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The flow of a compressible fluid with slip through a cylinder with an asymmetric local constriction has been considered both numerically, as well as analytically. For the numerical work, a particle-based method whose dynamics is governed by the multiparticle collision (MPC) rule has been used together with a generalized boundary condition that allows for slip at the wall. Since it is well known that an MPC system corresponds to an ideal gas and behaves like a compressible, viscous flow on average, an approximate analytical solution has been derived from the compressible Navier–Stokes equations of motion coupled to an ideal gas equation of state using the Karman–Pohlhausen method. The constriction is assumed to have a polynomial form, and the location of maximum constriction is varied throughout the constricted portion of the cylinder. Results for centerline densities and centerline velocities have been compared for various Reynolds numbers, Mach numbers, wall slip values and flow geometries.

The analysis of motion equations of flexible-link manipulators indicates that nonlinear terms and behavioral instability exist in their outcomes. In this system, when the differential equations possess structural time-dependent parameters, there is a higher chance for system instability, especially chaos in comparison to time-invariant ones. It is a point for abnormal behavior that is unpredictable and always accompanied by damages. Therefore, the physical parameters and system inputs that cause chaotic responses should be detected and prevented as much as possible. To this aim, the dynamic equations of flexible link manipulator with revolute-prismatic joints as a case of time-variant dynamic system are obtained by Hamilton’s principle with the assumption of nonlinear strains. The discretized motion equations are solved and the results are presented in the forms of bifurcation diagrams (for variation of torque/force amplitude), Poincare maps, phase-plane portraits, and the largest Lyapunov exponent. Finally, the obtained results are validated with the aid of the experimental setup. The results indicate that although the system response in low torque/force amplitudes is subharmonic (amplitude of 0.01) and quasi-periodic (amplitudes of 0.02-0.03), it becomes quasi-periodic/chaotic with a mild slope with respect to time in high amplitude values. Moreover, since the behavior of modal generalized coordinate is different from the rigid ones, the quasi-periodic and chaotic vibrations do not hurt the joints.

The quasi-steady translational motion of an incompressible couple stress fluid confined between two eccentric spherical surfaces is investigated. It is assumed that the motion is generated by allowing the internal sphere to translate with a constant velocity. The Stokesian assumption of low Reynolds numbers is considered so that the nonlinear terms of the equation of motion are neglected. The superposition principle is employed to construct the general solution of the governing equations in the presence of spherical surfaces using two spherical systems of coordinates relative to the centers of the two spherical surfaces. The no-slip boundary conditions on the spherical boundaries are applied. In addition, the conditions of vanishing couple stresses on the bounding surfaces are imposed. The boundary collocation technique is utilized numerically to satisfy the boundary conditions on the spherical surfaces. The drag force experienced by the fluid flow on the spherical particle is obtained and represented numerically through table and graphs. The numerical results show that the values of the drag force increases monotonically with the increase in the radii ratio. It increases also with the increase of the couple stress viscosity parameter. The obtained results are consistence with the results available in the literature for viscous fluids.

For general frame structures subjected to periodic loads, the motion equation of structures is generally expressed as a nonhomogeneous Mathieu-Hill equation in the form of matrices. The parametric resonance stabilities of these structures are generally determined by calculating the Lyapunov exponents (or the energy-growth exponents) of structural time-history responses. The process of stability analysis is complicated and time-consuming. This paper proposes a new (simplified) method to solve dynamic stability problems. The concept of the dynamic axial-force transfer coefficient is first presented. A new simplified formula is derived to calculate the parametric resonance stability boundaries of the frame structures by using the dynamic axial-force transfer coefficient. In the special cases in which the first static (elastic) buckling mode of the frame structure is approximately equal to the first vibration mode, the stability boundaries of the parametric resonances can be more easily obtained with the aid of commercial software. The stability boundaries calculated by the present method are in good agreement with the results of the existing numerical method and experiment. The present method is simpler and more effective than the existing methods for stability analyses of the parametric resonance and autoparametric resonance of general frame structures.

Nonholonomic mechanics, which relies on the method of Lagrange multipliers in the d’Alembert–Lagrange formulation of the classical equations of motion, is the standard description of the dynamics of systems subject to nonholonomic constraints. Vakonomic mechanics has been proposed as another possible approach to the dynamics of nonholonomic systems. Partial equivalence theorems have been found that provide conditions under which nonholonomic motions are also vakonomic motions. The aim of the present work is to show that these results do not cover at least one important physical system, whose motion brought about by vakonomic mechanics is completely inequivalent to the one derived from nonholonomic mechanics. For the rolling coin on an inclined plane, it is proved that no nontrivial solution to the equations of motion of nonholonomic mechanics can be obtained in the framework of vakonomic mechanics. This completes previous investigations that managed to show only that, for certain mechanical systems, some but not necessarily all nonholonomic motions are beyond the reach of vakonomic mechanics. Furthermore, it is argued that a simple qualitative experiment that anyone can perform at home supports the predictions of nonholonomic mechanics.

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

The objective of the paper is to study the dynamic characteristics of a repulsion mechanism applied in superconducting fault current limiter (SFCL). A novel fully coupled simulation was established, which involved modules of the flexible-body dynamics, the external circuits, motion equations, multi-body dynamics, transient electromagnetic fields and fatigue analysis. The dynamic characteristics including the velocity, the displacement curve, discharging current, Lorentz force, the stress and strain of each dynamic transmission part in the opening process, was calculated. The vulnerable parts were focused and their fatigue life was calculated. The deviation about discharging current and displacement curve of the fully-coupled simulation was experimentally validated to be less than 8%. Stress concentration positions of the vulnerable parts recommended by the simulation results are the same as fatigue failure positions in the experimental results, which verified the accuracy of the fully coupled simulation.

