Gyroscopic Research Articles

Analysis of Bahrain GP Circuit for Use in MotoGP with Account for Gyroscopic Effect of Wheels on Speed and Lean Angle

Motorcycle racing is an exhilarating and demanding sport that requires riders to master a variety of skills and techniques. Among these, counter-steering and leaning the motorcycle are paramount for achieving efficient and safe cornering. Unfortunately, a significant number of major accidents in motorcycle racing occur during the cornering phase, particularly near sharp corners on racing tracks. To mitigate these risks and ensure rider safety, this research project focuses on the dynamic analysis of the gyroscopic couple effect on motorcycle wheels with the aim of determining the safe angle of heel for more efficient and secure cornering.

Fault tolerance of a quadrotor via feedback linearization approach

A control algorithm is proposed to efficiently control the state, position, and height of a nonlinear dynamic model of a quadcopter. Based on feedback linearization, a state space model is presented for the system with the controller with a two-loop control structure designed and implemented in it. The inner and faster controller is responsible for adjusting the quadcopter height and angles, and the outer and slower controller is responsible for changing the desired figures of roll and pitch angles to control the system position. Whenever a rotor of the quadcopter rotor fails, the status and position of the system are converged and the system is stabilized. Simulation results based on different scenarios indicate the proper performance of the control system whenever there are external disturbances. Note that the gyroscopic effects because of<em> </em>the propeller rotation were not considered.

Dynamic analysis and validation of a multi-body floating wind turbine using the moving frame method

This research applies the moving frame method (MFM) to the multi-body dynamic analysis of an OC3 phase IV spar buoy with the NREL 5MW turbine. Further, it verifies previous results obtained through numerical comparisons with commercial software. The long-term goal is to lay the foundation for leveraging the MFM to create a self-contained software system for future analyses that can incorporate effects that are more sophisticated, when commercial codes fall short. In this first evidentiary phase, this project treats the floating turbine as a three-bodied system consisting of the platform (platform + tower), nacelle and rotor (hub + blades). Then the paper presents the MFM in a tutorial style—in the context of this problem’s resolution. The paper supplements the multi-body dynamic equations of motion obtained through the MFM with simplified and reduced hydrodynamic, aerodynamic and mooring loads to simulate the translational and rotational response of the floating turbine under various load conditions. The results closely approximate those found in previous work and, in the process, demonstrates MFM’s analytical advantage. Current results capture the coupled responses in all degrees of freedom and gyroscopic effects occurring when the platform pitches with the spinning rotor. The project thus provides an accurate model for the dynamics of the turbine and opens the door to inserting correct advanced hydrodynamics to validate the model further. The work presents simulations for the different load cases through a 3D web page using WebGL and the ThreeJS library. Users may download all software to verify the results. An undergraduate student conducted the work alone, demonstrating the ease of implementation of the MFM.

Centrifugal forces enable band gaps that self-adapt to synchronous vibrations in rotating elastic metamaterial

The rotation of mechanical systems can greatly change their modal and wave propagation response. In the classic example of the Foucault pendulum, the Coriolis force from the Earth’s rotation causes the pendulum path to appear deflected to an observer on Earth. More recently, Coriolis forces and gyroscopic effects from rotation have been shown to break time-reversal symmetry, induce non-reciprocity, and change the band structure of phononic crystals. However, rotation also introduces centrifugal forces that can considerably affect the vibration response through stress stiffening and spin softening effects. Although this is well studied in the field of rotor dynamics, the effects of centrifugal forces on the dynamic behavior of acoustic metamaterials (AMs) are still unexplored. In this paper, we study the dependence of torsional band gaps on rotational speed in a rotating shaft with attached local resonances. Different from previous studies, we explicitly consider the effects of stress stiffening in local resonances by introducing a stress stiffening function in a reduced order model of the rotating AM. We then calculate the stress stiffening function for beam-tip-mass resonators and show that these resonators have a resonant frequency that scales linearly with the rotational speed in the asymptotic limit, which in turn causes the band gaps of the AM to depend linearly on the rotational speed. Motivated by the presence of synchronous vibrations in rotating machinery, in which frequency is also linearly dependent on rotational speed, we show that this rotating AM supports band gaps that self-adjust to the synchronous vibration frequency, resulting in attenuation over a wide range of frequencies and thus rotational speeds. Finally, we validate the reduced order model of the rotating AM using 3D finite element analysis. This work contributes a foundational understanding of the effects of rotation in AMs and builds a connection between the AM and rotor dynamics communities, with the potential to introduce novel vibration control in rotor dynamics problems.

