Linearized Equations Of Motion Research Articles

Estimation of added effects and their frequency dependence in various fluid–structure interaction problems

This paper focuses on a fluid–structure interaction topic—the determination of added effects caused by fluid forces acting on a body, considering the standard linear equation of motion. We present various problems that assume small-displacement oscillations of single and multiple bodies in inviscid irrotational (potential) flow or viscous incompressible flow in both closed domain and external flow. For inviscid flow, effects of geometric parameters on the added effects were studied. The presented results extend results known from the literature. For viscous flow, frequency dependence of the added effects was studied for a wide range of frequency. The added effects were computed from data from numerical simulations of fluid flow, where the body oscillations were modeled using the dynamic mesh approach. Effects of the phase shift caused by the dynamic mesh were addressed. The added mass was compared with the corresponding value determined for inviscid flow where applicable. The results show strong dependence of the added effects on many parameters, making their proper computation challenging even for simplified cases.

Active polar ring polymer in shear flow-An analytical study.

We theoretically study the conformational and dynamical properties of semiflexible active polar ring polymers under linear shear flow. A ring is described as a continuous semiflexible Gaussian polymer with a tangential active force of a constant density along its contour. The linear but non-Hermitian equation of motion is solved using an eigenfunction expansion, which yields activity-independent, but shear-rate-dependent, relaxation times and activity-dependent frequencies. As a consequence, the ring's stationary-state properties are independent of activity, and its conformations and rheological properties are equal to those of a passive ring under shear. The presence of characteristic time scales by relaxation and the activity-dependent frequencies give rise to a particular dynamical behavior. A tank-treading-like motion emerges for long relaxation times and high activities, specifically for stiff rings. In the case of very flexible polymers, the relaxation behavior dominates over activity contributions suppressing tank-treading. Shear strongly affects the crossover from a tank-treading to a relaxation-dominated dynamics, and the ring polymer exhibits tumbling motion at high shear rates. This is reflected in the tumbling frequency, which displays two shear-rate dependent regimes, with an activity-dependent plateau at low shear rates followed by a power-law regime with increasing tumbling frequency for high shear rates.

No time to derive: unraveling total time derivatives in in-in perturbation theory

The in-in formalism provides a way to systematically organize the calculation of primordial correlation functions. Although its theoretical foundations are now firmly settled, the treatment of total time derivative interactions, incorrectly trivialized as “boundary terms”, has been the subject of intense discussions and conceptual mistakes. In this work, we demystify the use of total time derivatives — as well as terms proportional to the linear equations of motion — and show that they can lead to artificially large contributions cancelling at different orders of the in-in operator formalism. We discuss the treatment of total time derivative interactions in the Lagrangian path integral formulation of the in-in perturbation theory, and we showcase the importance of interaction terms proportional to linear equations of motion. We then provide a new route to the calculation of primordial correlation functions, which avoids the generation of total time derivatives, by working directly at the level of the full Hamiltonian in terms of phase-space variables. Instead of integrating by parts, we perform canonical transformations to simplify interactions. We explain how to retrieve correlation functions of the initial phase-space variables from the knowledge of the ones after canonical transformations. As an important first application, we find the explicit sizes of Hamiltonian cubic interactions in single-field inflation with canonical kinetic terms and for any background evolution, straight in terms of the primordial curvature perturbation and its canonical conjugate momentum, as well as the corresponding ones in the tensor sector, and the ones mixing scalars and tensors. We also briefly comment on quartic interactions. Our results are important for performing complete calculations of exchange diagrams in inflation, such as the (scalar and tensor) exchange trispectrum and the one-loop power spectrum. Being already written in a form amenable to characterize quantum properties of primordial fluctuations, they also promise to shed light on the non-linear dynamics of quantum states during inflation.

