Rotary Inertia Research Articles

Longtime dynamics of solutions for higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff models with higher-order rotational inertia and nonlocal damping

The Kirchhoff model is derived from the vibration problem of stretchable strings. This paper focuses on the longtime dynamics of a higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff system with higher-order rotational inertia and nonlocal damping. We first obtain the state of the model’s solutions in different spaces through prior estimation. After that, we immediately prove the existence and uniqueness of their solutions in different spaces through the Faedo-Galerkin method. Subsequently, we prove their family of global attractors using the compactness theorem. Finally, we reflect on the subsequent research of the model and point out relevant directions for further research on the model. In this way, we systematically study the longtime dynamics of the higher-order (m1,m2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(m_{1},m_{2})$\\end{document}-coupled Kirchhoff model with higher-order rotational inertia, thus enriching the relevant findings of higher-order coupled Kirchhoff models and laying a theoretical foundation for future practical applications.

Sliding-mode Active Disturbance Rejection Permanent Magnet Synchronous Motor Control System

Aiming at the permanent magnet synchronous motor with external perturbations and internal uncertainties such as external load, rotational inertia and armature resistance changes, a servo control system based on sliding mode self-immunity controller is designed. By constructing a mathematical model of the servo system of the permanent magnet synchronous motor, considering the internal uncertainty caused by the change of the system parameters and the external random disturbance as the "total disturbance", and designing the transition process and the linear expansion state observer to observe and compensate for it, the system response can track the input signal quickly and without overshooting, and the sliding-mode state feedback makes the closed-loop servo system achieve fast and stable control, and the sliding-mode Active Disturbance Rejection control controller can be used to control the system. The sliding mode state feedback enables the closed-loop servo system to realize fast and stable control, and its consistent stability is proved by the Lyapunov method. Simulations show that this sliding- mode Active Disturbance Rejection control control strategy improves the system response speed and load carrying capability compared with the traditional sliding-mode control.

Active Vibration Damping Using Rotor Torque Control in Electric Aircraft

Vibration damping in aerospace structures can decrease the likelihood of failures and instabilities and improve passenger comfort. This paper introduces the novel idea of damping vibration using the electric proprotors on aircraft without compromising flight control. The equations of motion of a cantilevered beam with a propeller at the tip driven by an electric motor are obtained using Hamilton’s principle, solved analytically in the frequency domain, and approximately in the time domain. Feeding back the beam tip angular rate to the motor torque is shown to asymptotically stabilize all transverse beam vibration modes. The overall vibration control consists of the inner rate feedback damping loop with an outer rotor speed control loop. Experimental frequency response and step response validate the models and show that the closed-loop damping in the first mode is three times higher than open loop with less than 1% rotor speed change for a 3% initial tip displacement. Theoretical results give good agreement with experiments. Parametric studies based on a 12 kg quadcopter indicate that if the rotor inertia is sufficiently large the proposed control can provide 8% damping with minimal impact on rotor speed.

Modeling and analysis of a macro-fiber piezoelectric bimorph energy harvester operating in d33-Mode using Timoshenko theory

Piezoelectric bimorph cantilevered beams are frequently employed in energy harvesting applications. The use of macro-fiber composites, which combine piezoelectric fibers with rectangular cross-sections and interdigitated electrodes, has significantly enhanced their performance by enabling operation in the 33-mode of piezoelectricity, thereby increasing conversion efficiency. In this article, the Timoshenko beam theory is applied to overcome the limitations of the Euler-Bernoulli theory in modeling transverse vibrations of harvesters with low slenderness ratios (length to thickness ratios). The mixture rule formulation is applied to determine the equivalent properties of the macro-fiber composite (MFC) structure. The electromechanical properties of a representative volume element (RVE) bounded by two successive interdigitated electrodes are coupled to the overall electro-elastic dynamic model of the structure using the Timoshenko theory. Frequency response functions for the power and tip vibration displacement of the MFC bimorph beam are determined from the model. Experimental data from the literature are used to validate the improved results predicted by the model developed in this article. A comparative analysis between the electromechanical responses predicted using a model based on Timoshenko and a model based on Euler-Bernoulli theory is conducted showing agreement, for harvesters with large slenderness ratios and discrepancies of up to 30% for low slenderness ratios. It has also been shown that the model based on Euler-Bernoulli theory overestimates both the mechanical and the electrical responses for low slenderness ratios as the shear and rotary inertia effects are not negligible in this case.