Centrifugal and gyroscopic effects on dynamic response of rotating cantilever beams under step loading

In this paper, a simplex finite element model for beams (FEMB) with a complete formulation is presented to study the dynamics of rotating cantilever beams subjected to distributed external loads. In our approach, the finite element method (FEM) and Timoshenko's beam theory is used. We will, particularly, examine the centrifugal and gyroscopic effects on the linear vibration of a rotating cantilever beam in stationary regime. In this model, extension, bending, and torsion degrees of freedom (DOF) are combined using a parameterization of the 3D motion of the beam by Euler's three-angle sequence. That allows identifying all the terms of the gyroscopic coupling in a more compact equations system. To ensure convergence, a sufficient number of finite elements are required in this model. The considered beam in the numerical simulation is pre-twisted and linearly tapered with a rectangular section. Results show that the dynamic centrifugal effect decreases the natural frequencies of the beam. Furthermore, the gyroscopic coupling induces rapid extension and torsion beatings when the beam undergoes a step bending load. Without damping, these beatings can persist and cause material fatigue.

Bearing Models for Advanced Ball Bearing Simulation

A comprehensive quasi-static and dynamic ball bearing model is described using many enhanced features, including novel calculations of the contact angle variations and ball–race sliding speeds in two directions, accounting for race curvature, pivoting effects and gyroscopic effects, appropriate lubricant rheology, and sliding forces (driving the balls against miscellaneous braking forces and moments such as, for example, the hydrodynamic rolling force). Ball–cage impact forces and cage-guiding ring forces are also considered, leading to a set of six dynamic differential equations to consider for each ball and three dynamic differential equations for the cage, where movement is assumed to be restricted to one plane. Quasi-static calculations can be undertaken by neglecting the cage and setting the accelerations of each degree of freedom of the ball to zero. New analytical or curve-fitted models are also provided using the mean values of sliding speed and viscosity to calculate sliding forces and moments, avoiding the use of contact slices with some useful relationships derived for calculating the rolling line locations, bearing torque, and dissipated bearing power. Several examples of calculated results are given and compared to available numerical or experimental results published in the literature. Finally, a new tool developed in-house is proposed for predicting many properties, particularly of interest at high speed, including the ball bearing torque, power loss, risk of smearing, and risk of cage failure.

Multi physical comprehensive design of a new type of FeCo based high speed permanent magnet motor

In order to solve the problem that the design of high-speed permanent magnet motor (HSPMM) is constrained by multi-physical fields, this paper starts with the comprehensive study of multi physical fields, introduces the finite element calculation model of rotor strength-rotor dynamics-electromagnetic field-fluid field and temperature field, carries out the comprehensive design of a 100 kW, 18 000 r/min FeCo based HSPMM, and gives the detailed and complete research process of the main parameters of HSPMM under multi-physical field constraints. First, the initial dimensions of the motor are obtained by electromagnetic and mechanical design theory. Secondly, the influence of design parameters on rotor stress is analyzed in detail, including the influence of bearing rigidity, impeller mass, rotor diameter, rotor core length and gyroscopic effect on critical speed. Subsequently, the electromagnetic characteristics of the motor are comprehensively studied, the cooling system is designed in the Flow-Heat coupling calculation model and the temperature field is studied. Finally, the prototype test verifies the correctness and validity of the comprehensive research method of multi physical field.