Prediction of the response of nonlinear dynamical systems under interacting parametric and external excitations is important in designing systems such as sensors, amplifiers or energy harvesters, to achieve the desired performance. This paper concerns the nonlinear forced Mathieu equation, with linear damping and a 2:1 ratio between the parametric and external excitation frequencies. The Method of Varying Amplitudes (MVA) is employed to derive approximate analytical expressions for the response of the system. Both single-term and double-term solutions are developed: it is seen that, employing the double-term approximation, the MVA can accurately predict the response of the system over a wide range of frequencies and system parameters, showing a maximum of 0.2% deviation from numerical results obtained by direct integration of the equation of motion. This is in contrast with most of the available theoretical approaches such as the conventional Method of Multiple Scales, which can predict the response accurately only for a narrow range of system parameters and excitation frequencies. Furthermore, it is seen that the response is bounded, and analytical expressions for the frequency and amplitude of the upper bound are developed: this is unlike other methods which predict unbounded response, unless nonlinear damping is considered. Analytical expressions for the response are developed, and results are verified with numerical results obtained from direct integration of the equation of motion. Numerical examples are presented, showing good agreement with results obtained by the MVA.

This study is concerned with the hydroelastic vibration of the flexible bottom plate of a cylindrical tank coupled with sloshing. The cylindrical tank is assumed to be placed vertically with the rigid wall. The matrix equations of motion describing the free-surface elevation coupled with the hydroelastic vibration of the bottom plate was derived by using the Lagrangian equation. The so-called non-dimensionalized added virtual increment factors were also calculated and compared to the previous results, and showed exact agreement. The numerical results show that only when the water is shallow and the bottom plate is very flexible, strong coupling exists in the bulging modes and asymmetric modes having one nodal diameter. However, if the fundamental frequency of the plate in contact with fluid is much higher than the fundamental frequency of the sloshing, the coupling between sloshing and the hydroelastic vibration of the bottom plate can be ignored.

The focus of the present paper is on investigating the nonlinear dynamics of the axially moving FG-CNTRC shells with different reinforcement distribution and the scale effects of CNTs in the subcritical regime of axial speed. The governing equations are derived in cylindrical coordinate utilizing the Hamilton principle by implementing the Donnell-Mushtari nonlinear shell theory and considering the mechanical properties of nanocomposite shells obtained from the extended rule of mixture. Two nonlinear coupled nonhomogeneous PDEs, a compatibility equation, and the motion equation in the radial direction are the result of applying in-plane airy stress function and continuity conditions on them. Then by substituting the flexural mode shape in the mentioned equations, the airy stress function is achieved. By the aid of Jordan conical form, the coupling of the second derivative of time in seven nonlinear nonhomogeneous ODEs resulted from applying the Galerkin method on the equilibrium equation in the radial direction is removed. Eventually, these ODEs are transformed into the Normal Form. The bifurcation analysis based on the frequency, the force, the damping ratio, and the velocity are carried out for circular cylindrical nanocomposite shells with four distribution types of SWCNT reinforcement and various volume fraction of CNTs. Four sorts of fixed points, including saddle nodes, pitchfork bifurcation, periodic doubling, and torus, have appeared in outlined parameters’ responses of nanocomposite circular cylindrical shells’ vibration. The Runge Kutta 4th order and pseudo arclength continuation as the numerical methods state the accuracy of the Normal Form Method.

A two-field formulation of the nonlinear dynamics of an elastic body is presented in which positions/orientations and the resulting velocity field are treated as independent. Combining a nonclassical description of elastic velocity that includes the convection velocity due to elastic deformation with floating reference axes minimizing the relative kinetic energy due to elastic deformation provides a fully uncoupled expression of kinetic energy. A transformation inspired by the classical Legendre transformation concept is introduced to develop the motion equations in canonical form. Finite element discretization is achieved using the same shape function sets for elastic displacements and velocities. Specific attention is brought to the discretization of the gyroscopic forces induced by elastic deformation. A model reduction strategy to construct superelement models suitable for flexible multibody dynamics applications is proposed, which fulfills the essential condition of orthogonality between a rigid body and elastic motions. The problem of expressing kinematic connections at superelement boundaries is briefly addressed. Two academic examples have been developed to illustrate some of the concepts presented.

A description of the original mathematical model of spatial oscillations of a four-axle autonomous traction module during its movement along straight and curved sections of the railway track is proposed. In this case, the design of a four-axle autonomous traction module is presented as a complex mechanical system, and the track is considered as an elastic-viscous inertial system. The equations of motion were compiled using the Lagrange method of the ІІ kind. For each of the solids, the kinetic energy is determined by the Koenig theorem. The potential energy component is obtained by the Clapeyron theorem, as the sum of the energies accumulated in the elastic elements of the system during their deformations. The dissipative energy was also taken into account when compiling the equations of motion. Generalized forces that have no potential, in this case, include the forces of interaction between wheels and rails, which are determined using the creep hypothesis. It is important to note that the model takes into account the forces in the bonds between the bodies of the system and the spatial displacements of the rigid bodies of the mechanical system, taking into account possible restrictions. Moreover, the mathematical model developed by the author is a system of differential equations of the 100th order. This mathematical model is the base for further theoretical studies of the dynamics of railway four-axle autonomous traction modules in single motion or when moving as part of a train. To solve the resulting system of differential equations, the author develops special software that allows for complex theoretical studies of spatial oscillations of a four-axle autonomous tractionmodule to determine the indicators of its dynamic loading and traffic safety.

The Lyapunov–Floquet (L–F) theory, a powerful result of Floquet theory, is one of the main tools for the stability analysis and study of nonlinear periodic systems. In this paper, Lyapunov–Floquet and modal transformations of the nonlinear approximation model of spacecraft relative motion, containing periodic state dependent coefficients (SDC), are developed. The transformations, advantageously, retain the stability characteristics of the original nonlinear relative motion dynamics. To provide a form to which averaging method can be applied, the nonlinear approximated equation of motion is scaled using the chief’s eccentricity. Through the application of L–F transformation, the linear part with periodic coefficients is reduced to time-invariant form and the transformed periodic nonlinear part has coefficients with true anomaly as the independent variable. The modal transformation reduced LF transformed equation into a form containing Jordan canonical form. Afterward, the averaging method is applied to the L–F transformed equation to describe the evolution of the motion in terms of the average values of the dynamical variables. Both the LF and modal transformed relative motion equations and the averaged equations, with chief in elliptical orbit, present forms that are useful and applicable for spacecraft guidance, navigation and control, formation flying, rendezvous and proximity operations.

