Rigid Hub Research Articles

Rigid–Flexible Coupled Modeling and Flexoelectric Control for Flexible Rotating Structures

The pursuit of lightweight structures has propelled spacecraft with large flexible appendages to the forefront of modern aerospace engineering. Flexoelectric actuators, with their inherent self-sensing and self-actuating capabilities, perfectly align with the demand for lightweight structures. This paper presents a dynamic model and a flexoelectric vibration control method tailored for spacecraft equipped with single-sided flexible appendages. The spacecraft system is a central rigid hub with flexible cantilever beam appendages that allow for single-axis rotation. The flexible beam has a flexoelectric element attached to it. Leveraging Hamilton’s principle, the equations of motion are derived, accounting for flexoelectric effects, the centrifugal stiffening effect, and the coupling between rigid body attitude and flexible beam vibration. A closed-loop proportional–derivative controller is crafted for attitude control of the system. Additionally, a flexoelectric patch is employed as an actuator for active vibration control of the flexible beam. The precision and efficacy of the proposed model and control scheme are meticulously validated through numerical simulation. The results indicate that the flexoelectric patch can effectively manage the vibration of the flexible beam, consequently impacting the attitude control performance of the system. Furthermore, the flexoelectric effect of the system under static conditions is scrutinized, and the influence of the flexoelectric actuator on the rigid–flexible coupling characteristics of the system is thoroughly analyzed. These analyses not only confirm the feasibility of flexoelectricity in rigid–flexible coupled systems, but also open avenues for applications and optimizations in intelligent control methods for flexible appendages.

General Dynamics for Single- and Dual-Axis Rotating Rigid Spacecraft Components

Deriving and propagating a spacecraft’s equations of motion is fundamental to describing its behavior accurately. These equations of motion depend on the spacecraft’s configuration, which includes any physical subsystem such as attitude control devices, solar panels, gimbals, etc. Prior work introduced the backsubstitution method to yield a modular and scalable formulation to develop complex spacecraft dynamics specific to rotating components attached to a rigid hub as effectors. This paper relaxes assumptions made in deriving effector components in prior work, such as mass properties and frame definitions. This produces a general architecture that uses common equations of motion for physically equal parts. The result is an analytical solution of a set of general rotating effector equations of motion that greatly expand the configuration space of spacecraft that can be simulated with the backsubstitution method. In contrast to prior work where the rotations are highly constrained, rigid-body components can rotate about one or two general axes, and the component mass distribution can be general, no longer requiring the component’s principal axis to align with the center of mass or hinge axis. A numerical software solution demonstrates and verifies how these effectors can mimic a range of dynamic spacecraft components.

PDE-based distributed adaptive fault-tolerant attitude consensus of multiple flexible spacecraft

In this article, a distributed adaptive fault-tolerant control scheme is presented for the attitude consensus of multiple flexible spacecraft formulated by partial differential equations (PDEs). Each spacecraft is supposed to be actuated by a control torque on the rigid hub and a control force on the tip payload. First, a finite-time distributed observer is constructed to estimate the virtual leader's state for each follower spacecraft under local communication. Then, an individual controller is exploited for each follower spacecraft, which consists of an adaptive boundary control torque law to track the recovered attitude and velocity and an adaptive boundary control force law to eliminate the elastic deformation. Particularly, the parametric adaptation laws are incorporated to compensate the actuator faults and perturbations. The attitude synchronization errors, velocity synchronization errors, and elastic deformations can asymptotically stabilize to zero under the introduced controller. This work owns the following two major novelties. One is introducing a unified distributed control framework for the PDE-based attitude consensus of multiple flexible spacecraft. Another is developing a novel parametric adaptation strategy to accommodate the actuator faults. Lastly, comparative simulations verify and highlight the main results.