Design, Analysis and Testing of Mechanism to Reduce The Impact on Drones in Failure

This work aims at proving the concept of an autorotation-based crash prevention system for drones. The work focuses on quadcopters, whereas the concept can be used for all multi-copters in general with slight modification in design. The system works on the principle of autorotation which is a commonly used method in helicopters for unpowered landing. The autorotation system causes a gyroscopic effect preventing the quadcopter from changing its orientation during fall and also provides drag that helps in reducing the fall velocity. This technique reduces the energy that needs to be absorbed by the quadcopters structure during impact by converting a part of the kinetic energy of fall to the rotational kinetic energy and also creating a profile normal to the local airflow to produce drag. To prove this concept, the terminal velocity of the fall is obtained by mathematical modelling and this is compared with practical results in order to verify the theoretical results. Experimentally, a substantial reduction in velocity was found and this proves that the autorotation mechanism indeed reduces impact during failure in drones.

Nonlinear Lyapunov-based control of a misaligned heteropolar magnetic bearing system

ABSTRACT The dynamic behavior of an active magnetic bearing (AMB) mounted rotor was investigated through an analytical study conducted as part of this research work. A biased AMB model was developed, in which six electromagnets were supplied with a bias current I0 , and their respective control currents. The magnetic load was obtained through a differentiated driving mode based on the virtual displacement principle. Motion equations were formulated, and simulations were performed to investigate the dynamic response of the misaligned AMB mounted on a high-speed rotor, considering rigid body motion, elastic motion, gyroscopic effect, shear deformation effect, and internal damping. Subsequently, a nonlinear novel direct Lyapunov-based controller derived from an energetic approach was applied and compared to an optimal regulator based on Hamilton-Jacobi-Bellman (HJB) equations. The Linear Quadratic Riccati (LQR) method required less energy, as expected, but the Lyapunov method was more efficient and demonstrated its ability to asymptotically control the plant instability as well as the misalignment effect. Finally, the method was applied to a milling machine spindle, and the controller was still able not only to handle the cutting forces generating disturbances but also to overcome the defect effect without any oscillation.

A new instantaneous contact based dynamic model of rolling element bearings with local defects

An accurate dynamic model is significant for the understanding of defect induced dynamic characteristics of rolling element bearings, which further contributes to effective diagnosis and prognosis of bearings. In the development of dynamic models of bearings, how to model the contacts between the defective component and normal component is of vital importance. Existing contact models use displacement excitation instead of actual roller-defective raceway contact and thus leave space for improvement. To fill in this gap, this paper proposes a new dynamic model of rolling element bearings with a defect on the inner or outer raceway in a rotating reference frame with the gyroscopic effect and gravity force considered. In the proposed model, the contact between the roller and the defective raceway is modeled based on the instant positions of the rollers, cage, inner ring and outer ring. The effectiveness of the proposed model is validated through simulation and experiment studies. The results have shown that compared with the existing models, the proposed model exhibits a more gradual load change in the stressing stage and the restressing stage when the roller passes through the defect, yielding a more accurate modeling of the real contact of bearing components.

Unbalance Response Analysis of a Rotor Kit with Two Identical Discs Located Between Bearings

In this paper, the dynamic behavior of a rotor kit with two identical disks located between the plain bearings was analyzed. Modal and harmonic analysis of this rotor kit configuration were performed in the Ansys software package. To calibrate the bearing parameters (stiffness and damping) in the numerical model, experimental measurements of the rotor kit with a disc mounted at the midspan of the shaft were performed. As a result of modal analysis, natural frequencies and models were obtained. Using the Campbell’s diagram, the critical speeds and the influence of the gyroscopic effects on the natural frequencies were determined. The responses of the rotor kit to different unbalance distributions were considered by harmonic analysis.