The non-relativistic equations of motion for a dipole in a stationary non-homogeneous electromagnetic field are derived and analysed. It is shown that they are Hamiltonian with respect to a certain degenerated Poisson structure. Described by them dynamics is complex because the motion of the centre of mass of the dipole is coupled with its rotational motion. The problem of the existence of linear in momenta first integrals which can be useful for the separation of rotational motion is discussed. The presence of such first integral appears to be related with a linear symmetry of electric and magnetic fields. Also results of search of quadratic in momenta first integrals for uniform and stationary electromagnetic fields are reported. Deriving equations of motion of a dipole in arbitrary stationary electromagnetic fields and analysis of described by them dynamics is important for the construction of electromagnetic traps for polar particles.

Variation in the shape of a magnetizable liquid drop suspended in another immiscible magnetizable liquid in the presence of a non-stationary uniform magnetic field is theoretically investigated in the case of high Reynolds numbers. The system of equations consists of the continuity equation for incompressible liquid, the equation of motion for an ideal incompressible liquid, and Maxwell’s equations in the quasi-stationary and ferrohydrodynamic approximations. To solve the problem, the representation of the magnetic field strength and the liquid velocity in form of a multipole expansion expressed in terms of irreducible tensors is used. In such an approach the magnetic field strength and the velocity can be sought for in the form of series with vector and tensor coefficients for which some relations are obtained that make it possible to determine the coefficients. Using these relations, the coefficients are sought in the form of asymptotic expansions in a parameter whose smallness ensures small deformations of the drop. The flow velocity, the magnetic field strength, and the shape of drop are found correct to terms of the first order with respect to the small parameter. The problems of forced and natural drop oscillations generated by switching on instantaneously-applied harmonically oscillating and rotating magnetic fields are solved.

Planetary gearboxes are widely used in the aerospace, automotive and renewable energy sectors. These gearboxes are the critical component in a system's reliability; hence, monitoring this component is essential. To study the vibration characteristics, a novel dynamic modelling approach using Lagrange's equations is presented in this paper. Newtonian equations of motion have been developed and solved using Lagrange's theorem, and modal analysis has been performed to estimate the dynamic characteristics of a single-stage planetary gearbox. The torsional stiffness and time-varying mesh stiffness have been considered in this paper for dynamic modelling. This study applies the perturbation method to solve the conditions of the system. The significant resonance frequencies of the individual components in the gearbox have been identified using coordinates based on eigenvectors. The finite-element analysis results have been considered for validation purposes and compared to the numerical model. The frequency range of the gearbox components' resonances and dynamic characteristics have been obtained from the FEA. The study shows that Lagrange's dynamic modes match with modes obtained from the finite-element modal analysis. In addition, the resonance frequencies produced by the sun and planet gears and the bearings are detected. Therefore, the results show a positive potential in gearbox fault diagnostics and characteristics frequencies studies for individual components in the gearbox.

The main target of this paper is to discuss the impacts of a vertical alternating current of electric field as well as the Hall current on peristaltic flow of an Eyring–Powell fluid. The streaming occurs via a uniform and non-uniform annulus having mild stenosis. The actions of radiative heat transfer and chemical reactions are imposed. The slip condition is applied for the velocity distribution. Meantime, the convective conditions are selected to describe the heat and mass transfer. Both the impacts of viscous dissipation and radiation in energy expression are assumed. Soret and Dufour feature yield the coupled differential systems. The presumptions of the long-wavelength and low Reynolds number are employed to approximate the governing equations of motion. The analytical solutions of these equations are based on utilizing the Homotopy perturbation technique. The effects of various physical parameters of the problem are discussed and drawn graphically via a set of figures. It may be noticed that the axial velocity increases with the reduction of electromagnetic parameters and Hartmann number. However, both of the Hall current parameter and the electrical Rayleigh number boost the fluid velocity. As well, it is observed that the Brickmann number and Dufour number give rise to the fluid temperature. Meanwhile, reverse effect is detected towards concentration for both of Schmidt number and chemical reaction parameter. The dual role effects of both the adverse temperature and the electrical Rayleigh number have been revealed for the electric potential distribution. The present consideration is very substantial in many medical implementations, such as the depiction of the gastric juice motion in the small intestine when an endoscope is inserted through it.

In this study, we present explicit equations of motion for general mechanical systems exposed to holonomic and nonholonomic constraints based on the Gibbs-Appell formulation. Without constructing the Gibbs function, the proposed method provides a minimal set of first-order dynamic equations applicable for multibody constrained systems. Transforming the Newton–Euler equations from physical coordinates to quasivelocity spaces eliminate constraint reaction forces from motion equations. In this study, we develop a general procedure to select effective quasivelocities, which indicate that the proposed quasivelocities satisfy constraints, eliminate Lagrange multipliers, and reduce the number of dynamic equations to degrees of freedom. Besides, we test the validity and efficiency of the proposed approach using three constrained dynamical systems as illustrative examples. Finally, we compare the simulation results with Udwadia–Kalaba, augmented Lagrangian, and other conventional methods.