Dynamic and stability analysis of crescent geometry-possessing textured journal bearing using nanolubricant

PurposeThis study aims to find the dynamic performance parameters of the journal bearing with micro geometries patterning the arc (crescent) shape textures provided in three specific regions of the journal bearing: the full, the second half and the increasing pressure region. The dynamic behavior of textured journal bearings has been analyzed by computing dynamic parameters and linear and non-linear trajectories.Design/methodology/approachThe lubricant flows between the bearing and journal surface are governed by Reynold’s equation, which has been solved by finite the element method. The dynamic performance parameters such as stiffness, damping, threshold speed, critical mass and whirl frequency ratio are examined under various operating conditions by considering various ranges of eccentricity ratios and texture depths. Linear and non-linear equations of motion have been solved with Ranga–Kutta method to get journal motion trajectories. Also, the impact of adding aluminum oxide and copper oxide nanoparticles to the base lubricant in combination with arc-shaped textures is analyzed to further see any enhancement in the performance parameters.FindingsThe findings demonstrated that direct stiffness and damping parameters increased to their maximum level with six textures in the pressure-increasing region when compared with the untextured surface. Also, nanoparticle additives showed improvements above the highest value attained with no inclusion of additives in the same region or quantity of textures.Originality/valueEngineers may design bearings with improved stability and overall performance if they understand how texture form impacts dynamic properties.

Bidirectional pendulum-type tuned mass damper inerter for synchronously controlling the alongwind and crosswind responses of super-tall buildings

In the context of slender super-tall buildings, which have a high height-to-width ratio and low damping, significant bidirectional vibrations can occur under strong winds. Traditional TMDI device is mainly focused on controlling unidirectional wind-induced vibration responses. To address this issue, the present study introduces a bidirectional pendulum-type tuned mass damper inerter (BPTMDI) designed to synchronously mitigate alongwind and crosswind responses. The study derives nonlinear equations of motion for a multiple degrees of freedom (MDOF) structure integrated with the BPTMDI, using the Euler-Lagrange method. These equations capture the three-dimensional motion characteristics of the BPTMDI. Additionally, linear equations of motion are derived for a simplified planar model. The time-domain responses of the MDOF structure coupled with the BPTMDI are calculated by dividing the system into linear and nonlinear subsystems and applying the state space method and Runge-Kutta method, respectively. The effectiveness of the BPTMDI is evaluated through time history response analysis of a 400 m tall case-study building subjected to bidirectional wind loadings. Results show that the BPTMDI can significantly reduce both alongwind and crosswind responses, with reductions of at least 29.75 % in displacement and 48.64 % in acceleration, regardless of wind direction. Furthermore, the BPTMDI demonstrates improved control efficacy compared to BPTMD systems. Lastly, the simplified planar model of the BPTMDI shows basic similarity to the nonlinear planar-spherical model.

Analytical identification of dynamic structural models: Mass matrix of an isospectral lumped mass model

AbstractCombining the accurate physical description of high‐fidelity mechanical formulations with the practical versatility of low‐order discrete models is a fundamental and open‐ended topic in structural dynamics. Finding a well‐balanced compromise between the opposite requirements of representativeness and synthesis is a delicate and challenging task. The paper systematizes a consistent methodological strategy to identify a physics‐based reduced‐order model (ROM) preserving the physical accuracy of large‐sized models with distributed parameters (REM), without resorting to classical techniques of dimensionality reduction. The leading idea is, first, to select a limited configurational set of representative degrees of freedom contributing significantly to the dynamic response (model reduction) and, second, to address an inverse indeterminate eigenproblem to identify the matrices governing the linear equations of undamped motion (structural identification). The physical representativeness of the identified model is guaranteed by imposing the exact coincidence of a selectable subset of natural frequencies and modes (partial isospectrality). The inverse eigenproblem is solved analytically and parametrically, since its indeterminacy can be circumvented by selecting the lumped mass matrix as the primary unknown and the stiffness matrix as a parameter (or vice versa). Therefore, explicit formulas are provided for the mass matrix of the ROM having the desired low dimension and possessing the selected partial isospectrality with the REM. Minor adjustments are also outlined to remove a posteriori unphysical effects, such as defects in the matrix symmetry, which are intrinsic consequences of the algebraic identification procedure. The direct and inverse eigenproblem solutions are explored through parametric analyses concerning a multistory frame, by adopting a high‐fidelity Finite Element model as REM and an Equivalent Frame model as ROM. Before mass matrix identification, modal analysis results indicate a general tendency of ROM to underestimate natural frequencies, with the underestimation strongly depending on the actual mass distribution of the structure. After the identification of the mass matrix and the elimination of unphysical defects, isospectrality is successfully achieved. Finally, extensions to prototypical highly massive masonry buildings are presented. The qualitative and quantitative discussion of the results under variation of the significant mechanical parameters provides useful insights to recognize the validity limits of the approximations affecting low‐order models with lumped parameters.