Modeling and nonlinear dynamic analysis of angular velocity system of 2-DOF aerial manipulator

This paper establishes the dynamic model of the angular velocity system of the 2-DOF (degree of freedom) aerial manipulator, describes the coupling relationship between the quadrotor aircraft and the manipulator, and provides the rotational speed relationship of each propeller during open-loop control. On this basis, the dynamic parameters, the nonlinear dynamics under static and dynamic conditions, and the Casimir energy of the system are studied in four parts. In the part of system dynamics parameters, the influence of manipulator configurations and postures on system moment of inertia, yaw motion, and lifting motion are analyzed, and their influence on gyroscopic effect is discussed. In the nonlinear dynamics of the system under static conditions, this paper analyzes the nonlinear dynamics of a typical aerial manipulator by the nonlinear dynamics methods. It reveals for the first time that the angular velocity of the system can produce dynamic behaviors such as sinks, chaotic oscillations, and period-doubling oscillations. The system is a non-autonomous system when it is under dynamic conditions. In this part, this paper considers the dynamic behaviors of the aerial manipulator in three situations: grasping operation, presence of external interference, and presence of time-varying yaw control quantity. It can be seen from the analysis that the dynamic conditions make the nonlinear dynamic behaviors of the system more complicated, which is worthy of further study. In the energy analysis, the Casimir energy of the three cases in this paper is calculated respectively when the sink, chaotic oscillation, and periodic oscillation are generated, and their mean, variance, and standard deviation are calculated. When the system is a sink, the three indexes are the smallest, the energy consumption is the least, and it is easier to control.

Coherent optical gyroscope on the basis of the bidirectional ultra-short pulse Erbium ring fiber laser

A gyroscopic effect has been studied in the bidirectional ultra-short pulse erbium-doped fiber ring laser hybridly mode-locked with a co-action of a Single-Walled Carbon Nanotube-based saturable absorber (SWCNT-SA) and Nonlinear Polarization Evolution introduced by means of the short-segment polarizing fiber. Owing to the Kerr nonlinearity contribution through self-phase modulation and self-steepening effects to the carrier-to-envelope phase slip of both clockwise (CW) and counterclockwise (CCW) solitons, wideband controllable tuning of a gyroscope bias point has been demonstrated by means of an appropriate adjustment of either intracavity polarization or pump power. Angular velocity detected ranges from 0.12 deg/s to 360 deg/s while the highest rotation sensitivity reaches 7 kHz/(deg/s) for 0.79 m2 single-coil ring gyroscope being in agreement with the calculated scale factor value. The gyroscope dead band with upper limit of ≈5.15 deg/s corresponding to the beat-note frequency of ≈15 kHz has been detected during clockwise rotation of the gyroscope. The highest gyroscope resolution of ≈0.01 deg/s=38 deg/h has been estimated on the basis of the bias point drift measurement.

Theory of gyroscopic effects for rotating objects

Scientists began to study gyroscopic effects at the time of the Industrial Revolution. Famous mathematician L. Euler described only one gyroscopic effect, which is the precession torque that does not explain other ones. Since those times, scientists could not explain the physics of gyroscopic effects, Recent studies and the method of causal investigatory dependency demonstrated, that the nature of gyroscopic effects turned out that be more sophisticated than contemplated by researchers. The external torque acting on the spinning objects generates the system of the eight inertial torques and their interrelated motions around axes presented in the 3D coordinate system. The interrelated torques and motions of the spinning disc were described by mathematical models, and validated by practical tests that explain the physics of the gyroscopic effects based on the kinetic energy conservation law. The inertial torques generated by the centrifugal, and Coriolis forces, the change in the angular momentum, and the dependent motions of the spinning object around axes constitute the fundamental principles of the gyroscope theory. The derived gyroscopic theory opened a new chapter in the dynamics of rotating objects of classical mechanics that should be presented in all word encyclopedias.