This paper discusses four different approaches that can be followed to derive the equations of motion for a fixed and symmetrical spinning top. Starting from the usual Euler equations in the body-fixed system, after manipulation it is shown that identical equations are derived for the space-fixe system as well. All the three Cartesian components of the angular momentum vector are calculated for both the body- and the space-systems and they are formulated so that they can be used for further numerical analysis. In addition to the classical set, the Euler equations are also easily derived using a rotating system originated at the pivot but not spinning. Moreover, Lagrange equations are derived and the latter are proven to be equivalent with the Euler equations. The best way among these four methods for teaching students is probably the instructor’s preference. Moreover, using commercial software, an adequately accurate numerical solution is derived. Not only the position of the spinning top is calculated but also the support forces at the pivot are predicted

The geometrical structure of constraints leads to the important conclusion that components of velocities and accelerations in the normal space of the constraint hypersurface are totally determined by the constraints. Orthogonal base vectors of the constraint’s normal and tangent spaces can be obtained by the QR decomposition with column permutation. A numerical method of closed-loop multibody systems with singular configurations is presented, in which the traditional hypothesis on independence of constraints is abandoned. Instead of correcting constraint violations at the end of each integration step, positions and velocities are modified to satisfy their constraint equations before they are used to form equations of motion. Such an approach can collaborate with any standard ODE solver. In order to systematically generate constraint equations and obtain the corresponding joint’s reaction forces, rotational and translational constraints of a closed-loop are standardized as part of six explicit equations, and a clear relationship between Lagrange multipliers and joint’s reaction forces is derived based on the principle of virtual power equivalence. The proposed method can dynamically identify independent constraints and modify the equations of motion accordingly. The numerical examples validated its effectiveness.

The problem of the passage of a monoharmonic plane sound wave through a thin composite rectangular plate hingly supported in the aperture of an absolutely rigid partition has been considered. In this work, two-dimensional equations of motion constructed by reduction of three-dimensional equations were used. These equations were based on a discrete structural model of multilayer plates deformation at small displacements and deformations. Additionally there were taken into account the internal friction of layer materials according to the Kelvin–Voigt model. The behavior of acoustic media is described by the generalized Helmholtz wave equations, composed with the introduction of the complex sound velocity according to Skudrzyk to take into account the dissipation of energy. An exact analytical solution of the formulated problem has been obtained. On this basis, this work studies the insulation properties and the stress-strain parameters of a composite plate reinforced with carbon textile depending on the incident sound wave frequency. In addition to this, the laws that represent the change in the sound pressure in an acoustic medium with a distance from the plate were established.

Parametric vibrations of the single-layered graphene sheet (SLGS) are studied in the presented work. The equations of motion govern geometrically nonlinear oscillations. The appearance of small effects is analysed due to the application of the nonlocal elasticity theory. The approach is developed for rectangular simply supported small-scale plate and it employs the Bubnov–Galerkin method with a double mode model, which reduces the problem to investigation of the system of the second-order ordinary differential equations (ODEs). The dynamic behaviour of the micro/nanoplate with varying excitation parameter is analysed to determine the chaotic regimes. As well the influence of small-scale effects to change the nature of vibrations is studied. The bifurcation diagrams, phase plots, Poincaré sections and the largest Lyapunov exponent are constructed and analysed. It is established that the use of nonlocal equations in the dynamic analysis of graphene sheets leads to a significant alteration in the character of oscillations, including the appearance of chaotic attractors.

This paper presents a method for the end effector motion control of a spatial three-link robot having elastic second and third links including measurement noises. In the derivation of equations of motion, not to face with complex equations of motion, each link is modeled as though the links are not connected and the restrictions on the links due to connecting them by joints are written as constraint equations. After that the Lagrange multipliers are eliminated and the constraint equations at the acceleration level are substituted into the equations of motion to reduce the number of equations. To handle the non-minimum phase property, the equations of motion of the elastic manipulator are divided as the equations corresponding to a pseudostatic equilibrium and the equations of the deviations from them. Definition of the pseudostatic equilibrium used in this study can be given as a hypothetical state in which the end effector velocity and the end effector acceleration possess their reference values while the elastic deflections are instantly constant. The advantages of this control method are that the elastic deflections and the control inputs required for the pseudostatic equilibrium are obtained by an algebraic method and the feedback stabilization control inputs for the deviation equations are determined without linearizing the dynamic equations. The required measurements are obtained from the strain gauges on the links, the encoders placed on the joints and the position sensors attached to the end effector. For each sensor, a low pass filter is used. Simulations are made with low and high values of crossover frequencies to show the positive and negative effects of filtering on the responses of the system.

We construct a Hamiltonian field theory model that combines hydrodynamics and thermodynamics. In this model, the continuity equation and the Euler equation for a potential flow are used as the Hamilton equations; and equations of state (or the Gibbs equations, which are equivalent to them ), as equations of second-class constraints. The Hamiltonian function density reduced to the surface of second-class constraints is the sum of densities of the kinetic and potential energies; free energy then plays the role of potential energy. The surface of second-class constraints is endowed with a natural symplectic structure. Canonical variables are also defined on the second-class constraint surface such that all physical variables can be expressed as functions of these variables. In particular, from the standpoint of the Hamiltonian formalism, entropy is interpreted as a generalized velocity — a Lagrange multiplier for the corresponding second-class constraint expressing the temperature as a function of the canonical variables at the last reduction stage. This multiplier is expressed in terms of the canonical variables at the last stage, yielding a nontrivial equation of motion for entropy. The model must be made more concrete by fixing the dependence of the specific free energy on its arguments. We choose the simplest nontrivial variant, the monatomic van der Waals gas whose atoms are in the ground state. The canonical Hamilton equations allow calculating the rate of change in the entropy of this dynamic system. For the physically interesting case where the system evolution leads to equilibrium, the entropy and its rate of change are functionals of the solution of dynamic equations for the density. A numerical solution of these equations gives a monotonic growth of the entropy (for a finite evolution time). The equation can be linearized to find the time asymptotics; an elliptic equation with the “wrong” sign of the analogue of the squared speed of sound, rather than a hyperbolic equation, is obtained for the asymptotic evolution of the density deviation from the equilibrium value. Thus, the time reversibility of the solution is lost.

In this study, we investigate the nonlinear resonant response of a magnetorheological (MR) damper–stay cable system with time delay, and the one-to-one internal resonance is considered. Based on Hamilton’s principle, the motion equations of the MR damper–cable system are derived, and the Galerkin method is applied to obtain the discrete model. Then, the method of multiple scales is applied to determine the modulation equations and the second-order solution of the nonlinear response of the MR damper–cable system. Following, the equilibrium solution and dynamic solution of the modulation equations are examined via the Newton–Raphson method and shooting method. The results show that the equilibrium solution may undergo Hopf bifurcation, resulting in the periodic solution. Moreover, the effects of the time delay and the inclination angle on the resonant response of the MR damper–cable system are investigated as well as those of the damper position. The numerical results show that the time delay increases the amplitudes of in-plane and out-of-plane modes and results in the more remarkable hardening behavior and relatively poor mitigation performance of the MR damper. However, the large time delay may suppress the complex chaotic modulation motion of the MR damper–cable system.