Response of a Stretched String Subjected to a Moving Mass

In this paper, the dynamics of a taut horizontal string with a constant velocity moving mass including its rotary inertia, is modelled. The equation of motion is solved using Galerkin’s approach, employing appropriate comparison functions. A discontinuity or jump in the trajectory of the mass has been established when the mass is about to leave the string. The consideration of rotary inertia in the model is found to affect the spatial location of the jump in the trajectory of the moving mass.

Timoshenko-Ehrenfest Beam-Based Reconfigurable Elastic Metasurfaces for Multifunctional Wave Manipulation.

Herein, a Timoshenko-Ehrenfest beam-based reconfigurable elastic metasurface is introduced that can perform multifunctional wave phenomena within a single substrate, featuring high transmission in the ultrabroadband frequency range. Conventional elastic metasurfaces are typically limited to specific purposes and frequencies, thereby imposing significant constraints on their practical application. The approach involves assembly-components with various geometries on a substrate for reconfigurability, enabling to easily control and implement multifunctional wave phenomena, including anomalous-refraction, focusing, self-acceleration, and total-reflection. This is the first study on elastic metasurfaces to theoretically analyze the dispersion relation based on the Timoshenko-Ehrenfest beam theory, which considers shear deformations and rotational inertia. The analytical model is validated by demonstrating an excellent agreement with numerical and experimental results. The findings include full-wave harmonic simulations and experimentally visualized fields for measuring various wave modulations. Furthermore, the practicality of the system is verified by significantly enhancing the piezoelectric energy harvesting performance within the focusing configuration. It is believed that the reconfigurable elastic metasurface and analytical model based on the Timoshenko-Ehrenfest beam theory have vast applications such as structural health monitoring, wireless sensing, and Internet of Things.

Free Vıbratıon of a Carbon Nanotube-Reınforced Nanowıre/Nanobeam wıth Movable Ends

ObjectiveThe purpose of this work is to investigate the size-dependent free vibrations of carbon nanotube-reinforced nanowires/nanobeams with movable ends. For this purpose, two movable end conditions are considered for carbon nanotube-reinforced nanowires/nanobeams with different carbon nanotube distributions. The size effect is addressed with the modified couple stress theory, which includes a material length scale parameter.MethodsIn this study, a solution approach based on the Fourier sine series and Stokes' transformation is used. With the help of this approach, both rigid and movable end conditions can be investigated. Firstly, equation sets consisting of infinite series and movable end parameters are derived. Then, eigenvalue problems are obtained for the free vibration of carbon nanotube-reinforced nanowires/nanobeams. The eigenvalues of these problems give the frequencies of the carbon nanotube-reinforced nanowires/nanobeams with movable ends.ConclusionsThe eigenvalue problems obtained in this study can be used to examine both rigid and movable end conditions. The accuracy of the problems obtained for solution is proven by various comparison studies. Then, a series of analyses are carried out for nanowire/nanobeam reinforced with carbon nanotubes, including both the size effect and the rotary inertia effect. When the frequencies of nanowire/nanobeams reinforced with carbon nanotubes are examined, it is understood that the material length scale parameter, carbon nanotube distribution, carbon nanotube volume fraction, rotational and lateral spring parameters can cause significant changes in free vibration.

