Showing papers on "Angular displacement published in 2020"

A Highly Sensitive Planar Microwave Sensor for Detecting Direction and Angle of Rotation

Abstract: This article presents a technique based on a modified complementary split-ring resonator (CSRR) to detect angular displacement and direction of rotation with high resolution and sensitivity over a wide dynamic range. The proposed microwave planar sensor takes advantage of the asymmetry of the sensor geometry and measures the angle of rotation in terms of the change in the relative phase of the reflection coefficients. The sensor consists of a movable modified CSRR (the rotor) and a microstrip line with a circular defect in the ground plane (the stator). By selecting the substrate material and the rotor thickness, the sensor can be designed to work at different operating frequencies. A theoretical model of the sensor is proposed and is followed by a detailed numerical analysis involving equivalent circuit simulations, full-wave computations, and measurement results. Using positioning error estimation and air-gap analysis, a technique based on phase-change measurements is found to be better than those based on magnitude measurements alone. The maximum sensitivity for measuring the angular rotation is found to be a 4.3° change in the relative phase of the reflection coefficient per 1° of rotation. The sensor has an angular measurement range from −90° to +90°. The sensor—a stator fabricated on a 0.5-mm-thick Rogers RT5880 substrate and three rotors fabricated on a 1.5-mm-thick Rogers RT5880, a 1-mm-thick FR4, and a 0.5-mm-thick Rogers RT5880—can effectively detect the direction of rotation, measure the angle of rotation and angular velocity with reasonable accuracy.

A new Gantry-Tau-based mechanism using spherical wrist and model predictive control-based motion cueing algorithm

Abstract: The 3 degree-of-freedom Gantry-Tau manipulator with the addition of the spherical wrist mechanism which is called Gantry-Tau-3R is designed as a high-G simulation-based motion platform (SBMP) with the capability of generating the large linear and angular displacement. The combination of both parallel and serial manipulator in newly designed Gantry-Tau-3R mechanism improves the ability of the mechanism to regenerate larger motion signals with higher linear acceleration and angular velocity. The high-frequency signals are reproduced using the parallel part of the mechanism, and sustainable low-frequency accelerations are regenerated via the serial part due to the larger rotational motion capability, which will be used through motion cueing algorithm tilt coordination channel. The proportional integral derivative (PID) and fuzzy incremental controller (FIC) are developed for the proposed mechanism to show the high path tracking performance as a motion platform. FIC reduces the motion tracking error of the newly designed Gantry-Tau-3R and increases the motion fidelity for the users of the proposed SBMP. The proposed method is implemented using Matlab/Simulink software. Finally, the results demonstrate the accurate motion signal generation using linear model predictive motion cues with a fuzzy controller, which is not possible using the common parallel and serial manipulators.

High-Fidelity Hexarot Simulation-Based Motion Platform Using Fuzzy Incremental Controller and Model Predictive Control-Based Motion Cueing Algorithm

Abstract: The real vehicle motion signals cannot be applied for the simulation-based motion platforms (SBMPs) due to the restricted workspace of the SBMPs. The motion cueing algorithms (MCAs) have been used to regenerate the same motion feeling of the real vehicle with consideration of the SBMP's limited workspace. The idea of the model predictive control (MPC) is used recently to find the optimal value of the input signal with consideration of the SBMPs’ limitations. The hexarot manipulator is developed as a mid-sized SBMP with high capability of generating the angular displacement compared to hexapod SBMP. It is the first time that hexarot manipulator is employed as an SBMP with controller unit due to its high workspace limitation, especially in roll angle. In addition, the main contribution of this research is to design and develop a fuzzy incremental controller to configurate the hexarot SBMP using MPC-based MCA to improve the motion feeling of the high-frequency signals compared with the traditional PID controller. The proposed method is verified using MATLAB software. Finally, the results demonstrate the right follow of the motion signals using MPC-based MCA, which is not possible using the common hexapod manipulators.

Subdivided Error Correction Method for Photoelectric Axis Angular Displacement Encoder Based on Particle Swarm Optimization

Abstract: To improve the subdivision accuracy of the absolute photoelectric shaft encoder, a particle swarm optimization (PSO) algorithm is proposed for error correction. The error is generated by the subdivision of the grating Moire fringe signal, and a Moire fringe photoelectric signal data acquisition system is built based on a digital signal processor. First, based on the principle of measuring the angular displacement with grating diffraction interference, a mathematical model for the Moire fringe photoelectric signal is established, and then the influence of signal deviation on the subdivision error is noted. Second, the compensation model of the signal subdivision error is established. Then, the PSO algorithm is introduced, and the inertial weight and speed factor in the algorithm are optimized. Seven undetermined parameters in the signal model are identified using an optimization algorithm. Finally, according to the subdivision error compensation model, the data processing system can automatically correct the subdivision error caused by the direct current (dc) offset, orthogonality error, and sinusoidal error. Taking the 23-bit photoelectric encoder as the experimental object, the actual static subdivision error can be reduced by at least 0.30–0.46”. In the encoder operation, if the correction preprocessing is performed on all the grating Moire signals, then the subdivision accuracy of the encoder can be improved, and its spatial adaptability and angular reliability will also be improved.