Three-dimensional dynamics of a novel triply-gyroscopic fluid-conveying pipe system

Gyroscopic characteristics extensively exist in motional components, which can lead to coupling of motions along different directions. However, the vibration and stability mechanism of complicated gyroscopic structures still remains unclear. This paper proposes a novel class of triply-gyroscopic system — fluid-conveying pipes conducting simultaneous rotating and spinning motions. Based on a Rayleigh beam model, the coupled differential equations governing in-plane and out-of-plane flexural vibrations and axial vibration are deduced, which fully account for the rotating gyroscopic force, spinning gyroscopic force and fluid gyroscopic force. The Galerkin technique is applied to discretize the governing equations, and the characteristic frequency method is further utilized for solution. Via an in-depth dimensionless analysis, the resonant frequency and stability evolution with the three motions is drawn, and its dependence on the triple gyroscopic and centrifugal effects is revealed. The impacts of significant geometrical and physical properties are also discussed. Three-dimensional (3D) backward and forward whirling shapes of the pipe are simulated, where the rotating motion is involved as novelty. Nonplanar configurations and ‘traveling wave’ vibrations are observed as a result of gyroscopic effects. The study will contribute to in-depth understanding of gyroscopic dynamics, and provide theoretical and designing base for the engineering gyroscopic structures.

Radiation-Induced Magneto-Optic Fluctuations in Spaceborne Fiber-Optic Gyroscope

Radiation-induced magneto-optic fluctuations (RI-MOF) is a kind of environmental compounding effect of fiber optic gyroscope (FOG) in space application. Here, we suggested a model of RI-MOF and experimentally tested it in irradiated polarization-maintaining fibers (PMFs) and polarization-maintaining photonic crystal fibers (PM-PCFs). The test results reveal that the main sources of RI-MOF are the radiation-induced change of the Verdet constant and the modal birefringence. In a spacecraft with good thermal control conditions, an evaluation method based on the RI-MOF model can be summarized and implemented to suppress the environmental disturbance on the performance of FOG.

On the modal characteristics and stability of translating cable: Application to an overall line of cable transport systems

Moving cables are used for many applications in engineering structures such as cable-supported transport systems that involves mobile cables suspended between supports for moving attached vehicles. Advantages of such systems in terms of cost, footprint and energy efficiency make its urban integration as one of the solutions to meet the need for new mobilities. Because of the inherent cable’s flexibility and the modification of dynamic properties during the cable movement, such systems are sensitive to the effects of perturbations and self-sustaining oscillations. The methods commonly used to describe cable dynamics are restricted to small static sag case with a predominant vertical dynamics. To accurately analyse a multi-segment mobile cable, this paper proposes an hybrid analytical–numerical strategy to deal with a general mechanical elementary model of a cable in unsteady translation formulated in curvilinear abscissa derived from the formalism of continuum mechanics. An assembly procedure permits to deal with a more complex system considering a cable loop in contact with intermediary supports and punctual loads according to different boundary conditions, considering the coupling between the vertical dynamics activated at the local scale of a span and the longitudinal dynamics activated at the scale of the whole cable loop. The translation speed and acceleration gyroscopic effects on the system’s complex modes are studied and design rules are provided to guide an optimal conception avoiding strong oscillations of the system.

A Numerical Analysis for Ball End Milling Due to Coupling Effects of a Flexible Rotor-Bearing System Using GPEM