Free-Vibration Analysis for Truncated Uflyand–Mindlin Plate Models: An Alternative Theoretical Formulation

Plates are flat structural elements whose thickness is small in relation to the size of the surface. Their use may include engine foundations, reinforced concrete bridge elements or parts of various floating structures. Consequently, knowledge of their mechanical behavior under static and dynamic loads is of primary importance in engineering applications and of interest from a structural point of view. As a result, numerous works existing in the literature have investigated the mechanical properties of plates using various plate models, such as Reissner’s theory, Levinson’s theory, Kirchhoff’s theory and Mindlin’s theory, and their static and dynamic behavior has been examined. In the present paper the truncated Uflyand–Mindlin plate equation is proposed. According to Uflyand–Mindlin theory, an alternative theoretical formulation is presented for the free-vibration analysis of plates, and the equations of motion and the general corresponding boundary conditions are derived. This paper develops the truncated Uflyand–Mindlin plate equation, i.e., without the fourth-order derivative, by means of the direct method and variational formulation. The first-order shear deformable plate theory developed by Elishakoff, which takes into account rotational inertia and shear deformation and does not include a fourth-order time derivative, is variationally derived here. This derivation complements that performed by Mindlin some 70 years ago. The innovative aspect of the suggested strategy is that variational and direct methods for studying plate dynamics are analogous. Finding the third equation of the reduced Uflyand–Mindlin equations, the accompanying boundary conditions and their mathematical resemblance are the goals of the presented formulations. In order to solve the dynamic equilibrium problem of a truncated Uflyand–Mindlin equation via a variational formulation, it is demonstrated that the differential equations and the corresponding boundary conditions have the same form as those found using the direct technique. This paper successfully completes this task. Finally, in order to validate the effectiveness and correctness of the proposed procedure, a numerical example of the case of a plate simply supported at all four ends is proposed.

Multimodal vibration damping of a three-dimensional circular ring coupled to analogous piezoelectric networks

Analogous piezoelectric networks have been shown to be effective for multimodal vibration attenuation in structures, including beams, plates, and rings. Previous studies for rings have only accounted for in-plane transverse vibration attenuation and disregarded the out-of-plane vibration modes. Furthermore, these previous numerical models and experiments have only been studied on thin rings, which ignore the effects of shear deformation and rotary inertia. As a result, these networks are not suitable for attenuating vibrations in thick rings. This study enhances on the previous electrical networks, considering both shear deformation and rotary inertia for both in-plane and out-of-plane vibrations. A new passive network topology is developed for the out-of-plane dynamics of thick rings, and the existing passive analogous network of in-plane vibration of thin rings is enhanced by considering the effects of shear deformation and rotary inertia. Combined, these new networks are capable of multimodal vibration damping of a thick ring in three-dimensions, encompassing primarily six types of vibration modes: the inextensional bending modes, the extensional modes, the thickness-shear modes, the coupled twist-bending modes, the torsional modes, and the transverse thickness-shear modes. By using piezoelectric elements to couple two separate analogous passive electrical networks derived from both the in-plane and out-of-plane governing equations of a ring and optimizing the internal resistance in each unit cell, it becomes possible to replicate the dynamics and effectively attenuate different types of vibration modes. This study serves as a theoretical foundation for implementations of passive vibration attenuation in ring structures.

Research on high fidelity modelling and optimum designing of an adaptive cycle engine's starting process

An adaptive cycle engine is an evolutional aero-engine that can achieve low fuel consumption and high efficiency by adjusting the thermodynamic cycle; however, few studies have been conducted on adaptive cycle engine performance at low speeds. A high-fidelity ground starting model of an adaptive cycle engine is established by correcting the low speed characteristics of the components, while considering the effects of rotor inertia, volume, combustion efficiency, and the thermal inertia of the combustor and turbine. By employing a physically enhanced prediction method, three operating states of the compressor at low speeds are obtained: compressor, stirrer, and turbine state. The matching mechanism of the adaptive cycle engine and the detailed influence of the variable geometries on the engine's performance during the starting process are explored. The study shows that the third bypass duct has little effect on the engine starting time, and opening the mode select valve can reduce the starting time. Compared with the initial settings, the starting time decreases by 3.98 s after optimization with the variable geometries. The detailed impact of the variable geometries on the engine's starting performance provides an engineering reference value for the control strategies.