Vibration Study of Rotating Rigid Hub and a Flexible Composite Beam under Simultaneous Primary-internal Resonance

In this paper, the vibration of a composite system consisting of a rotating rigid hub and a flexible thin-walled beam is considered and studied. The equation of motion is derived in the previous work of Warminski and Latalski. The method of multiple scale technique has been applied to obtain frequency response equations near the simultaneous internal and primary resonance case in the absence of the acceleration of the hub. The vibration stability at this resonance case is investigated from the frequency response equations and studied using Liapunov’s methods. The effects of changes in selected structural parameters on the vibrating system behavior are investigated and studied numerically. Through the performed studies of the effects of changes in selected system a shift of the steady state amplitudes and the multi-valued of the bent curves are observed and the steady state amplitudes of the beam and hub have decreasing in the instability regions for natural frequencies. Finally, a comparison with the papers of previously published work is reported.

Description of an HAWT hub enabling teetering about two out‐of‐plane axes with modeling showing significant reductions in cyclic behavior and fatigue due to turbulence and wind shear

AbstractA three‐bladed, dual‐axis teetering hub (3T) has been developed and compared to a three‐bladed rigid hub (3R) using computer modeling. Both hubs were fitted to a hypothetical 5‐MW offshore wind turbine mounted on an offshore floating platform. Cyclic analysis showed the 3T hub significantly reduced the impact of wind shear on the blades and virtually eliminated the impact of both turbulence and wind shear on the main shaft bearing and yaw bearing. Lifetime fatigue studies for the two hubs were performed using a rainflow‐counting program at various levels of turbulence and wind shear. Studies showed the significant reductions in cyclic variation provided by the 3T hub resulted in significant reductions in lifetime fatigue in comparison to the 3R hub. Simulations with the 3T hub showed an increase in turbulence caused a very small increase in blade fatigue but no increase in fatigue for the main shaft bearing and yaw bearing. In contrast, fatigue studies with the 3R hub showed an increase in turbulence caused significant damage to the blades and a combined increase in turbulence and wind shear caused extremely significant damage to the blades. This paper introduces a recently patented hub design that is superior to the ball‐and‐socket design described in a prior paper by this author.

Size-Dependent Rigid–Flexible Coupling Dynamics of Functionally Graded Rotating Moderately Thick Microplates

Micro air vehicles, which are typical small-sized rotating-motion systems, have seen major advancements in recent years. To provide some theoretical basis for developing and producing micro air vehicles, this study establishes a novel rigid–flexible coupling dynamic model for functionally graded (FG) moderately thick rectangular microplates attached to a central rotating rigid hub based on the modified couple stress theory and first-order shear deformation theory. The proposed model incorporates nonlinear coupling term of in-plane deformation to reflect the dynamic stiffening effect caused by rotational motion. Material characteristics of the FG microplate have a linear power-law distribution along the thickness axis. Further, the discrete form dimensionless coupling dynamic equations and their numerical solutions are obtained by combining the Euler–Lagrange equation and the Chebyshev–Ritz method. Convergence and comparative studies are carried out to demonstrate the accuracy and validity of the proposed model. Thereafter, the influence of material length scale parameter, rotational speed, gradient index, and aspect ratio on the frequency of the microplates is investigated. Numerical results reveal that couple stress and dynamic stiffening effects both enhance the rigidity of the microplates, whereas the gradient index decreases the rigidity. Nonlinear coupling term which leads to significant differences in frequency value and trace line can’t be ignored for rotative structure. In-plane motion and its coupling terms play a significant function for the moderately thick or thick microplates. The increase of rotational speed and gradient index will reduce the size dependency of the microplate. Furthermore, the frequency trajectory steering and corresponding mode transition phenomenon are graphically represented.

Modular incorporation of deformable spine and 3D neck musculature into a simplified human body finite element model