In this paper, the tool-tip responses for ball end milling, due to the coupling effects of a flexible rotor-bearing system, are investigated numerically. The milling machine tool spindle is modelled as the flexible rotor-bearing system. The critical speeds, natural modes, and unbalance responses of the system are calculated by applying the generalized polynomial expansion method. This generalized polynomial expansion method expresses the displacement as a series formed by the product of generalized coordinates and axial coordinate polynomials. According to the dynamic cutting force obtained by some scholars in the past, combined with the characteristics of the flexible rotor, the dynamic response of the tool-tip for ball end milling is numerically analyzed. The responses, including time histories, orbits, and FFT diagrams, are plotted to analyze the dynamic behaviors of the tool-tip. The coupling effects of the flexible rotor-bearing system on the system for ball end milling are first studied using the generalized polynomial expansion method. Unlike previous studies, the natural frequency varies with spindle speed and which of the different modes are included in the tool-tip response depends mainly on the spindle speed. Thanks to the gyroscopic effect, the critical speeds and responses of tool-tips can be discussed with respect to various spindle speed and tool flutes. The natural modes are accurately determined, and will excite critical speeds for certain modes, including forward and backward modes, thereby significantly affecting tool-tip response. In addition, the cutting force component associated with the tool-tip response affects the rotor-bearing system parameters, complicating the issue. Milling at higher spindle speed (2160–19,950 rpm), an important new result is found that the tool-tip oscillates with the cutting-force frequency, accompanied by a longer period vibration of the first backward mode of the rotor-bearing system. It can also be seen from the frequency spectrum analysis that, as the spindle speed increases, the peak amplitude of the first backward mode becomes larger. Milling at lower spindle speed (960, 1320 rpm), the in-plane vibration trajectory of the tool-tip gradually expands outwards clockwise around the origin until a stable loop is reached. This is because only the first backward mode of the rotor-bearing system is excited. Considering the coupling effect of the rotor-bearing system to perform the vibration analysis of the milling machine system, the parameters of the system can be designed or the spindle speed can be selected to avoid severe vibration during machining.

Gyroscopic effect evaluation and resonance speed prediction of complex high-speed rotor system based on energy

Making full use of gyro effect to adjust resonance speed distribution is of great significance for avoiding resonance design of high-speed rotor system. Existing rotor dynamics research often regards gyro effect term as an implicit quantity to be solved in dynamic equation of complex rotors, but does not carry out quantitative evaluation on its gyroscopic effect. Therefore, this paper decomposes kinetic energy components of the rotor, deduces equilibrium relationship between modal kinetic energy and potential energy of the rotor system at a specific speed, defines the ratios of gyro kinetic energy to lateral vibration energy/elastic potential energy in modal vibration as parameters to evaluate the degree to which the gyro effect affects resonance characteristics. Based on this, the gyro effect of a typical complex high-speed rotor system is evaluated. The derivation process also shows that, modal shape directly determines the gyro energy ratios, ultimately determines the change amplitude of resonance speed. Then a method to predict resonance speed under gyro effect based on modal shape is proposed, and experimental verification is carried out. The results show that, it is feasible and concrete to understand gyroscopic effect from the perspective of system energy equilibrium and transformation.

Dynamic modeling of GTF star gear-rotor coupling system considering structural flexibility

This paper establishes a dynamic model of Geared turbofan (GTF) star gear-rotor coupling system, in which the flexibility of shaft, disk, shell, diaphragm coupling, ring gear, and carrier are all taken into consideration. Additionally, an internal meshing model for flexible ring gear is developed, and the matrix reduction techniques is introduced to enhance the computational efficiency. Based on the dynamic model, the modal characteristics of the coupling system are extracted, and the influence of structural flexibility are investigated. Considering gyroscopic effect of the rotor, the Campbell diagram and the critical speeds are obtained. In addition, the vibration amplitudes under unbalanced excitation and time-varying mesh stiffness (TVMS) excitation are calculated and compared. The results of modal characteristics analysis indicate that many of the star gear train models are greatly affected by structural flexibility. Besides, the counter-rotating between the low-pressure rotor and the fan rotor results in frequency veering in the Campbell diagram, which is rarely observed in parallel gear-rotor system. The vibration responses indicate that unbalanced excitation of the rotor has relatively less effect on the gear vibration, while the low frequency excitation at star gear train is capable of exciting resonant vibration of the rotor.