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load acts on the other along a rectangular section at the initial moment of time. Integral Laplace and Fourier transforms are applied sequentially to the equations of motion and boundary conditions in contrast to traditional approaches when integral transforms are applied to solutions' representations through harmonic functions. This leads to a one-dimensional vector homogeneous boundary value problem with respect to unknown displacement's transformants. The problem was solved using matrix differential calculus. The original displacement field was found after applying inverse integral transforms. For the case of stationary vibrations a method of calculating integrals in the solution in the near loading zone was indicated. For the analysis of oscillations in a remote zone the asymptotic formulas were constructed. The amplitude of vertical vibrations was investigated depending on the shape of the load section, natural frequencies of vibrations and the material of the medium.

On the basis of all commutation relations of the $$\kappa $$ -deformed phase space incorporating the $$\kappa $$ -Minkowski space-time, we have derived in this paper an extended first approximation of both Maxwell’s equations and Lorentz force in doubly (or deformed) special relativity (DSR). For this purpose, we have used our approach of the special relativistic version of Feynman’s proof by which we have established the explicit formulations of electric and magnetic fields. As in Fock’s nonlinear relativity (FNLR), the laws of electrodynamics depend on the particle mass which therefore constitutes a common point between the two extended forms of special relativity. As one consequence, the corresponding equation of motion contains two different types of contributions. In addition to the usual type, another one emerges as a consequence of the coexistence of mass and charge which are coupled with the $$\kappa $$ -deformation and electromagnetic field. This new effect completely induced by the $$\kappa $$ -deformed phase space is interpreted as the gravitational-type Lorentz force. Unlike FNLR, the corrective terms all depend on the electromagnetic field in DSR.

Based on Lagrangian mechanics, use of velocity constraints as a special set of quasi-velocities helps derive explicit equations of motion. The equations are applicable to holonomic and nonholonomic constrained multibody systems. It is proved that in proposed quasi-spaces, the Lagrange multipliers are eliminated from equations of motion; however, it is possible to compute these multipliers once the equations of motion have been solved. The novelty of this research is employing block matrix inversion to find the analytical relations between the parameters of quasi-velocities and equations of motion. In other words, this research identifies arbitrary submatrices and their effects on equations of motion. Also, the present study aimed to provide appropriate criteria to select arbitrary parameters to avoid singularity, reduce constraints violations, and improve computational efficiency. In order to illustrate the advantage of this approach, the simulation results of a 3-link snake-like robot with nonholonomic constraints and a four-bar mechanism with holonomic constraints are presented. The effectiveness of the proposed approach is demonstrated by comparing the constraints violation at the position and velocity levels, conservation of the total energy, and computational efficiency with those obtained via the traditional methods.

This paper deals with periodic motions and their stability of a flexible connected two-body system with respect to its center of mass in a central Newtonian gravitational field on an elliptical orbit. Equations of motion are derived in a Hamiltonian form, and two periodic solutions as well as the necessary conditions for their existence are acquired. By analyzing linearized equations of perturbed motions, Lyapunov instability domains and domains of stability in the first approximation are obtained. In addition, the third- and fourth-order resonances are investigated in linear stability domains. A constructive algorithm based on a symplectic map is used to calculate the coefficients of the normalized Hamiltonian. Then, a nonlinear stability analysis for two periodic solutions is performed in the third- and fourth-order resonance cases as well as in the nonresonance case.

A rigid-body dynamic model of multi-stage planetary roller screw mechanism (multi-stage PRSM) is proposed in this paper. The structure of multi-stage PRSM is introduced and the motion analysis is presented. The total kinetic energy of the mechanism is calculated. The rotation of the screws and carriers is chosen as generalized degrees of freedom. The generalized forces and motion equations of multi-stage PRSM are derived using the Lagrange method. The transient and steady-state behaviours of multi-stage PRSM are simulated, followed by an analysis of the influence of friction coefficients and thread pitches on the motion and forces acting on the multi-stage PRSM. Taking a two-stage PRSM as an example, the simulation results show that the friction coefficient between screw #1 and screw #2 has a slight effect on efficiency and rotational velocity ratios of carriers to screws. When the sum of the pitches of screws is a constant, the axial component of contact force between screw #1 and roller #1 decreases with the increase in the pitch of screw #1.