An analytical model for the analysis of vibration and energy flow in a clamped stiffened plate using integral transform technique

Based on Kirchhoff thin plate and Mindlin thick plate theories, the vibration and energy flow characteristics of clamped stiffened plate are studied by using the analytical model constructed by finite integral transform method. The results show that the energy flow characteristics of the stiffened plate at the beam/plate coupling interface depend on the position of the rib in the vibration modes of the plate. The effects of shear deformation and rotatory inertia on the energy flow across the beam/plate coupling interface of the stiffened plate are further investigated. It is found that the inclusion of rotatory inertia of the beam and plate in the model only affects the energy flow component controlled by the moment coupling but not that controlled by the shear force coupling. Whilst the inclusion of the shear deformation of the beam and plate mainly causes a decreased amplitude of the energy flow for the mode group where the beam is located away from both the nodal and antinodal lines of modes, in addition to the shear deformation of the plate which also leads to an increased amplitude of the energy flow component controlled by the shear force coupling for the mode group where the beam locates at the antinodal line of modes. The understanding of energy flow characteristics of the stiffened plate at the beam/plate interface is essential to effectively control the noise and vibration problems of structures such as transformer tanks and machine covers.

Finite element modeling of the rotational dynamics of a single magnetic particle in a strong magnetic field and liquid medium

We have performed a comparative computational study of magnetic particle alignment in a strong magnetic field and ambient viscous fluid using the first-principles Navier–Stokes equation as well as numerical modeling using the finite element method (FEM). FEM solution has been compared with the solution of the unsteady rotations of a magnetic particle in a viscous fluid during the alignment process described with nonlocal integro-differential equations (IDEs) for torque and angular velocity. The assumption of nonlocality comes from the history term of acceleration torque with a non-Basset kernel function, which has its origin in the presence of vortices in the flow of a particle fluid environment due to its unsteady rotation and ambient fluid inertia and friction. The flow vortices of the ambient fluid are explicitly shown in the solution of the FEM model. Moreover, the solution of the time evolution of the alignment angle and angular velocity for the FEM model are in good agreement with an IDE model which we have recently developed. The obtained results are justification for interchangeability of the FEM and IDE models, which may have important consequences for large-scale simulations of magnetic microparticles.

Dynamic performance of multi‐layered composite sandwich open cylindrical shells with double‐layered soft cores

AbstractIn this paper, the dynamic performance of Multi‐layered Sandwich Composite Cylindrical Open Shells (MSCCOS) with double‐layered soft cores is analyzed for the first time by adopting the First‐order Shear Deformation Theory (FSDT). For accomplishing this aim, vibration differential equations of MSCCOS are derived based on Hamilton's variance principle. Then, the closed‐form Navier solution is introduced to obtain the numerical results of differential equations. Subsequently, the effectiveness and accuracy are discussed by several comparisons. Ultimately, the dramatic effect of different structural and material parameters on natural frequencies and associated loss factors is investigated and described vividly. Because shear deformation and rotary inertia of all layers are considered in vibration differential equations, thin and moderately thick MSCCOS could be researched, and some interesting discoveries for the dynamic performance of MSCCOS are derived.Highlights A vibration model of five‐layered composite open shell was obtained. The optimal structure parameters maximizing the damping properties are shown. Dynamics properties of five‐layered composite open shell are presented.

