Rayleigh Dissipation Research Articles

Geometrical description and Faddeev-Jackiw quantization of electrical networks

In lumped-element electrical circuit theory, the problem of solving Maxwell's equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.

Dynamic modeling and analysis of viscoelastic hard-magnetic soft actuators with thermal effects

Hard-magnetic soft materials (HMSMs) are a class of magnetoactive smart polymers capable of sustaining a high residual magnetic flux density and can undergo large actuation strains under an external magnetic excitation. Due to these exceptional characteristics, soft actuators based on HMSMs hold enormous potential for remote-controlled applications. The temperature and viscoelasticity considerably influence the performance of these materials during the operation. This work aims to develop an analytical framework for modeling the dynamic behavior of a planar hard-magnetic soft actuators (HMSA) considering temperature and viscoelastic effects. The constitutive behavior of the viscoelastic HMSA is described by employing an incompressible neo-Hookean model in conjunction with a Zener rheological model and the Rayleigh dissipation function. The dynamic governing differential equations of motion are derived by utilizing the non-conservative form of the Euler–Lagrange equation. This study delves into the collective influence of temperature and viscoelastic properties on the stability, periodicity, and resonance characteristics of nonlinear vibrations exhibited by HMSM-based planar actuator subjected to dynamic magnetic loading, presenting the findings through time-history responses, Poincaré maps, and phase-plane plots. The presented results can help in the efficient and robust design of HMSM-based actuators and can also serve as an initial step toward the development of advanced actuators exposed to dynamic loading under variable temperatures for diverse applications in the fields of engineering and medicine.

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

Finger flow modeling in snow porous media based on lagrangian mechanics

Practical formulation for finger flow is lacking even though such a non-Darcy flow is commonly observed in homogeneous snow porous media. This study generalized physics for porous media flow based on the Lagrangian Mechanics; for instance, Darcy formula was theoretically derived minimizing the energy loss by solving the Euler-Lagrange Equation with Rayleigh dissipation function. This least energy loss (LEL) principle was then used to elucidate the finger flow formation in homogenous media. It was found that the inactive saturation (non-contributing void fraction) plays an important role in finger flow formation. This new theory also linked Darcy-Richards capillary flow and Newtonian viscous flow theories. Predicted flow area fractions by the LEL model were verified by the cold room laboratory experiment using well-controlled, homogeneous snow sample, as performed and documented by Katsushima et al. (2013).

Nonlinear Vibration and Dynamic Buckling of Complex Curved Functionally Graded Graphene Panels Reinforced with Inclined Stiffeners

This paper presents a new semi-analytical approach for the nonlinear vibration and dynamic responses of functionally graded graphene platelet-reinforced composite (FG-GPLRC) panels with complex curvature reinforced by orthogonal and/or inclined stiffeners in the thermal environment resting on the nonlinear viscoelastic foundation with the nonlinearities of von Kármán in the framework of the higher-order shear deformation theory (HSDT). The three curvature types of complexly curved panels are considered to be cylindrical, sinusoid, and parabola panels. Orthogonal and/or inclined stiffeners are modeled utilizing an improved smeared stiffener technique. The stress function form is estimated by applying the like-Galerkin method for complexly curved panels. By applying the Lagrange function and Euler–Lagrange equation, the nonlinear equations of motion of the panels are obtained. The viscous damping effects of the viscoelastic foundation are considered by utilizing the Rayleigh dissipation function. Numerical results are examined utilizing the Runge–Kutta method to acquire the time-deflection curves, and by utilizing the Budiansky–Roth criterion, the critical dynamic buckling loads are determined. From the investigated results, it is possible to evaluate the nonlinear vibration and buckling dynamic responses of three forms of panels with stiffeners.

Nonlinear vibration and dynamic buckling responses of stiffened functionally graded graphene‐reinforced cylindrical, parabolic, and sinusoid panels using the higher‐order shear deformation theory

AbstractThe nonlinear vibration and dynamic responses of functionally graded graphene‐reinforced composite (FG‐GRC) laminated cylindrical, parabolic, and sinusoid panels stiffened by FG‐GRC stiffeners in the uniformly distributed temperature variation are presented in this paper. An improved smeared stiffener technique is used to model the added stiffnesses of stiffeners to the total stiffnesses of panels. The higher‐order shear deformation shell theory (HSDT) with the geometrical nonlinearities of von Kármán is applied to establish the governing formulations. The stress function form is estimated using the approximated technique for complex curvature panels. Lagrange function and Euler‐Lagrange equations are applied, and the Rayleigh dissipation function is taken into account to obtain the nonlinear equation of motion. Numerical examples are investigated using the Runge‐Kutta method to obtain the dynamic responses of panels, and the critical dynamic buckling loads of panels are considered using the Budiansky‐Roth criterion. Some significant remarks on the nonlinear vibration and dynamic buckling responses of three types of stiffened panels can be recognized from the numerical examples.

