Linearized Equations Of Motion Research Articles

Quantum Properties of Coherently Driven Three-Level Atom Coupled to Vacuum Reservoir

A three-level laser with an open cavity and a two-mode vacuum reservoir is explored for its quantum properties. Our investigation begins with a normalized order of the noise operators associated with the vacuum reservoir. The master equation and linear operators' equations of motion are used to determine the equations of evolution of the atomic operators' expectation values. The equation of motion answers are then used to calculate the mean photon number, photon number variance, and quadrature variance for single–mode cavity light and two–mode cavity light. As a result, for γ=0, the quadrature variance of light mode a is greater than the mean photon number for two-mode cavity light. As a result, for the two-mode cavity light, the maximum quadrature squeezing is 43.42 percent.

Linear and nonlinear vibrations of strain gradient beams with rough surface

This paper aims to study the effect of surface roughness on the linear and nonlinear free vibration behaviors of microbeams based on the strain gradient theory. Linear and nonlinear equations of motion and boundary conditions are derived by using the Hamilton’s principle. A closed-form solution of strain gradient beams (SGBs) is obtained by solving a general sixth-order differential equation. Furthermore, by utilizing the linear mode and Galerkin’s method, an exact solution for nonlinear vibration is also obtained for SGBs. A detailed parametric study is performed to study the influences of surface roughness, scale parameter, beam thickness, and boundary condition on the linear and nonlinear vibration behaviors of SGBs. The present study may provide help for optimal designing and fabricating micro-/nano-resonators.

Dynamic Analysis of a Curved Beam With Tuning of Elastic Modulus and Mass Density in Circumferential Direction

Abstract Deformation and stress fields in a curved beam can be tailored by changing its mechanical properties such as the elastic modulus/mass density, which is typically done using functionally graded materials (FGM). Such functional gradation can be done, for instance, by using particles or fiber-reinforced materials with different volume fractions along the beam length. This article presents in-plane vibrations of functionally graded (FG) cantilevered curved beams. Both semi-analytical and finite element modeling are employed to find natural frequencies and mode shapes of such beams. The natural frequencies obtained from the analytical solution and finite element analysis are in close agreement with an error of 6.2% when the variance of material properties gradation is relatively small. In the analytical approach, the direct method is employed to derive the governing linear differential equations of motion. The natural frequencies and mode shapes are obtained using the Galerkin and the finite element methods. First, three natural frequencies and corresponding mode shapes are analyzed for different elastic modulus/mass density distribution functions. Furthermore, the natural frequencies of FG curved beams with a crack are also investigated. Our results indicate that larger cracks near the clamped side of the beam significantly decrease the first natural frequency. In the second and third vibration modes, cracks located in the area with a maximum moment result in the lowest natural frequency values. However, the second and third natural frequencies of the cracked curved beam are not affected by the presence of a crack, if the crack is located at the nodal points of the curved beam.

On-Chip Microwave Frequency Combs in a Superconducting Nanoelectromechanical Device.

Nanomechanical resonances coupled to microwave cavities can be excited, measured, and controlled simultaneously using electromechanical back-action phenomena. Examples of these effects include sideband cooling and amplification, which are commonly described through linear equations of motion governed by an effective optomechanical Hamiltonian. However, this linear approximation is invalid when the pump-induced cavity microwave field is large enough to trigger optomechanical nonlinearities, resulting in phenomena like frequency combs. Here, we employ a niobium-based superconducting electromechanical device to explore the generation of microwave frequency combs. We observe the formation of combs around a microwave resonant frequency (3.78 GHz) with 8-MHz frequency spacing, equal to the mechanical resonant frequency. We investigate their dynamics for different optomechanical parameters, including detuning, pump powers, and cavity decay rates. Our experimental results show excellent agreement with numerical modeling. These electromechanical frequency combs can be beneficial in nanomechanical sensing applications that require precise electrical tracking of mechanical resonant frequencies.

Dynamics of active polar ring polymers.

The conformational and dynamical properties of isolated semiflexible active polar ring polymers are investigated analytically. A ring is modeled as a continuous Gaussian polymer exposed to tangential active forces. The analytical solution of the linear non-Hermitian equationof motion in terms of an eigenfunction expansion shows that ring conformations are independent of activity. In contrast, activity strongly affects the internal ring dynamics and yields characteristic time regimes, which are absent in passive rings. On intermediate timescales, flexible rings show an activity-enhanced diffusive regime, while semiflexible rings exhibit ballistic motion. Moreover, a second active time regime emerges on longer timescales, where rings display a snake-like motion, which is reminiscent to a tank-treading rotational dynamics in shear flow, dominated by the mode with the longest relaxation time.