Розглядається задача про поширення квазілембових хвиль у попередньо деформованому нестисливому пружному шарі, що взаємодіє з півпростором в'язкої стисливої рідини. Дослідження проводиться на основі тривимірних лінеаризованих рівнянь теорії пружності скінченних деформацій для нестисливого пружного шару і тривимірних лінеаризованих рівнянь Нав’є—Стокса для півпростору в'язкої стисливої рідини. Застосовуються постановка задачі та підхід, засновані на використанні представлень загальних розв’язків лінеаризованих рівнянь для пружного тіла і рідини. Отримано дисперсійне рівняння, яке описує поширення нормальних хвиль у гідропружній системі. Побудовано дисперсійні криві квазілембових хвиль у широкому діапазоні частот. Проаналізовано вплив скінченних початкових деформацій у нестисливому пружному шарі, а також півпростору в'язкої рідини на фазові швидкості, коефіцієнти загасання, дисперсію квазілембових мод і поверхневу нестійкість гідропружного хвилеводу. Числові результати представлені у вигляді графіків і дано їх аналіз. The problem of the propagation of quasi-Lamb waves in a pre-deformed incompressible elastic layer that interacts with the half-space of an viscous compressible fluid is considered. The study is conducted on the basis of the three-dimensional linearized equations of elasticity theory of finite deformations for the incompressible elastic layer and on the basis of the three-dimensional linearized Navier–Stokes equations for the half-space of a viscous compressible fluid. The problem formulation and the approach, which are based on the utilization of representations of the general solutions of the linearized equations for an elastic solid and a fluid, are applied. Next, applying the Fourier method, we arrive at three eigenvalue problems for the equations of motion of the elastic body and the fluid. Solving them, we find the eigenfunctions. Substituting the general solutions into the boundary conditions, we obtain a homogeneous system of linear algebraic equations for the arbitrary constants. From the condition for the existence of a nontrivial solution, we derive the dispersion equation. It describes the propagation of normal waves in the hydroelastic system. The dispersion curves for quasi-Lamb waves over a wide range of frequencies are constructed. The effect of the finite initial deformations in an elastic layer, the thickness of the elastic layer, and the half-space of viscous compressible fluid on the phase velocities, attenuation coefficients, and dispersion of quasi-Lamb modes are analyzed. It follows from the graphical material presented above that, in the case of compression with 0.54, i.e., with a 46–percent reduction in the length of the highly elastic incompressible body, the phase velocities of the surface waves (Stoneley waves and Rayleigh waves) vanish. This indicates that the surface instability develops at 0.54 for a highly elastic incompressible non-Hookean body initially in a plane stress-strain state. We should point out that these figures agree with results obtained earlier in the theory of stability and correspond to the critical value of the con-traction parameter. In the case of highly elastic incompressible bodies, the linearized wave theory makes it possible to study not only general and several specific wave processes, but also the conditions under which the surface instability begins in elastic bodies and hydroelastic systems. It also follows from the graphs that the viscous fluid slightly affects the surface instability of hydroelastic systems. The numerical results are presented in the form of graphs, and their analysis is given.

Using the Fourier transform, integral representations of solutions to boundary value problems of heat conduction and diffusion in a two-phase region with a moving interface are obtained. The proposed approach makes it possible to obtain the equation of motion of the interface without the need to first find the temperature and (or) concentration fields. This makes it possible to study the stability of the interface with respect to disturbances in its shape. The validity of the proposed approach is demonstrated by the example of self-similar growth of a spherical crystal in a supercooled melt and crystallization of the melt on a substrate of the same substance. On the basis of the obtained equation, which determines the rate of self-similar motion of the interface, the features of the kinetics of crystallization of the melt on the substrate are analyzed. The conditions of applicability of the developed approach to the solution of boundary value problems of heat conduction and diffusion in regions separated by a moving boundary are briefly discussed.

The longstanding question of how stochastic behaviour arises from deterministic Hamiltonian dynamics is of great importance, and any truly holistic theory must be capable of describing this transition. In this review, we introduce the influence functional formalism in both the quantum and classical regimes. Using this technique, we demonstrate how irreversible behaviour arises generically from the reduced microscopic dynamics of a system-environment amalgam. The influence functional is then used to rigorously derive stochastic equations of motion from a microscopic Hamiltonian. In this method stochastic terms are not identified heuristically, but instead arise from an exact mapping only available in the path-integral formalism. The interpretability of the individual stochastic trajectories arising from the mapping is also discussed. As a consequence of these results, we are also able to show that the proper classical limit of stochastic quantum dynamics corresponds non-trivially to a generalised Langevin equation derived with the classical influence functional. This provides a further unifying link between open quantum systems and their classical equivalent, highlighting the utility of influence functionals and their potential as a tool in both fundamental and applied research., 36 pages and 6 figures. Significantly expanded text, especially sections on quantum influence functional. Accepted to Eur. Phys. J. Spec. Top

Due to the increasing usage of nanostructures in nanotechnology and nanodevice, the following article aims to investigate the free vibration and buckling behaviours of a Timoshenko functionally graded nanobeam under to thermal and magnetic environment. The nanoscale and microstructure effects of FG nanobeam are included to classical mechanics using nonlocal strain gradient theory. The gradation of material properties throughout the beam thickness is described by power-law function. The material properties are assumed to be temperature dependent. Considering the thermal and Lorentz forces, the equations of motion of the functionally graded Timoshenko nanobeam are obtained using the strain gradient and nonlocal elasticity theories. The transverse Lorentz force induced by the horizontal magnetic field vector is derived using Maxwell’s equations. External compressive axial and transverse point loads are included in the formulation and the motion equations are solved using a Navier-type approach. The effects nonlocal size scale, and strain gradient microstructure influence, thermal loadings and magnetic field intensities on the free vibration, transverse bending and buckling behaviours of the functionally graded nanobeam are presented. The following model can be used as benchmark to analyse the nanobeam structure under thermomagnetic field using a finite element or any other numerical method.

This study is dedicated to investigate the nonlinear dynamics of a system composed of a flexible rotor shaft carrying a longitudinally dispositioned unbalanced rigid disc. The less-attended multi-modal nonlinear vibration analysis of rotor structures, considering the effect of geometric and inertial nonlinearities due to the large deflection of the flexible structure, is investigated in this study. To this end, the flexible shaft is modelled using the Euler–Bernoulli beam theory. In order to model the unbalanced rigid disc, it is considered as combination of a symmetric disc with an additional eccentric lump mass. The equations of motion of the system are derived using the extended Hamilton principle. Assuming the inextensional shaft and considering high twisting rigidity of the shaft, the flexural-flexural vibrations of the system of rotor shaft-disc are studied in this paper. The equations of motion are discretized using the Galerkin method exploiting various reduced order models including single-mode, two-mode, and three-mode approximations. In order to obtain the steady-state response of the system, two methods are used in this study. The first method is to use the semi-analytical modified complex averaging technique along with the numerical arc-length continuation method. The second method is using the direct integration in MATLAB. The nonlinear dynamics of different systems such as a system consisting only a flexible shaft and a system of flexible shaft carrying an unbalanced rigid disc at different positions along the shaft are investigated. The results show that geometric nonlinearity of the flexible shaft is dominating the inertial nonlinearity and hence, the system demonstrates a hardening nonlinear behaviour at the vicinity of the first two resonances. Besides, it is shown that dispositioning the unbalanced disc along the flexible shaft results in change in the nonlinear dynamics of the system at various modes. It is illustrated that if the unbalanced disc is located at the node of one of the modes of the vibration of the system, that mode is not excited, as expected. The results of the paper reveal that the longitudinal disposition plays a key role in the instability analysis of the system of rotor shaft-disc, as well as the radial unbalance. In the second part of the paper, nonlinear vibration of the system of a flexible rotor shaft with a longitudinally dispositioned unbalanced rigid disc is mitigated using nonlinear energy sink (NES). Two different conditions of the unbalanced disc are considered for the system. For each case, the NES is designed to control the undesired nonlinear vibration of the system close to the first two natural frequencies. The results show that the designed NESs are capable of controlling the system with different dynamic responses including periodic, quasi-periodic, chaotic, and also the system with unstable dynamic response that may lead to collapse of the system.