Dual Quaternions Representation of Lagrange's Dynamic Equations

Abstract This paper introduces for the first time, the Lagrange's dynamic equations in dual number quaternion form. Additionally, Rayleigh's dissipation function in dual quaternion form is introduced here allowing for the accounting of dissipative (non-conservative) forces such as motion through a viscous fluid, friction, and spring damping force. As an example, dual quaternions are used here to derive the Lagrange dynamic equations of a robot manipulator.

Viscoelastic Effects on the Nonlinear Oscillations of Hard-Magnetic Soft Actuators

AbstractThe hard-magnetic soft materials (HMSMs) belong to the magnetoactive category of smart polymers that undergo large actuation strain under an externally applied magnetic field and can sustain a high residual magnetic flux density. Because of these remarkable characteristics, HMSMs are promising candidates for the remotely controlled actuators. The magnetic actuation behavior of the hard-magnetic soft actuators (HMSAs) is considerably affected by the viscoelastic material behavior of HMSMs. In this article, we aim at developing an analytical dynamic model of a typical planar model of HMSAs concerning the viscoelasticity of HMSMs. A Zener rheological model in conjunction with an incompressible neo-Hookean model of hyperelasticity and Rayleigh dissipation function is employed for defining the constitutive behavior of the viscoelastic HMSA. The governing equations of dynamic motion are deduced by implementing the nonconservative form of the Euler–Lagrange equation. The established dynamic model is utilized for providing preliminary insights pertaining to the effect of the viscoelasticity on the nonlinear oscillations of the actuator. The phase–plane portraits, Poincaré maps, and the time–history response are plotted to investigate the stability, resonant behavior, and periodicity of the actuator. The results and inferences reported here should provide the initial step toward the design and the development of modern actuators for diverse futuristic applications in the medical and engineering fields.

Vortex Dynamics in a Hybrid Aligned Nematic Microvolume with an Orientational Defect

The aim of this theoretical paper is to investigate the physical mechanism responsible for the appearance of vortex flow in a hybrid aligned nematic (HAN) microvolume with an orientational defect, excited by a temperature gradient ∇T. This was done in the framework of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation. We have carried out a numerical study of the system of hydrodynamic equations including director reorientation, fluid flow v, and the temperature redistribution across the HAN microvolume under the influence of ∇T, when the HAN microvolume is heated from above. Calculations show that, due to the interaction between the gradient of the director field ∇n^ and ∇T, the HAN microvolume settles down to a stationary complex vortex flow regime.

Energy Consumption Analysis of a Rolling Mechanism Based on a Five-Bow-Shaped-Bar Linkage

To reveal the relationship between the center of mass (CoM) trajectory of a closed five-bow-shaped-bar linkage and its energy consumption, this paper presents a trajectory planning method based on the workspace of the CoM. Using different height points located on the symmetric centerline of the workspace of the CoM as via points, the CoM trajectory is planned by combining cubic polynomials with Bézier curves based on quadratic Bernstein polynomials. Herein, the system energy consumption is obtained by integrating the product of generalized velocity and generalized force versus time, where the generalized force is calculated by Lagrange’s equation including the Rayleigh dissipation function. Then, two schemes of dynamic rolling are proposed to compare, and the theoretical results show that the system consumes less energy under the sinusoid scheme when the via point height is lower and the via point of higher height is more suitable under the modified trapezoidal curve scheme. Furthermore, this paper combines the locomotion simulation software to design the locomotion of the mechanism’s CoM trajectory under two schemes in detail and verifies the correctness of the theoretical results.

A continuum model based on Rayleigh dissipation functions to describe a Coulomb-type constitutive law for internal friction in woven fabrics

A continuum model intended to provide predictions for the response of a woven fabric that includes the effects of friction between fibers is proposed. Specifically, we consider a macroscopic formulation in which the fabric weave is composed of two orthogonal families of continuously distributed yarns. The elastic behavior of the planar fabric is characterized by a second-gradient formulation, incorporating the capacity of the fibers to resist a bending deformation. Particular care is devoted to modeling the action of preventing fiber overlapping through a potential energy barrier. The frictional sliding effect of warp threads interwoven with the weft yarns is introduced through a Rayleigh dissipative function that can be appropriately shaped to consider a Coulomb-type law. Spinning friction of yarns belonging to different families also is conceived when a relative rotation between fibers is present to generalize the dissipation phenomenon involved in the considered sheet. Numerical simulations of the proposed model are provided and discussed.