Seismic optimization of variable friction pendulum tuned mass damper with hysteretic damping characteristic

A variable friction pendulum-tuned mass damper (VFP-TMD) with hysteretic damping characteristics was proposed. The mechanism of the proposed VFP-TMD demonstrated linear behavior and nonlinear properties. Further, the fully nonlinear (FN) and partial linear (PL) equations of motion of the structure with the VFP-TMD system were derived. The VFP-TMD demonstrated stable and excellent seismic mitigation capacities and stroke limitation functionality. Based on the fixed-point theory, the closed-form exact solutions to design parameters of an undamped structure with a VFP-TMD system subjected to the ground motion were presented for different performance indices. The parametric studies showed that optimal designed VFP-TMD can be perfectly tuned in the frequency domain, which proved the applicability of closed-form solutions. To better understand VFP-TMD with hysteretic damping, constant friction pattern and multi-stage variable friction pattern were proposed for comparison. In contrast with constant friction and multi-stage variable friction, VFP-TMD with hysteretic damping characteristic was significantly optimized by the closed-form solutions. To verify the effectiveness of the closed-form solutions on VFP-TMD, twenty sets of real earthquake records were conducted which exhibited the superiority of linear variable friction patterns on seismic mitigation simultaneously. Results showed that the optimized VFP-TMD had an “infinitely large bandwidth”, which enabled it to reach its full potential according to the excitation amplitude.

Lyapunov equation in open quantum systems and non-Hermitian physics

The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.

Surprisal analysis of diffusion processes

We report a preliminary step in the application of information theory motivated surprisal analysis to irreversible processes but having linear equations of motion. Here we aim to describe the dynamics of varied Smoluchowski diffusion processes. In surprisal analysis, the probability density over the sample space is written as an exponential function of linear combination of observables that pose as constraints to the density evolution. The coefficients of this expansion can be interpreted as Lagrange multipliers that arise in the maximum entropy formalism. Known solutions of diffusion processes can be reproduced exactly using such a description. An exponent that is a linear combination of observables can also provide a close approximation for the dominant behavior of the solution of some other interesting examples for which there are no known exact results.

Nonlinear free and forced vibration analysis of sandwich cylindrical panel with auxetic core and GPLRC facing sheets in hygrothermal environment

In this study, the free and forced vibration analysis of composite sandwich panel subjected to the harmonic force excitation in the hygrothermal environment is investigated. The sandwich panel is composed of three-phase composites with polymer/Graphene platelet/fiber skins at the top and bottom surfaces and double-V auxetic honeycombs core layer with a negative Poisson’s ratio. Governing equations of motion within the framework of higher-order shear deformation theory (HSDT) and von Kármán nonlinearity are obtained. Applying the generalized differential quadrature method (GDQM) and homotopy perturbation technique, respectively the linear and nonlinear equations of motion results in solving the vibration problem. In the numerical illustration, at first validation of the present formulation is carried out by comparing the numerical results with those available in the open literature. Then the effects of several parameters such as geometric parameters of double-V auxetic core, force excitation, GPL volume fraction, and different boundary conditions on the nonlinear frequencies–amplitude of sandwich panel is studied. Finally, the important findings of this research indicate that the ratio of core thickness, inclined angle, and thickness to inclined length have significant effects on nonlinear frequencies–amplitude response. Based on the results of this article designers can development of different parts of aircraft.

Roll–Yaw Lock: Aerodynamic Motion Instabilities of Floating Offshore Wind Turbines

Abstract This article presents an investigation of a newly discovered motion instability phenomenon for floating wind turbines, first observed in numerical aero-hydro-servo-elastic simulations. The source of instability is identified as antisymmetric aerodynamic stiffness coupling terms in roll and yaw caused by the turbine thrust force. Analytical expressions for the roll–yaw and yaw–roll stiffness coupling terms are derived, using the actuator disk analogy of the rotor plane. It is demonstrated that a two degrees-of-freedom linear equation of motion without explicit external forcing qualitatively captures the instability. Based on this model, an analytical stability criteria is derived. An intuitive physical explanation of the phenomena is provided, considering the energy flow of the system. An important finding is that dissipative forces, in the form of aerodynamic and hydrodynamic damping, are needed to fully explain the observed instability. It is demonstrated, counterintuitively, that increased damping in yaw reduces the stability margin. This is explained by the fact that dissipative forces result in a phase difference between the harmonic roll and yaw motion in the coupled roll–yaw modes.