EHD instability of a cylindrical interface separating two couple-stress fluids

This article is an attempt at examining the axi-symmetric and asymmetric streaming flows described by the CSF framework. A liquid that has microfibers implanted in it, like a fiber-reinforced composite substance, is so-called CSF. It is a system that consists of an endless vertical cylindrical interface that separates the two CSF structure. The CSFs are increasingly growing significant in modern manufacturing and technology, necessitating greater research into these fluids. An axial EF acts over the cylindrical contact in addition to the influence of CSF. The VPT is employed for the sake of convenience to minimize mathematical complexity. Combining the elementary linear equations of motion and the proper linear related BCs is the major procedure of the linear technique. A collection of physically dimensionless numbers is produced using a non-dimensional process. Subsequently, the requirements for hypothetical linear stability are developed. With the aid of the Gaster's theorem, the MS is applied in computing the dispersion relationships. After carefully examining a variety of effects on the stability investigation of the system at issue, it has been shown that the system is more unstable when a porous material is present than it would be without one. The resulting axisymmetric disturbance situation is more unstable. The linear techniques are depicted throughout a number of graphs.

Comparative analysis on the performance of different types of input‐ and command‐shaping controllers in minimizing payload residual vibration of an overhead crane with an inclined supporting track

AbstractReducing the effects of external disturbance on overhead crane systems is crucial, as they can impair the controller performance and cause excessive vibrations or oscillations of the payloads. One such external disturbance is the inclination of the supporting track of the crane trolley, which causes the system dynamics model to change. An open‐loop control strategy is widely utilized to control the payload sway motion and generally does not require any alterations in the physical structure of a system or the installation of sensors and/or actuators. Input and command shaping are two common open‐loop control techniques applied to control overhead cranes. In this paper, the effect of moving an overhead crane system along an inclined supporting track is investigated. In addition, the ability of different types of input‐ and command‐shaping control schemes to suppress the residual vibrations due to trolley track inclination is demonstrated. Two types of input‐shaping controllers, which are double‐step, zero vibration, and one command waveform (WF) shaper based on a trigonometric function, are used and tested. A linear equation of motion of the overhead crane resting on an inclined surface is developed to simulate the overhead crane and payload motion. The effectiveness of the different types of open‐loop controllers to suppress residual vibrations is verified by both simulation and experimental results. In addition, a new WF command shaper is proposed and designed to overcome track inclination while eliminating payload residual vibration. A comprehensive comparative analysis, both numerically and experimentally, is performed on the new proposed shaper to measure its effectiveness in handling inclination when compared to other types of open‐loop controllers. The new shaper outperforms other controllers in eliminating payload residual vibration for a wider range of inclination angles.

Research on PID Control of Quadrotor Drones Based on MATLAB

Four-rotor UAV is a very practical and widely used UAV. In this paper, the development status and future trend of Fourrotor UAV at home and abroad are introduced, and then the transformation matrix from ground coordinates to airframe coordinates is deduced by using Euler equation. In the process of dynamic modeling, the plane position reference coordinate system and rotation angle reference coordinate system are selected to analyze the external forces and moments on the body, and the linear motion equation and the angular motion equation are written in parallel. On the basis of UAV dynamics model, the classical PID control method is used to control the attitude of the inner loop and the position of outer loop . According to simulation results, the flight controller can keep the UAV flying stably.