High-Precision Angular Speed Tracking Control of Gimbal System With Harmonic Reducer

In this article, a composite controller based on an extended state observer (ESO) and sliding-mode control (SMC) is presented in order to deal with the angular speed fluctuation caused by the kinematic error of a harmonic reducer in the gimbal system of a control moment gyro. First, based on the analysis of the kinematic error derived from the relationship between the angular speed of the motor and that of the load, a second-order gimbal system model is established based on the Lagrange function and the Rayleigh dissipation function. Second, a three-order linear ESO is designed to observe the uncertain disturbances of the gimbal system, and the disturbances are rejected via the proposed composite controller combining the ESO with SMC. Third, we prove the overall system stability by using the Lyapunov approach and the final value theorem. Finally, both simulation and experimental results verify the effectiveness of the proposed method in attenuating gimbal system disturbances.

Global asymptotic stability of input-saturated one degree-of-freedom Euler–Lagrange systems with Rayleigh dissipation under nonlinear control

ABSTRACT In this manuscript, the regulation of one degree-of-freedom Euler–Lagrange systems subject to input saturation is addressed. In particular, the design and analysis of a nonlinear static state feedback controller is presented. As a result, it is proven via Lyapunov's direct method that, in the presence of Rayleigh dissipation, the closed-loop equilibrium point is globally asymptotically stable with a strict Lyapunov function. Since saturation occurs in the system which contains the actuator model, the proposed control law is unconstrained and can be simplified to a proportional-derivative with desired gravity compensation algorithm. As a by-product global asymptotic stability is also proven for the case where Rayleigh dissipation is null. Numerical simulations on a crank-slider mechanism are presented. Moreover, experimental results on a DC-DC buck power converter are also shown and confirm the viability of our approach.

Dynamic analysis of flexible-link manipulator in underwater applications using Gibbs-Appell formulations

The dynamic model of a flexible robotic chain by considering its motion in a fluid medium is developed in this study. The use of flexible manipulators in fluid applications offers several advantages, but the fluid-manipulator interaction forces, as well as the elasticity of each link, complicates the modeling process. To tackle this issue, the dynamic effects of the fluid on the performance of flexible arms are identified and expressed mathematically so that the resulting interaction during the robot motion can be modeled simultaneously. In this regard, the dynamic interaction between the links and the fluid as a result of link flexibility is modeled as a distributed load along the arm length. In addition to the structural damping of the links, the effect of fluid damping along with its added mass affects the performance of the considered robotic system. Thus, compared to a flexible robotic arm in the air or a rigid robotic arm in the fluid, the flexible deformation of the robot links during motion is affected by the fluid flow. The recursive Gibbs-Apple formulation is used to derive the equations of motion of an N-link robotic chain. Not only does this formulation reduce the computational complexity compared to similar algorithms, but it also allows us to consider the effect of forces exerted on the joints and robot links in the form of the Rayleigh dissipation function. The final equations of motion are simulated using MATLAB for a two-link flexible arm. Simulations of robot links are conducted for different elasticities, different initial conditions, and different media with different viscosities. According to the obtained results, the deformation of flexible manipulators decreases from 5 mm (in the air) to 3 mm (in the fluid), while the displacement in water is 10% of that in the air under similar conditions.

Surfactant-dependent contact line dynamics and droplet spreading on textured substrates: Derivations and computations

We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. Our derivations base on Onsager’s principle with Rayleigh dissipation functionals for either the viscous flow inside droplets or the motion by mean curvature of the capillary surface. We also prove the Rayleigh dissipation functional for viscous flow case is stronger than the one for the motion by mean curvature. After incorporating the textured substrate profile, we design a numerical scheme based on unconditionally stable explicit boundary updates and moving grids, which enable efficient computations for many challenging examples showing significant impacts of the surfactant to the deformation of droplets.

Nanofluidics of nematic liquid crystals in hollow capillaries.

The aim of this paper is to investigate the response of a homogeneously aligned nematic nanosized hollow cavity (HANNHC) confined between two charged horizontal coaxial cylinders and subjected to both a radially applied electrostatic field E, arising from the surface charge density κ and the temperature gradient ∇T set between these cylinders. This was done within the framework of an extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation. The physical mechanism responsible for the excitation of the hydrodynamic flow in the HANNHC is based on the interaction of the director and temperature gradients and the static electric field. Calculations show that under the influence of both the ∇T and E, a stationary flow u^{st} is excited in the HANNHC in the horizontal direction. It is shown that the electric force enforced by the flexoelectric polarization plays a crucial role in the excitation of u^{st} between these cylinders.