Inertial issues in renewable energy integrated systems and virtual inertia techniques

The global proliferation of renewable energy drastically altered the characteristics of power systems. Integration of clean energy sources reduces the inherent rotational inertia, making the system precarious and susceptible to various disturbances. The major challenges encountered are fast frequency fluctuations, voltage fluctuations, high rate of change of frequency (RoCoF), and frequency nadir. In order to address and adapt to a future low-inertia scenario, it is crucial to understand the effect of inertia on various parameters. This paper introduces a comprehensive review of the fundamental aspects of inertia and challenges that arise due to the reduction in inertia. Researchers have tackled this issue by employing various virtual inertia (VI) emulation techniques, which also have been extensively reviewed in the literature along with their merits, limitations, and recent developments. The impact of RES penetration on system dynamics is analyzed by simulating an IEEE-9 bus system with renewable energy source (RES) in MATLAB/Simulink. Furthermore, a three-phase fault is also introduced, to emphasize the effect of reduced inertia by observing the rotor angle and frequency deviation. The results validate that RES integration and fault location are observed to have a significant impact on stability parameters, making them extremely unstable.

Dynamic behavior of pin-free beam with torsional spring and hub at pinned end

This paper described the free vibration of a Euler-Bernoulli beam and discovered the first five dimensionless natural frequencies and mode shapes of motion for the pin-free beam model with a helical spring and rotating mass on the pinned side of the beam. The effect of changing the dimensionless torsional stiffness ratios was investigated, as was the effect of changing the dimensionless rotational inertia torque ratios Concerning the vibration properties of the beam. The mathematical equations that regulate the pin-free beam model were constructed using the exact analytical solution, establishing the Admissible Function and the natural frequency equation for the basic pin-free beam system. The approximate Rayleigh-Ritz approach was then used to determine the first five dimensionless natural frequencies, as well as the movement pattern for the pin-free beam model with a helical spring and a rotating mass on the side of the hinge. The results demonstrated that modifying the beam's natural frequencies is dependent on the ratios of the additional torsional spring parameters as well as the ratios rotational moment of inertia. The threshold system frequencies increase when the spring parameters are the major factor, and decrease when the inertia moment parameters are the main factor. Both play a role in influencing natural frequency to varied degrees. Unifying the final distribution of parameters might lead to optimizing the impact of parameters on the natural frequencies of the system.

An analytical investigation on nonlinear vibrations and stability of Timoshenko pipes conveying two-phase flow

The primary objective of this study is to conduct a comprehensive investigation on the nonlinear free vibration and stability analysis of the multilayer graphene reinforced pipe conveying liquid–gas two–phase flow based on the Timoshenko beam model. The Halpin–Tsai micromechanics model is employed which provides a means to evaluate and analyze the mechanical behavior and performance of the multilayer FG reinforced pipe. According to Von–Kármán strain relations and applying the Hamilton's principle, the governing equations of motion are derived. The Galerkin technique is operated to convert the partial differential equation (PDE) to the ordinary differential equation (ODE). The homotopy analysis method is employed to solve the fourth order nonlinear differential equation. Closed–form expressions for the nth nonlinear frequency and time history as well as the critical flow velocity based on the Timoshenko theory are achieved. The results demonstrate that neglecting the effects of rotary inertia and shear deformation may cause some errors by the increment of the initial amplitude. Furthermore, the critical flow velocity, depends not only on the mechanical and geometrical properties of the pipe but also on the initial mode shape excitation. The results indicate that by the increment of flow velocity, liquid density and slenderness ratio, the nonlinear frequency decreases, on the other hand, by the increment of the initial amplitude, the nonlinear frequency increases.