First-Order Approximate Mei Symmetries and Invariants of the Lagrangian

In this article, the formulation of first-order approximate Mei symmetries and Mei invariants of the corresponding Lagrangian is presented. Theorems and determining equations are given to evaluate approximate Mei symmetries, as well as approximate first integrals corresponding to each symmetry of the associated Lagrangian. The formulated procedure is explained with the help of the linear equation of motion of a damped harmonic oscillator (DHO). The Mei symmetries corresponding to the Lagrangian and Hamiltonian of DHO are compared.

On the stability of the equilibrium of the double pendulum with follower force: Some new results

The article deals with the problem of the stability of the equilibrium of a double pendulum under the action of a follower force. The latter is assumed to be imperfect, that is, the direction of its action does not coincide with the axis of the second link of the pendulum. The relationships between the masses of the pendulums, the stiffnesses of the hinges, and the damping coefficients are assumed to be arbitrary. Conditions of stability in the absence of damping and for a weakly damped system are obtained and analyzed. These conditions are formulated for the linearized motion equations in the form of an estimate of the dimensionless parameter corresponding to the value of the critical load. The influence of damping ratio and stiffness ratio on the permissible value of this parameter has been studied. It is shown that near the boundary of the region of flutter instability, the value of the critical load can be increased by decreasing the stiffness of the hinge at the attachment point of the pendulum.

Combined Quadrotor Autopilot System and Differential Morphing System Design

The aim of this article is to design and model a quadrotor with autopilot system and differential morphing using the Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm. Along with differential morphing, quadrotor modelling and control was also done. Although it is simple in structure, it has a complex structure in terms of model and control. Newton-Euler method was used for the dynamic model. Non-linear motion equations have been converted to linear motion equations. The full quadrotor model was drawn in the Solidworks program. Mass and moment of inertia information was obtained from this model. Simulation model was created by using state space model approach in Matlab / Simulink environment. Proportional integral derivative (PID) algorithm was used as the control structure. Differential morphing and PID coefficients rates were determined with SPSA. With the optimization method, determining the differential morphing rate, PID coefficients and applying it to the quadrotor provides a very innovative method.

Stress analysis and thermoelastic instability of an annular functionally graded rotating disk

The in-plane behavior, linearized transverse vibration, and stability of a thin heated Functionally Graded (FG) rotating annular disk are investigated. The inner edge of the disk is clamped and heated to a constant temperature. The outer edge of the disk is free. A convective heat transfer is considered acting on the disk due to rotation in an environment with still air. The coefficient of thermal expansion, thermal conductivity, and Young’s modulus vary radially according to the same power law. Assuming linear in-plane strains, the in-plane stresses are obtained and used to solve the linear equation of transverse motion to determine the critical speeds and stability characteristics. The results show that both the material property distribution and the heat convection coefficient have a significant effect on both the critical speed and stability characteristics. Moreover, heating a rotating FG disk can cause the modes with higher number of nodal diameters be excited before modes with lower ones. A convective coolant heat transfer acts as a strong stabilizer for the disk. Results suggests that an FG rotating disk with material properties increasing from the center is favorable to keep the disk stable and stress levels low if a continuous air supply is provided.

Impact of non-synchronous whirl on post-resonance backward whirl in vertical cracked rotors

Certain heavy-duty industrial machinery uses vertical rotating shafts. Such rotating systems typically undergo transient operation phases to reach their nominal speeds; recurrent passage through resonance generally occurs during startup and coast down operations. The Non-synchronous whirl between the shaft rotational speed and its whirling in the whirl orbit in the vicinity of the resonance zones, significantly affecting the breathing mechanism of a propagating transverse crack. Therefore, the appearance of Non-synchronous whirl during the passage through resonance rotational speed activates the crack breathing mechanism. To verify this observation, Jeffcott rotor and finite element models with induced breathing cracks are considered here for geometrically symmetrical and asymmetrical vertical rotor configurations. The mathematical models associated with these systems yield linear time-varying equations of motion incorporating the angular acceleration rate. The appearance of a Non-synchronous is found activating the crack breathing mechanism, which immediately excites the post-resonance backward whirl even with small crack depths in the rotor system. Accordingly, the excitation of post-resonance backward whirl can be considered as a strong indicator of an early crack propagation phase in vertical rotors.

Computer modeling via a numerical method for the cross-section effect on the forced vibration of FG nonuniform and imperfect microtubes

This paper examines the nonlinear forced vibrational behaviors of axially functionally graded (AFG) cylindrical microscale tubes under external dynamic stress, including the imperfect porosity. Using the Hamilton principle, the first-order shear deformation beam theory is linked with the modified couple stress theory (MCST) and the nonlinear von-Kármán theory to get the nonlinear governing equations and associated boundary conditions. For cross-sections that vary throughout the tube length, both uniform and non-uniform functions involving the three kinds are assumed. Furthermore, the microtube is constructed of functionally graded materials that vary smoothly throughout the tube length, and the porosity distribution is in the radial direction, resulting in two-dimensional functionally graded (2D-FG) structures with both porosity and material distributions. The nonlinear results are obtained after solving the linear motion equations using the generalized differential quadrature technique (GDQM) and the iteration approach. Finally, the effects of several factors on the natural and resonance frequencies are investigated, including the cross-section, FG parameter, porosity parameter, amplitude, external excitation frequency, and small-scale effect.