Refined Geometrically Nonlinear and Linear Equations of Motion of an Elongated Rod-Type Plate

Refined Geometrically Nonlinear and Linear Equations of Motion of an Elongated Rod-Type Plate

Prediction and characterization of three-dimensional multi-mode coupled galloping of multi-span ice-accreted transmission conductors

This study presents a nonlinear analysis framework of three-dimensional galloping of multi-span ice-accreted transmission line conductors suspended by insulators. The vertical, lateral and torsional conductor displacements are described by modal displacements with mode shapes of single span conductors, and the insulators are modeled as pendulums. The aerodynamic forces on the conductor are modeled using quasi-steady theory. The linear equations of motion with respect to the static equilibrium are also derived. The analysis framework is applied to two- and three-span four-bundled ice-accreted transmission conductors with complex eigenvalue analysis and response history analysis. A harmonic balance approach is also proposed for a direct calculation of steady-state galloping response. The accuracy and efficiency of the proposed frameworks are verified using the results estimated from a finite element model. The similarities and differences of galloping characteristics, i.e., initiation condition, steady-state amplitude, dynamic tension, between single- and multi-span conductors are investigated. The role of suspension insulators on multi-span galloping is clarified. The longitudinal motions of insulators result in reduced vertical modal frequencies of conductor, thus lead to different galloping characters. The galloping of multi-span conductors with closely spaced modal frequencies can be initiated from multiple modal branches with a beating phenomenon.

A modified streamwise body force model of fan with distorted inflow for rapid propulsion-airframe integrated simulation

Streamwise Body Force Model (SBFM) could be used to simulate the force of blade on the airflow, resulting in rapid propulsion-airframe integrated simulation. However, when subjected to inlet distortion, the upstream flow field of fan stage is redistributed, which causes inaccurate prediction of fan stage performance. As inspired by the upstream influence of compressor, this paper aims to present a modification strategy for SBFM method to predict the compressor performance under circumferential inlet distortion without any knowledge of compressor geometry. Based on the linearized motion equation, the Upstream Influence Model (UIM) is introduced to predict the upstream flow field redistribution. Then the theoretical Mach number at Aerodynamic Interface Plane (AIP) position is calculated and selected to determine the corresponding body force coefficients based on the functional relationship between body force coefficients and Mach number, thus the upstream influence of compressor could be accurately quantified and the Modified Streamwise Body Force Model (MSBFM) could be established. Two studied cases are calculated with different methods and the upstream flow fields are analyzed. The prediction error of MSBFM method for compressor adiabatic efficiency is less than 3%, and the calculation efficiency is improved 20 times under the condition of ensuring computing accuracy. The MSBFM method has the potential for rapid propulsion-airframe integrated simulation.

Effects of uniform and periodic magnetic fields at the nonlinear stability of three magnetic fluids in porous media

The current manuscript tackles the interaction between three viscous magnetic fluids placed on three layers and saturated in porous media. Two of them fill half of the spaces above and below a thin layer that lies in the middle region. All layers are laterally extended to infinity in both horizontal directions. All fluids move in the same horizontal direction with different uniform velocities and are driven by pressure gradients. The system is stressed by tangential stationary/periodic magnetic fields. The viscous potential theory is used to simplify the mathematical procedure. The motion of the fluids is described by the Brinkman–Darcy equations, and Maxwell equations are used for the magnetic field. The work has been motivated in light of the engineering applications like petroleum products manufacturing and the electromagnetic field effect that can be used to control the growth of the perturbation and then the recovery of crude oil from the pores of reservoir rocks. The nonlinear technique typically relies on solving linear equations of motion and presenting the nonlinear boundary conditions. The novelty of the problem concerns the nonlinear stability of the double interface under the impact of periodic magnetic fields. Therefore, the approach has resulted in two nonlinear characteristic differential equations governing the surface displacements. Accordingly, the development amplitudes of surface waves are designated by two nonlinear Schrödinger equations. Stability is theoretically analyzed; the nonlinear stability criteria are derived, and the corresponding nonlinear stability conditions are explored in detail. The analysis reveals the resonance as well as the non-resonance situations. Approximate bounded solutions of the perturbed interfaces are estimated. It is found that the thickness of the intermediate layer is represented as a function of time and plotted. The impact of different parameters on the stability profile is investigated. For the middle layer, it is observed that magnetic permeability and viscosity have a stabilizing effect. By contrast, basic streaming and permeability have a destabilizing influence. Due to different saturation amplitude at each case, the analysis of the periodic case shows that the lower interface is much more stable than the upper one.