Shear-Driven Mechanism of Temperature Gradient Formation in Microfluidic Nematic Devices: Theory and Numerical Studies

The purpose of this paper is to show some routes in describing the mechanism responsible for the formation of the temperature difference ΔT at the boundaries of the microfluidic hybrid aligned nematic (HAN) channel, initially equal to zero, if one sets up the stationary hydrodynamic flow vst or under the effect of an externally applied shear stress (SS) to the bounding surfaces. Calculations based on the nonlinear extension of the classical Ericksen–Leslie theory, supplemented by thermomechanical correction of the SS σzx and Rayleigh dissipation function while accounting for the entropy balance equation, show that the ΔT is proportional to the heat flux q across the HAN channel and grows by up to several degrees under the influence of the externally applied SS. The role of vst=ust(z)i^ with a sharp triangular profile ust(z) across the HAN in the production of the highest ΔT is also investigated.

Observability, controllability and stability analysis of discrete time engineering dynamic systems by means of Lagrangian, Hamiltonian and dissipative functions in discrete forms

Purpose The purpose of this paper is to analyze the dynamical state of a discrete time engineering/physical dynamic system. The analysis is performed based on observability, controllability and stability first using difference equations of generalized motion obtained through discrete time equations of dissipative generalized motion derived from discrete Lagrange-dissipative model [{L,D}-model] for short of a discrete time observed dynamic system. As a next step, the same system has also been analyzed related to observability, controllability and stability concepts but this time using discrete dissipative canonical equations derived from a discrete Hamiltonian system together with discrete generalized velocity proportional Rayleigh dissipation function. The methods have been applied to a coupled (electromechanical) example in different formulation types. Design/methodology/approach An observability, controllability and stability analysis of a discrete time observed dynamic system using discrete equations of generalized motion obtained through discrete {L,D}-model and discrete dissipative canonical equations obtained through discrete Hamiltonian together with discrete generalized velocity proportional Rayleigh dissipation function. Findings The related analysis can be carried out easily depending on the values of classical elements. Originality/value Discrete equations of generalized motion and discrete dissipative canonical equations obtained by discrete Lagrangian and discrete Hamiltonian, respectively, together with velocity proportional discrete dissipative function are used to analyze a discrete time observed engineering system by means of observability, controllability and stability using state variable theory and in the method proposed, the physical quantities do not need to be converted one to another.

Two shear driven flow regimes in microfluidic nematic devices: Tumbling and laminar

The purpose of this study is to show how the tangential component of the shear stress (SS) σzx applied to the boundary of the microsized nematic channel containing a temperature gradient ∇T effects the character of director field n̂ evolution to its stationary orientation n̂st. Calculations based on extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the stress tensor and Rayleigh dissipation function, with accounting the entropy balance equation, show that due to coupling among the SS σzx,∇T, and ∇n̂ in confined hybrid aligned nematic (HAN) channel the horizontal nematic flow v may be excited. Calculations also show that there is a difference in the evolution of n̂ to its stationary orientation n̂st, in the “laminar” case of confined nematic phase, composed of 4-cyano-4′-pentylbiphenyl(5CB) molecules, when -γ2/γ1>1, and in the “tumbling” case of confined nematic phase, composed of 4-cyano-4′-octylbiphenyl(8CB) molecules, when -γ2/γ1<1, respectively. It has been also shown the possible way how to set up a temperature difference ΔT=Tup-Tlw across the HAN microfluidic channel, initially equal to zero, under the action of σzx applied to the boundary of the HAN channel.

Energy dissipation and behavioral regimes in an autonomous double pendulum subjected to viscous and dry friction damping

The nonlinear dynamics of a system of a compound double pendulum are experimentally analyzed and numerically verified. The experiments are carried out for a combination of initial angles in the range 0 ⩽ θ 0, ϕ 0 ⩽ 180°, with zero initial angular velocity and with an increment of 1°. The motion of the pendula is captured in the video and the individual pendula are tracked using image processing. The energy dissipation in the system is evaluated using the Rayleigh dissipation function to include non-conservative forces as well, as a lumped parameter in the Lagrangian equation. Furthermore, the experimental and the numerical results are brought to agreement by tuning the dissipation factors. The linear, nonlinear and chaotic behavior of the system is classified in the coordinate plane. The behavioral regime represents the dependency of the nature of the system on the total energy and the difference in instantaneous amplitudes of the pendula. Uncertainty in the behavior of the system is observed in the transition region from the nonlinear to the chaotic regime due to the sensitivity of the system to the initial conditions. The dependence of the dissipation factors on the angular displacements is represented as a surface plot, which highlights the dominance of viscosity in the higher amplitude oscillations and the dominance of dry friction in the lower amplitude oscillations. This phenomenon is established by observing the nature of decay in the response plots of the pendula.