used to determine the fertility score. Main results and the role of chance In individuals with 28-day cycles, CVM showed significantly higher quality during the ovulatory phase, marked by increased elasticity (p = 0.00001), amount (p = 0.00001)), and lubricative texture (p &amp;lt; 0.0001), when compared to both the follicular and luteal phases. However, variations in individuals with longer, irregular cycles of over 35 days length, were less pronounced. Age was found to significant positive correlation with overall CVM quality (r = 0.06, p &amp;lt; 0.0001), while no significant correlations were observed with smoking (r = 0.11, p = 0.23), alcohol consumption (r = 0.13, p = 0.17), or BMI (r = 0.006, p = 0.41)). Zero-shot learning for segmentation showed improved accuracy in detecting and extracting CVM from images, with precision of 0.867 and F1 score of 0.929, highlighting the model’s ability to correctly distinguish between CVM patterns corresponding to fertile and infertile status. The intersection over union score suggested consistent model performance. Overall, CVM emerged as a scalable fertility biomarker in typical menstrual cycles, with AI models showing potential for enhancing prediction accuracy and scalability. Limitations, reasons for caution The study’s limitations include its reliance on self-reported CVM data and the potential bias in the foundation models as there was no fine-tuning, due to a small dataset (n = 13). Additionally, the effectiveness of CVM as a biomarker in irregular menstrual cycles and subfertility requires further exploration. Wider implications of the findings Automated analysis of CVM can be a low-cost, scalable (potentially to millions, via mobile application), non-invasive method for fertility awareness. Our findings highlight the potential of AI in empowering fertility awareness, especially in low-resource settings with little access to reproductive healthcare, thus contributing to global health equity. Trial registration number n/a

Physicochemical Studies on Amino Acid Based Metallosurfactants in Combination with Phospholipid.

Dicarboxylate metallosurfactants (AASM), synthesized by mixing N-dodecyl aminomalonate, -aspartate and -glutamate with CaCl2, MnCl2 and CdCl2, were characterized by XRD, FTIR, and NMR spectroscopy. Layered structures, formed by metallosurfactants, were evidenced from differential scanning calorimetry and thermogravimetric analyses. Solvent-spread monolayer of AASM in combination with soyphosphatidylcholine (SPC) and cholesterol (CHOL) were studied using Langmuir surface balance. With increasing mole fraction of AASM mean molecular area increased and passed through maxima at ~60 mol% of AASMs, indicating molecular packing reorganization. Systems with 20 and 60 mol% AASM exhibited positive deviations from ideal behavior signifying repulsive interaction between the AASM and SPC, while synergistic interactions were established from the negative deviation at other combinations. Dynamic surface elasticity increased with increasing surface pressure signifying formation of rigid monolayer. Transition of monolayer from gaseous to liquid expanded to liquid condensed state was established by Brewster angle microscopic studies. Stability of the hybrid vesicles, formed by AASM+SPC+CHOL, were established by monitoring their size, zeta potential and polydispersity index values over 100 days. Size and spherical morphology of hybrid vesicles were confirmed by transmission electron microscopic studies. Biocompatibility of the hybrid vesicles were established by cytotoxicity studies revealing their possible applications in drug delivery and imaging.

Rebound Exercises in Rehabilitation: A Scoping Review.

The term "trampoline" was coined in 1969, introducing a dynamic feedback mechanism for exercise. Rebounding exercise on a mini-trampoline utilizes an elastic surface supported by springs and gravity, potentially reducing cumulative trauma from repetitive loading. This type of physical activity provides enjoyable and engaging exercise for adolescents, especially those who are overweight, thereby reducing the likelihood of injuries associated with exercise. Mini-trampoline exercises enhance blood circulation, oxygen delivery, and bone health, impacting lower limb strength, balance, motor performance, blood glucose levels, executive function, physiological markers, and overall quality of life. The study focused on examining the overall impact of rebounding exercises in the field of rehabilitation. Its main goal was to assess how these exercises affect the rehabilitation process and different health measures. By investigating the comprehensive influence of rebounding exercises, the study aimed to determine their effectiveness in aiding physical and functional recovery, targeting specific rehabilitation goals, and enhancing overall health outcomes. We systematically reviewed medical literature databases such as PubMed, MEDLINE, Scopus, Google Scholar, and EBSCO. We included research articles, systematic reviews, meta-analyses, clinical trials, case studies, and observational studies published in English up to 10 years before the review's cutoff in December 2023. We considered participants across all age groups. Articles not in English were excluded from the review. The outcome measures were body composition, waist-hip ratio, Bruininks-Oseretsky test for motor proficiency, reaction time, insulin resistance, lipid profile, blood cholesterol level, forced expiratory volume in one second, and forced vital capacity, bone health indicators, blood lactate level, balance, strength: repetitive maximum, brief pain inventory (short form). A total of 11 reports met these criteria. In conclusion, this review provides a thorough look into the use, challenges, and future potential of rebound exercises in rehabilitation and fitness. Despite their wide-ranging applications, issues such as insufficient research, equipment variability, and safety concerns persist. Advancement requires more research for evidence-based guidelines, improved equipment design and safety measures, and collaboration among researchers, clinicians, and manufacturers. Overcoming challenges and fostering innovation can establish rebound exercises as a valuable tool in rehabilitation and fitness.

Influence of mass conservation cavitation boundary on transient performance of water-lubricated bearings

Given the current research gap regarding cavitation phenomena and startup conditions in water-lubricated bearings (WLBs), this study offers an innovative approach by integrating the mass conservation boundary condition proposed by Jakobsson, Floberg, and Olsson (known as the JFO boundary condition) with a transient mixed lubrication model of WLBs. It also considers the impact of elastic deformation and surface roughness peaks, thus establishing a transient startup model of WLBs that incorporates the cavitation effect. Furthermore, the dynamic behaviors of this boundary condition are contrasted with the commonly used Reynolds boundary condition during the startup process of the bearings, as well as under eccentric and step load impacts. The findings demonstrate that the JFO boundary condition, accounting for cavitation effects, significantly influences the transient performance of WLBs. The presence of a cavitation region diminishes the damping of the bearing system following step load impacts, leading to increased overshoot and adjustment time while weakening the system's self-adjusting ability against step loads. This study offers valuable insights for theoretical analysis of ship propulsion systems operating under complex conditions.