The interaction between atoms and molecules and intense laser pulses is typically described in either the velocity gauge, length gauge or in the Kramers-Henneberger frame. Here, certain relations between these forms are discussed. In particular expansions of the solution to the time-dependent Schrodinger equation in adiabatic states are considered. Such expansions in adiabatic states could be expected to be attractive for intense long-wavelength infrared laser pulses, where the external field changes on a slow timescale compared with the electron motion. It is shown that an expansion in velocity gauge adiabatic states gives the equations of motions that follow from an expansion in field-free states and by expressing the interaction operator in the length gauge. Likewise, it is shown that an expansion in the Kramers-Henneberger frame adiabatic states gives the equations of motion following an expansion in field-free states and using the velocity gauge interaction operator. Finally, an alternative form for the laser-matter interaction operator is considered, which is similar to the velocity gauge, but shifts the canonical momentum operator by the impulse associated with the Kramers-Henneberger interaction operator, rather than that of the electric field as is the case in the velocity gauge.

We construct an exactly solvable model of a linear harmonic oscillator with a coordinate-dependent mass in a uniform gravitational field. This model is placed in an infinitely deep potential well with the width $$2a$$ and corresponds to the exact solution of the angular part of the Schrodinger equation with one of the Hautot potentials. The wave functions of the oscillator model are expressed in terms of Jacobi polynomials. In the limit $$a\to\infty$$ , the equation of motion, wave functions, and energy spectrum of the model correctly reduce to the corresponding results of the ordinary nonrelativistic harmonic oscillator with a constant mass. We obtain a new asymptotic relation between the Jacobi and Hermite polynomials and prove it by two different methods.

10.1119/10.0002362.1Solving the equations of motion for a cylinder with elliptical cross section rolling down an inclined plane constitutes a rather challenging problem. Using Lagrangian mechanics, we have derived the equations of motion analytically and the resulting differential equations are solved numerically. Using a CNC milling machine, we manufactured a solid elliptic cylinder and recorded its dynamics using a high-speed video camera for various inclinations of the plane. We compared theory and experiment by extracting the center of mass coordinates and the angular velocity from the video recording. Limits imposed by the support force reaching zero and insufficient friction to prevent slipping were examined both experimentally and theoretically.

We consider an inertial (large Reynolds number) gravity current (GC) released from a lock of length $$x_0$$ and height $$h_0$$ into an ambient fluid of height $$Hh_0$$ which propagates along a channel with permeable floor and/or walls. The cross section (CS) of the container is defined by the general $$-f_1 (z) \le y \le f_2 (z)$$ for $$0 \le z \le Hh_0$$ , while the top and the bottom are at $$z = Hh_0$$ and $$z=0$$ (z is the physical coordinate measured from the bottom of the channel). We use Boussinesq assumption and formulate the shallow-water equations of motion. These equations include a parameter $$\lambda$$ which reflects the ratio between the typical time of propagation of the current over a length $$x_0$$ to the typical time of the drop of the interface due to porous boundary over a height $$h_0$$ . To solve the problem we employ the method of finite-differences. It is demonstrated that initially, the effect of a porosity of the floor or the walls on the distance of propagation of the GC is insignificant, but the “slumping” stage is absent. During the advanced stages, the porosity slows down the current and decreasing the effective distance of propagation. We illustrate the methodology for flow in symmetric channels with typical power-law CSs $$f_1(z) = f_2(z) = 0.5(c+bz^{\alpha })$$ , where $$b,c, \alpha$$ are non-negative constants and show that for the identical characteristics of the porous material, the presence of the porous floor has more significant effect on the velocity of the propagation of the current than the presence of the permeable walls in all tested forms of containers. Special attention is given to triangular CS channels $$f_{1}(z)=f_{2}(z) = bz$$ with permeable walls. We show that in such containers the volume of the current decreases like $$\exp \left( -\sqrt{1+\frac{1}{b^2}}\lambda t\right)$$ (where t is time) and for $$Hh_0 \gg 1$$ its run-out length can be expressed by a compact analytical formula.

In this paper, a Lie coordinate-free torque based Euler–Lagrange equations of motion are developed for a 3-D link (3-DOF) robot. Intentional torque and jerky torque (non-intentional torque) are considered as the inputs to the dynamic profile of the robot. The jerky torque is modelled as a superposition of compound Poisson processes, which is a unique feature. The state vector of the robot, i.e., angular position and angular velocity vector, is thus a Markov process whose transition probability generator can be expressed in terms of the rate of the compound Poisson process that defines the jerky torque. Proof of frame invariance is provided to support the coordinate-free robot dynamics profile. Noise-free measurement is investigated as an ideal case. Angular position measurement is considered with white Gaussian noise. Further, an implementable finite-dimensional EKF approximate to Kushner–Kallianpur filter is obtained to estimate the robot state vector. Finally, the simulations are implemented on commercially available Omni bundle robot.

We find periodic solutions of the Coulomb d-dimensional (d = 1, 2, 3) equations of motion for two identical negative point charges in the field of four identical positive point charges fixed at vertices of a rectangle. Systems of this kind have equilibrium configurations. Periodic solutions are obtained with the help of the Lyapunov center theorem.