Sound characteristics of disordered granular disks: effects of contact damping

We investigate numerically the sound properties of disordered dense granular packings in two dimensions. Employing linear equations of motion and excluding contact changes from our simulations, we demonstrate time evolution of sinusoidal standing waves of granular disks. We varied the strength of normal and tangential viscous forces between the disks in contact to explore the dependence of sound characteristics such as dispersion relations, attenuation coefficients, and sound speeds on the contact damping. For small wave numbers, the dispersion relations and sound speeds of acoustic modes are quite insensitive to the damping. However, a small dip in the phase speed of the transverse mode decreases as the viscous force in normal direction increases. In addition, the dispersion relation of the rotational mode differs qualitatively from the theoretical prediction for granular crystals. Therefore, disordered configurations with energy dissipation play a prominent role in sound properties of granular materials. Furthermore, we report how attenuation coefficients depend on the contact damping and quantify how they differ from the prediction of lattice theory. These improved relations, based on our numerical results, can in future be compared to advanced theories and experiments.

Nonlinear stability of two dusty magnetic liquids surrounded via a cylindrical surface: impact of mass and heat spread

The current article examines a nonlinear axisymmetric streaming flow obeying the Rivlin–Ericksen viscoelastic model and overloaded by suspended dust particles. The fluids are separated by an infinite vertical cylindrical interface. A uniform axial magnetic field as well as mass and heat transmission (MHT) act everywhere the cylindrical flows. For the sake of simplicity, the viscous potential theory (VPT) is adopted to ease the analysis. The study finds its significance in wastewater treatment, petroleum transport as well as various practical engineering applications. The methodology of the nonlinear approach is conditional primarily on utilizing the linear fundamental equations of motion along with the appropriate nonlinear applicable boundary conditions (BCs). A dimensionless procedure reveals a group of physical dimensionless numerals. The linear stability requirements are estimated by means of the Routh–Hurwitz statement. The application of Taylor’s theory with the multiple time scales provides a Ginzburg–Landau equation, which regulates the nonlinear stability criterion. Therefore, the theoretical nonlinear stability standards are determined. A collection of graphs is drawn throughout the linear as well as the nonlinear approaches. In light of the Homotopy perturbation method (HPM), an estimated uniform solution to the surface displacement is anticipated. This solution is verified by means of a numerical approach. The influence of different natural factors on the stability configuration is addressed. When the density number of the suspended inner dust particles is less than the density number of the suspended outer dust particles, and vice versa, it is found that the structure is reflected to be stable. Furthermore, as the pure outer viscosity of the liquid increases, the stable range contracts, this means that this parameter has a destabilizing effect. Additionally, the magnetic field and the transfer of heat don’t affect the nature of viscoelasticity.

Nonlinear vibration characteristic of FGM sandwich cylindrical panel with auxetic core subjected to the temperature gradient

This study deal with the nonlinear vibration of sandwich panel, composed of functionally graded material (FGM) skins and double U auxetic core subjected to heat conduction. The nonlinear equations are obtained by applying Hamilton’s principles in framework higher order shear deformation theory (HSDT) and von Karman’s nonlinear theory. Linear and nonlinear equations of motion are solved by applying the differential quadrature method (DQM) and homotopy perturbation technique, respectively. 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 auxetic core, power-law exponent, different boundary conditions, and temperature gradient on the nonlinear frequencies of sandwich panel are 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 frequency response. Based on the results of this article designers can development of different parts of aircraft.

Questions on calculation of primordial power spectrum with large spikes: the resonance model case