Spinal injuries are a concern for automotive applications, requiring large parametric studies to understand spinal injury mechanisms under complex loading conditions. Finite element computational human body models (e.g. Global Human Body Models Consortium (GHBMC) models) can be used to identify spinal injury mechanisms. However, the existing GHBMC detailed models (with high computational time) or GHBMC simplified models (lacking vertebral fracture prediction capabilities) are not ideal for studying spinal injury mechanisms in large parametric studies. To overcome these limitations, a modular 50th percentile male simplified occupant model combining advantages of both the simplified and detailed models, M50-OS + DeformSpine, was developed by incorporating the deformable spine and 3D neck musculature from the detailed GHBMC model M50-O (v6.0) into the simplified GHBMC model M50-OS (v2.3). This new modular model was validated against post-mortem human subject test data in four rigid hub impactor tests and two frontal impact sled tests. The M50-OS + DeformSpine model showed good agreement with experimental test data with an average CORrelation and Analysis (CORA) score of 0.82 for the hub impact tests and 0.75 for the sled impact tests. CORA scores were statistically similar overall between the M50-OS + DeformSpine (0.79 ± 0.11), M50-OS (0.79 ± 0.11), and M50-O (0.82 ± 0.11) models (p > 0.05). This new model is computationally 6 times faster than the detailed M50-O model, with added spinal injury prediction capabilities over the simplified M50-OS model.

Dynamic model for hydropower generators with floating rotor rim

For rotordynamic analysis of hydropower units, the generator is treated as a rotating rigid body. However, previous studies have confirmed that certain designs of generators are elastic, so the complex geometry of generators cannot be considered rigid. This work produced a model of hydropower generators with floating rotor rims, consisting of a rigid hub and a flexible rotor rim coupled with flexible connections. The model takes into account the influence of centrifugal and Coriolis effects, and the electromagnetic interaction between rotor and stator. The model also reproduces the dynamics of the generator with static and dynamic eccentricities. A generator prototype was employed to test the model, showing its different applications. Once validated by empirical data, this model could be used when designing generators.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

Nonlinear dynamic characteristics of horizontal-axis wind turbine blades including pre-twist

This paper focuses on the dynamic modeling and vibration analysis of horizontal axis wind turbine (HAWT) blades with emphasis on the effects of the pre-twist, rotational rigid body motion and coupling among the different blade's degrees of freedom. The blade is considered as a rotating pre-twisted beam mounted on a rigid hub with a constant angle relative to the plane of rotation, so-called precone angle. Kinetic energy of the blade based on its elastic as well as rigid motion together with its strain energy and generalized forces resulting from gravitational, centrifugal and aerodynamic loadings are obtained and used to derive new coupled dynamic equations by means of Hamilton's principle. The weak formulation is constructed and the Rayleigh-Ritz method (RRM) is adopted to extract a reduced order model (ROM) for the blade's vibration analysis. The developed nonlinear ROM is linearized around the steady-state solution and the blade's dynamic characteristics are obtained. The reference wind turbine blade called NREL 5-MW HAWT blade is selected as the case-study and effects of pre-twist, centrifugal force, angular velocity-depended terms, nonlinearities and transverse static loads on its dynamic characteristics are investigated. Results are shown to be in good agreement with available data in the literature.

Investigation of an Influence Function Model as a Self-Rotating Wheel Polishing Tool and Its Application in High-Precision Optical Fabrication

A new and patented polishing tool (ZL2020102387137) called a Self-rotating Wheel Polishing Tool (SWPT) was built, and its tool influence function (TIF) was investigated in this study. The polishing wheel is an innovative two-layer structure: a rigid hub inside and a flexible polishing pad outside. By using finite element analysis, the dynamic contact characteristics between the polishing wheel and the workpiece were studied, and the theoretical TIF was modeled. Due to the influence of friction resistance, the TIF is not circular, but oval. We then ran material removal experiment, and it was found that the experimental TIF and the theoretical TIF are very close and both are close to the Gaussian shape. Finally, optical fabrication was performed. After four times of about 3 h fabrication, the form error converged from PV-1.434λ (λ = 632 nm), RMS-0.308λ to PV-0.144λ, RMS-0.009λ, and PV and RMS converged by 90% and 97%, respectively. The form accuracy achieved the expected target of RMS-0.02λ, which proves that the SWPT has the characteristics of high convergence rate and high fabrication accuracy. The SWPT has a broad application prospect in the field of high-precision optical fabrication.