This paper analyses the stability of a waveboard, the skateboard consisting in two articulated platforms, coupled by a torsion bar and supported of two caster wheels. The waveboard presents an interesting propelling mechanism, since the rider can achieve a forward motion by means of an oscillatory lateral motion of the platforms. In this paper, the system is described using a multibody model with nonholonomic constraints. To study the stability of the forward upright motion with constant speed, the equations of motion are linearized with respect to a reference motion. The employed numerical linearization procedure is valid for general multibody systems with holonomic and nonholonomic constraints. In practice, the employed approach makes use of the Jacobian matrix, which is expressed in terms of any of the main design parameters of the waveboard. This paper introduces a sensitivity analysis of the eigenvalues with respect to the forward speed, the casters’ inclination angle, the torsional stiffness of the torsion bar, the aspect ratio of the toroidal wheels and the mass of the human rider. Lastly, a summary with the influence of these design parameters on the external actuations exerted by the rider in the waveboard maneuvering is shown.

Assuming non-Fourier thermal effects, Tzou's dual-phase-lag model has been applied to introduce the governing heat conduction equation in the presented mathematical model. Moreover, in order to design a well-posed stable dual-phase-lag model, the governing time fractional dual-phase-lag heat equation has been established by introducing conductive temperature and thermodynamical temperature, satisfying the two-temperature theory. Due to the application of phase-lags the heat conduction equation became hyperbolic. The corresponding governing equations of motion and stresses have been considered in two-dimensional bounded spherical domain. The spherical boundaries are assumed to be traction free. The Laplace and the Legendre integral transforms have been applied to obtain the analytical solutions of conductive and thermodynamical temperatures, displacement components and thermal stresses. The Gaver-Stehfest algorithm has been employed to achieve the time domain inversions of Laplace transforms numerically, satisfying the Kuznetsov convergence criteria. Classical, fractional and generalized thermoelasticity theories has been recovered theoretically and numerically as well for various fractional orders and phase-lags values.

Although quasi-Keplerian discs are among the most common astrophysical structures, computation of secular angular momentum transport within them routinely presents a considerable practical challenge. In this work, we investigate the secular small-inclination dynamics of a razor-thin particle disc as the continuum limit of a discrete Lagrange-Laplace secular perturbative theory and explore the analogy between the ensuing secular evolution -- including non-local couplings of self-gravitating discs -- and quantum mechanics. We find the 'quantum' Hamiltonian that describes the time evolution of the system and demonstrate the existence of a conserved inner product. The lowest-frequency normal modes are numerically approximated by performing a Wick rotation on the equations of motion. These modes are used to quantify the accuracy of a much simpler local-coupling model, revealing that it predicts the shape of the normal modes to a high degree of accuracy, especially in narrow annuli, even though it fails to predict their eigenfrequencies., Comment: 9 pages, 4 figures

This article presents an innovative control architecture for tilt-rotor quadcopters with H-configuration transporting unknown, sling payloads. This control architecture leverages on a thorough analysis of the aircraft's equation of motion, which reveals gyroscopic effects that were not fully characterized and were disregarded while synthesizing control algorithms in prior publications. Furthermore, the proposed control architecture employs barrier Lyapunov functions and a novel robust model reference adaptive control law to guarantee a priori user-defined constraints on both the trajectory tracking error and the control input, despite poor information on the aircraft's inertial properties and the presence of unknown, unsteady payloads. Flight tests involving a quadcopter pulling an unmodeled cart by means of a thin rope of unknown length, which is slack at the beginning of the mission, verify the effectiveness of the theoretical results.

This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, perturbed Kepler Hamiltonians which perturbation is of the form ∑ j = 2 N A j ∕ r j with A j constant. Although there are many applications of these Hamiltonians in Physics, Astronomy and Astrodynamics, we focus our interest on a particular case in the core of Artificial Satellite Theory, the Cid’s radial intermediary. For this problem, we integrate the equations of motion in two different ways, by means of the elliptic P -Weierstrass function and by using the Krylov–Bogoliubov averaging method to integrate a perturbed harmonic oscillator. In this case, the resulting solution is given in terms of the classical Kepler’s equation, with no need of introducing more complex generalized Kepler’s equation.

This paper focuses on the study of a new type of a planar, circular restricted three-body problem with an attractive artificial electrostatic field (E-field) at collinear libration points. For instance, this attractive field can be generated by an orbiting spacecraft located at the Mars-Phobos L1 libration point and an electrostatic capsule launched from Phobos. The feasibility of the proposed retrieval system is discussed from the aspect of local space weather Debye length. The attractive E-field splits the collinear libration point into two new collinear points, and the greater the E-field potential and the Debye length the greater the distance between the new libration points and the “old” original libration point. The new equilibrium positions caused by the action of the E-field have been found, and an instability of these new libration points has been proven. A new Jacobi integral in analytical form is obtained and equations of motion are derived for the restricted problem of three bodies taking into account the E-field. A numerical simulation shows the impact of the E-field potential on capsule capture in the small vicinity of the Mars-Phobos L1 libration point. This work expands the classic three-body problem filling with new content. The obtained results can be applied, for example, to study an opportunity of delivering the Phobos samples using Coulomb interaction of bodies in space.

We present a succinct way to quantize Maxwell's equations. We begin by discussing the random nature of quantum observables. Then we present the quantum Maxwell's equations and give their physical meanings. Due to the mathematical homomorphism between the classical and quantum cases [1], the derivation of quantum Maxwell's equations can be simplified. First, one needs only to verify that the classical equations of motion are derivable from a Hamiltonian by energy-conservation arguments. Then the quantum equations of motion follow due to homomorphism, which applies to sum-separable Hamiltonians. Hence, we first show the derivation of classical Maxwell's equations using Hamiltonian theory. Then the derivation of the quantum Maxwell's equations follows in an analogous fashion. Finally, we apply this quantization procedure to the dispersive medium case. (This article is written for the classical electromagnetic community assuming little knowledge of quantum theory, an introduction of which can be found in [2, Lecs. 38 and 39].)