Inflationary models predicting a scale-dependent large amplification of the density perturbations have recently attracted a lot of attention because the amplified perturbations can seed a sizable amount of primordial black holes (PBHs) and stochastic background of gravitational waves (GWs). While the power spectra in these models are computed based on the linear equation of motion, it is not obvious whether loop corrections are negligible when such a large amplification occurs during inflation. In this paper, as a first step to discuss the loop corrections in such models, we use the in-in formalism and calculate the one-loop scalar power spectrum numerically and analytically in an illustrative model where the density perturbations are resonantly amplified due to oscillatory features in the inflaton potential. Our calculation is technically new in that the amplified perturbations are numerically taken into account in the in-in formalism for the first time. In arriving at our analytical estimates, we highlight the role that the Wronskian condition of perturbations, automatically satisfied in our model, plays in obtaining the correct estimates. In addition, the analytical estimates show that the contribution originating from the quantum nature of the perturbations in the loop can be dominant. We also discuss the necessary conditions for subdominant loop corrections in this model. We find that, for the typical parameter space leading to the \U0001d4aa(107) amplification of the power spectrum required for a sufficient PBH production, the one-loop power spectrum dominates over the tree-level one, indicating the breakdown of the perturbation theory.

3D beam-envelope approach for an asymmetric beam in a high intensity linac with a solenoid channel

Presented here is a theory model based on the beam-envelope approach to determine the global behavior of a charged beam propagating through a high intensity linac with a solenoid channel, namely, the orbital evolution of the root-mean-square beam size in the transverse and longitudinal direction. The model is applied to the Gaussian particle distribution, where the motion of an individual particle is governed by linear focusing/defocusing forces in the solenoid channel and RF fields, except for space-charge forces. In this model, highly nonlinear effects that originated from space-charge forces disappear by averaging over the harmonic motions such as the betatron motion and phase motion in the moving direction and the Gaussian distribution. All particles making up a 3D Gaussian bunch obey the same linear equations of motion where all information of nonlinearity in the space-charge forces and the particle distribution are carried in. The theory is validated using Gaussian macro-particle simulation and one of the widely used photonic integrated circuit codes, TraceWin. As an example, the model is applied to the case of an international fusion materials irradiation facility superconducting linac with solenoid guiding and half wave RF acceleration. The results are compared with results obtained by this model and other two approaches.

Nonlinear free and forced vibration analysis of the sandwich composite cylindrical panel with auxetic core and GPL-reinforced facing sheets subjected to the temperature gradient

This study deals with the nonlinear free and forced vibration of a sandwich panel, composed of Graphene platelet-reinforced composite (GPLRC) skins and re- entrant auxetic core subjected to heat conduction and harmonic force excitation. The nonlinear equations are obtained by applying Hamilton’s principles in the framework’s higher-order shear deformation theory and von-Karman’s nonlinear theory. Linear and nonlinear equations of motion are solved by applying the differential quadrature method and homotopy perturbation technique, respectively. 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 auxetic core, force excitation, GPL volume fraction, different boundary conditions and temperature gradient on the nonlinear frequencies- amplitude of sandwich panel are 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.

A cost-effective algorithm for the co-simulation of unsteady and non-linear aeroelastic phenomena

Increasingly flexible structures force engineers and scientists to improve simulation capabilities incorporating an adequate description of non-linear phenomena. Furthermore, many critical operating conditions of aeronautical and civil structures are associated with unsteady phenomena, which makes it necessary to tackle this kind of problems. A critical aspect to be analyzed is the stability of the system: the determination of, not only the critical flutter speed, but also the post-critical behavior, is fundamental to guarantee the safety of designs and determine mitigation strategies. In this work, a numerical technique that allows simulating the non-linear and unsteady behavior of highly flexible structures immersed in a fluid at high Reynolds numbers is presented. This problem is addressed using a co-simulation approach, coupling the Unsteady Vortex Lattice Method (as the aerodynamic model) with the Finite Element Method using non-linear beams and rigid bodies (as the structural dynamics model). The coupling is achieved through a weak interaction algorithm. The resulting method has a very good compromise between versatility and computational cost, and is not restricted to periodic motions or linear equations of motion. Nonlinearity is included through the possibility to represent large displacements and rotations of the structure, a visco-elastic material behavior, and a nonlinear relation between the aerodynamic loads and the incidence of the incoming flow. The domains considered in the submodels and the temporal and spatial discretizations used for each one are essentially different; the interaction process must take into account these aspects. The computational implementation is carried out by combining a general-purpose structural dynamics code with an aerodynamic tool oriented to the analysis of three-bladed horizontal axis wind turbines. Examples to validate the code and show the potential of the proposed method are presented in the Results section.