Nonlinear longitudinal–bending–twisting vibrations of extensible slowly rotating beam with tip mass

Nonlinear longitudinal–bending–twisting vibrations of a slowly rotating extensible beam are studied in the paper. The beam model is based on extended Euler–Bernoulli theory with shear deformation neglected. Geometrical nonlinearities, caused by large oscillations, couple longitudinal vibrations, bending in two directions and twist. The beam is fixed to a rigid hub at a given preset angle and carries mass attached at the beam’s tip. The governing partial differential equations and dynamical boundary conditions are derived from the extended Hamilton principle. The model is solved analytically up to the third order perturbation by the multiple time scale method applied directly to the partial differential equations. The influence of angular velocity of the hub, tip mass and other structural beam parameters on nonlinear vibrations around selected resonance zones is presented.

Study of Tweel Non-Pneumatic Tires

Abstract: Non-Pneumatic Tyre (NPT) is an airless tire that supports the weight of the car. Although tires manufactured of solid rubber exist, they don't provide a smooth ride even though they have enough compliance if used in conventional vehicles. The NPT discussed here is mainly divided into three parts. Deformable spokes that support the downward weight, a rigid hub, Tread made out of rubber which comes into contact with the road and Reinforced shear band also plays a vital role. The properties of Non-Pneumatic Tyres like contact pressure, rolling resistance, and weight carrying capacity can be varied by differentiating the dimensions or the manufacturing materials used to produce NPT. Many kinds of research are taking place all over the world to make NPT an alternative option to the traditional pneumatic tire over a decade. This paper integrates an overview of the research works that were carried out to develop and improve NPT. Keywords: Pneumatic, Tweel, Honeycomb structure, Hyperelastic

Static analysis model of a non-pneumatic tire with elastic spokes contacting with rigid ground

An analytical model of a non-pneumatic tire is proposed to study the static deformation responses of a non-pneumatic tire in contact with a rigid ground. The tire consists of a shear band which is formed by an annular beam, and elastic spokes that connect the shear band to the rigid hub of the tire. The shear band is modeled using a Timoshenko beam. The spokes are modeled by linear springs, which are distributed evenly in circumferential direction. Governing equations of the model were derived using a theoretical analysis. The shear band static deformation was obtained based on the discussion of the relationship between spoke stiffness and the parameters of the shear band. A finite element model was developed to verify the accuracy of the model. As a part of the results from this study, a parametric analysis of quantities of interest for the tire is presented, which can be used in improving the optimal design of non-pneumatic tires. This scheme offers a holonomic solution for the complicated differential equations and gives a computationally efficient tool for rapid analyzing and designing such systems.

Vibration, synchronization and localization of three-bladed rotor: theoretical and experimental studies

Numerical and experimental methods in free and forced vibrations of the rotating structure consisting of the rigid hub and three flexible beams are considered. Firstly, the system of four mutually coupled dimensionless differential governing equations is presented and then forced response of the system as well as synchronization phenomenon are investigated. Next, the finite elements method is used to design the rotating structure and analyse complex dynamic response. During the numerical calculations symmetric, as well as de-tuned rotor are analyzed. Finally, results obtained from ordinary differential equations and numerical simulations are compared with experimental tests.

Comparison of the Satellite Attitude Control System Design using the H∞ Method and H∞/MLI with Pole Allocation Considering the Parametric Uncertainty

This paper presents the comparison of the Attitude Control System (ACS) design for a rigid-flexible satellite with two vibrations mode, using the traditional H∞method and the H∞/LMI with pole allocation considering the parametric uncertainty. In the ACS design is important take into account the influence of thestructure’s flexibility, since they can interact with the satellite rigid motion, mainly, during translational and/or rotational manoeuvrer, damaging the ACS. Usually the mathematics model obtained from the linearization and/or reduction of the rigid flexible model loses information about the flexible dynamical behaviour and introduces some uncertainty. The satellite model is represented by a flexible beam connected to a central rigid hub considering a setof parametric uncertainties. Simulations results have shown that the control law designed by the H∞/LMI method has better performance and it is more robust than H∞method since the first was able to support the action of the uncertainty perturbation and to control the rigid flexible satellite attitude and suppressing vibrations.