Nonlinear EHD Instability of Two-Superposed Walters’ B Fluids Moving through Porous Media

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across the interface. Additionally, the structure is pervaded via a uniform, normal electrical field in the absence of superficial charges. The nonlinear scheme basically depends on analyzing the linear principal equation of motion, and then applying the appropriate nonlinear boundary-conditions. The current organization creates a nonlinear characteristic equation describing the amplitude performance of the surface waves. The classical Routh–Hrutwitz theory is employed to judge the linear stability criteria. Once more, the implication of the multiple time scale with the aid of Taylor theory yields a Ginzburg–Landau equation, which controls the nonlinear stability criteria. Furthermore, the Poincaré–Lindstedt technique is implemented to achieve an analytic estimated bounded solution for the surface deflection. Many special cases draw upon appropriate data selections. Finally, all theoretical findings are numerically confirmed in such a way that ensures the effectiveness of various physical parameters.

Thermoelastic non-stationary fields in a rigidly fixed plate

In this article, a new closed solution of the axisymmetric dynamic problem of the theory of thermoelasticity is constructed for a rigidly fixed circular isotropic plate in the case of temperature changes on its face surfaces. The mathematical formulation of the problem includes linear equations of motion and thermoelasticity in the spatial formulation with respect to the components of the displacement vector, as well as the function of temperature change. The study of non-self-adjoint equations is carried out in an unrelated statement. Initially, we consider the initial boundary value problem of thermoelasticity without taking into account the deformation of the plate, and at the next stage, we study the problem of elasticity theory under the action of a given (defined) temperature change function. To solve the problems, we use a mathematical apparatus for separating variables in the form of finite integral transformations: Fourier, Hankel, and generalized integral transformation (CIP). In this case, at each stage of the study, the procedure is performed to bring the boundary conditions to the form that allows you to apply the appropriate transformation.

Modeling the motion of large vessels due to tsunami-induced currents

This paper describes a model developed to predict the general behavior of large vessels under energetic tsunami conditions using established regional-scale tsunami models. The methodology is based on a two-way coupled approach to account for the interaction between vessels and flow, which becomes increasingly important in restricted waters. A shallow water hydrodynamic solver is modified to include vessels as horizontal pressure distributions, and friction coefficients are adjusted accordingly at the locations of vessels to account for the effects of skin friction. Adequacy of these modifications are then tested against a benchmark case designed in OpenFOAM®, and comparisons demonstrate the modified shallow water solver's ability to provide a reasonable realization for energetic flow passing a floating object. A vessel transport model coupled with the hydrodynamic solver is based on the linear equations of ship motion with three degrees of freedom (surge, sway and the yaw). Finally, the model is equipped with a collision solver founded on the concept of the conservation of momentum and impulse. Results from two large scale applications of the developed tool are presented. These analyses revealed that the flow in and around the ports is indeed strongly affected by the presence of vessels. Also, it is observed that the model can approximate the ship's behavior, but at the same time the model results are very sensitive to the initial choice of the input parameters due to the chaotic nature of the process.

Harmonic balance analysis of output characteristics of free-standing mode triboelectric nanogenerators

Harmonic balance analysis of the output characteristics of free-standing mode triboelectric nanogenerators (FS-TENG) is conducted in this study. Considering the contact-impact force between the free-standing layer and fixed electrodes, the piecewise linear differential equation of motion of the layer is derived. Combined with an equivalent circuit differential equation, the electromechanical coupling model of the FS-TENG with nonlinear variation in the electrode distance is established. The harmonic balance method and alternate frequency/time domain method are adopted to obtain the steady-state output characteristics of the FS-TENG. The arc-length continuation method is utilized to track the multisolution region owing to bifurcation. The stability of harmonic balance solutions is determined through an eigenvalue analysis in the frequency domain. Both numerical integration and dynamic tests are conducted to verify the accuracy of the harmonic balance analysis. The influence of the nonlinear variation of electrode distance on the output characteristics of the FS-TENG is further examined through comparison with the existing model. Based on these results, the effects of vibration excitation amplitude, contact stiffness, and structural damping on the output characteristics of the system are discussed. The harmonic balance analysis in this study will contribute to the structural optimization design of the FS-TENG and further promote the development of the FS-TENG for engineering practicability.