The Influence of Additional Virtual Synchronous Generator Technology in VSC-MTDC Systems with Wind Power on System Frequency

Introduction: A frequency control strategy is proposed based on additional virtual synchronous generator technology for voltage source converter-based multi-terminal high voltage direct current systems with wind power. Method: This strategy addresses the system's inertia reduction and frequency stability issues caused by integrating large amounts of wind power through multi-terminal DC transmission. Firstly, the virtual synchronous generator mathematical model is constructed based on the system structure. Secondly, for the problem of zero rotational inertia of voltage source converter in a flexible DC transmission system, based on the P-U droop control method of the converter station, additional virtual synchronous control generation technology is applied to simulate the P-f droop characteristics of the synchronous generator by adding virtual rotational inertia, so that the converter has the inertial response of synchronous generator to realize primary frequency regulation. Result: Finally, the simulation is verified on the PSCAD/ EMTDC platform with an example of a three-terminal parallel MTDC transmission system. Conclusion: the analyzed results demonstrate that the virtual synchronous generator control strategy is very valuable and useful for improving the frequency performance of the system.

Grey wolf optimizer based-natural frequency optimization of laminated composite shells carrying distributed mass

In this study, a grey wolf optimizer is proposed to find the optimum stacking sequence to maximize the fundamental frequency of shells. The vibration analysis of the shells is done using first-order shear deformation theory and a nine-node isoparametric element with five degrees of freedom per node. A mass lumping scheme with rotary inertia is considered in the finite element formulation to include the rotary inertia effect. The design variable considered is the fiber angle orientation of the shells. The present approaches are compared with previously published results using layerwise optimization theory, the Ritz method, and classical lamination theory. A comparison study is also performed, comparing the GWO-based finite element model against six different metaheuristic approaches, and it demonstrates that the Grey Wolf Optimizer consistently outperforms these competing algorithms. The optimum fiber angle orientation of shells carrying distributed mass for various boundary conditions and shell types is analyzed for the first time, to the best of the author’s knowledge. This novel discovery will benefit future studies.

A comparative analysis of the vibrational behavior of various beam models with different foundation designs

This article discusses the modal behavior of elastically constrained beams under various types of foundations and provides insights into the effects of different factors on the eigenfrequencies of beams. Numerical and analytical techniques, specifically the Galerkin finite element method (GFM) and the separation of variables, are utilized to determine the eigenfrequencies and mode shapes of beams. Modal analysis of Timoshenko, shear, Rayleigh, and Euler-Bernoulli beams that are elastically constrained and resting on Winkler, Pasternak, and Hetényi foundations, considering non-classical boundary conditions, is included in the study. The effects of factors such as flexural rigidity, transverse modulus, and Winkler foundation constant on natural frequencies of different beam models are investigated. The proposed method efficiently converges to the exact solution without shear locking in the stiffness element. The results demonstrate that the natural frequencies of the beam rise because of the shear layer, flexural rigidity, and foundation constant. Furthermore, the Hetényi elastic foundation affects the natural frequency of the beam, depending on the relative values of beam stiffness and foundation stiffness. Additionally, incorporating both shear deformation and rotary inertia has a greater impact on the eigenfrequencies of Euler-Bernoulli beams compared to incorporating only one of these effects. The findings of this work provide valuable insights into the behavior of beams under different foundation conditions and have potential applications in the design and optimization of structures incorporating beams, thereby enhancing the understanding of beam analysis.

Dynamical behavior of a spinning tapered Rayleigh beam subjected to an axial load

As a first attempt, the vibration and stability analysis of a spinning tapered beam subjected to an axial load are investigated. The radius of the cross-section of the beam is assumed to vary exponentially in the longitudinal direction. The tapered beam is modeled in Rayleigh beam assumption with rotary inertia, eccentricity, and gyroscopic effects. Based on the Hamilton principle, the governing equations of motion of the spinning tapered beam with an axial force are obtained. Also, a Galerkin discretization scheme is employed to obtain the natural frequency and critical divergence rotating speeds of the system. As a result, the instability thresholds and stability evolution of the system are acquired. Furthermore, a detailed parametric study is conducted to evaluate the effects of the slenderness ratio, taper parameter, axial load, and eccentricity ratio on the dynamical behavior of the system. The results show that the natural frequency and critical divergence rotating speed of the tapered beam are larger than those of the circular-section beam. In addition, the axial load, taper parameter, and eccentricity are the key factors that influence the stability of the system.