Nonlinear dynamic behavior of rotating blade with breathing crack

This study aims at investigating the nonlinear dynamic behavior of rotating blade with transverse crack. A novel nonlinear rotating cracked blade model (NRCBM), which contains the spinning softening, centrifugal stiffening, Coriolis force, and crack closing effects, is developed based on continuous beam theory and strain energy release rate method. The rotating blade is considered as a cantilever beam fixed on the rigid hub with high rotating speed, and the crack is deemed to be open and close continuously in a trigonometric function way with the blade vibration. It is verified by the comparison with a finite element-based contact crack model and bilinear model that the proposed NRCBM can well capture the dynamic characteristics of the rotating blade with breathing crack. The dynamic behavior of rotating cracked blade is then investigated with NRCBM, and the nonlinear damage indicator (NDI) is introduced to characterize the non-linearity caused by blade crack. The results show that NDI is a distinguishable indicator for the severity level estimation of the crack in rotating blade. It is found that severe crack (i.e., a closer crack position to blade root as well as larger crack depth) is expected to heavily reduce the stiffness of rotating blade and apparently result in a lower resonant frequency. Meanwhile, the super-harmonic resonances are verified to be distinguishable indicators for diagnosing the crack existence, and the third-order super-harmonic resonances can serve as an indicator for the presence of severe crack since it only distinctly appears when the crack is severe.

Nonlinear vibrations of an extensional beam with tip mass in slewing motion

Dynamics of a rotor composed of a flexible beam attached to a slewing rigid hub is presented in the paper. Dynamics of the structure is studied for a slender beam model, based on extended Bernoulli–Euler theory, which takes into account a nonlinear curvature, coupled transversal and longitudinal oscillations and non-constant angular velocity of the hub. Moreover, to demonstrate a general case for dynamical boundary conditions, lumped mass fixed at the beam tip is added. The partial differential equations (PDEs) are derived from Hamilton principle of the least action. The analytical solutions of the PDEs are obtained by the multiple time scale method applied directly to PDEs. Forced vibrations around selected resonance zones are studied and the influence of beam rotation, preset angle, hub radius, tip mass is presented. Hardening and softening phenomena, respectively for the first and the second mode, are obtained for various angular velocity values.

Comparison of Neck Injury Criteria Values Across Human Body Models of Varying Complexity.

Due to the severity and frequency of cervical spine injuries, the neck injury criterion (Nij) was developed to provide a quantitative relationship between forces and moments of the upper neck with accompanied injury risk. The present study was undertaken to evaluate differences in calculated Nij for the Global Human Body Model Consortium’s detailed and simplified average 50th percentile male models. The simplified model is a computationally light version of the more detailed model and therefore it is of interest to achieve similar Nij values between the two models. These models were compared in 15 match paired conditions of rigid head impact and a mixture of seven full body rigid hub and sled pulses, for 44 total simulations. Collectively, Nij values of the simplified model were found to exhibit a second-degree polynomial fit, allowing for a conversion to the prediction of the detailed model. Correlates were also derived for impact and inertial loading cases individually, for which the latter may be the subject of future work. The differences in Nij may be attributed to a variety of modeling approach differences related to neck muscles (attachment location and morphometric implementation), localization of head mass within the M50-OS, head geometry, and intervertebral joint space properties. With a primary focus on configurations in the anterior-posterior direction, there is a potential limitation in extensibility to lateral loading cases. In response to the relatively low Nij values exhibited, future work should evaluate the appropriateness of the established critical intercepts of Nij for computational human body models.

Generalized Reissner analysis of large axisymmetric deflections of thin circular and annular plates

A generalized Reissner theory for axisymmetric problems of thin circular and annular plates is applied to various problems. The plates are assumed to be linearly elastic, and shear deformation is neglected. Large deflections, rotations, and strains are allowed. The plate edges may be clamped or simply supported, movable or immovable. The types of loading considered are uniform pressure, central concentrated load, uniform edge moment, full ponding, and central rigid hub pushed downward (with or without an elastic foundation). In some cases, the effects of Poisson's ratio and the radius-to-thickness ratio are investigated, and the maximum deflections are compared to those based on von Kármán theory and Reissner theory.