An Angular-Displacement Microwave Sensor Using an Unequal-Length-Bi-Path Transversal Filtering Section

Abstract: A class of angular-displacement microwave sensor based on a microwave signal-interference transversal filtering section (TFS) with unequal-electrical-length paths is reported for the first time. It consists of a bi-path TFS composed of two in-parallel transmission-line segments with distinct lengths, which is modified by adding a rotational open-circuit-ended stub as the rotor to a curved section as the stator of one of its line segments. In this manner, the spectral positions of the transmission zeros, which are produced in its filtering transfer function through destructive signal-energy interactions, are modified with the rotation of this additional stub. This allows single/multi-band angular-displacement microwave-sensing capabilities in terms of transmission-zero inter-spacing variation for passbands/stopbands and stopband-peak-level modification to be attained by means of the mechanical rotation of the rotor. The theoretical operational principles of this type of bi-path-TFS-based angular-displacement microwave sensor and design curves for specific values of its electrical parameters are provided. Moreover, for experimental validation purposes, a proof-of-concept prototype of the devised TFS-based microwave-sensor approach operating at 961 MHz with 180° dynamic range is prototyped in microstrip technology and measured. A comparative analysis of the developed angular-displacement microwave sensor with the state-of-the-art is also presented.

An Angular Displacement Sensor Based on Microwave Transversal Signal Interference Principle

Abstract: A novel rotational angle microwave sensor with a high dynamic range and sensitivity for detecting angular displacement is presented. It exploits the spectral separation between two notch frequencies or transmission zeros that are generated in its transfer function, and whose value varies as a function of the rotation angle to be measured. The sensor is based on a bi-path transversal-signal-interference (TSI) circuit configuration that is implemented on microstrip technology, where a rotor comprised of an open-circuit-ended stub is loaded onto a stator of this transversal circuit with equal-length electrical paths. The mechanical rotation of the rotor changes the contact point of the open-circuit-ended line to the stator, thus effectively modifying one of the overall paths of the stator and, hence, the inter-transmission-zero separation. Design guidelines for the synthesis of the proposed rotation sensor with a specified dynamic range and sensitivity are provided. Furthermore, a proof-of-concept microstrip-substrate-based prototype with an angular rotation range of 180° is developed at 920 MHz and tested to validate this approach experimentally.

A Novel Approach to Separate Geometric Error of the Rotary Axis of Multi-axis Machine Tool Using Laser Tracker

Abstract: It is highly desirable to enhance machining accuracy for multi-axis machine tool, in which the geometric accuracy of rotary axis is a main contributing factor. Thus, how to achieve the fast and accurate identification of each geometric error of rotary axis, as well as its correction and compensation has become a key issue. This paper presents a novel method of geometric error separation of the rotary axis by means of laser tracker. For this method, the direction variation of the vectors composed by some adjacent measuring points during the rotation of turntable is measured, and rotary axis’s six geometric errors including three linear displacement and three angular displacement errors can be accurately identified on the basis of the mapping relationship between the vector’s direction variation and each geometric error. Meanwhile, the multi-station and time-sharing measurement is adopted based on GPS principle, aiming to overcome the effect of angle measurement error using laser tracker. Eventually, the geometric error separation mathematical model on rotary axis with this novel method is established, and the corresponding measurement algorithms containing the base station calibration and the measuring point determination based on the hybrid genetic algorithm, as well as each geometric error separation algorithm are deduced respectively. Furthermore, the numerical simulations are conducted to examine the validity of the derived algorithms. Results of the comparative experiment demonstrate that high-efficiency and high-precision measurement for the geometric error of the rotary axis can be accomplished by the proposed approach.

A Compact and High-Precision Capacitive Absolute Angular Displacement Sensor

Abstract: The present work proposes a compact capacitive absolute angular displacement sensor design that adopts a stator and rotor configuration with a double capacitive ring structure, where the outer ring is composed of n measurement periods that function as a fine measurement component, providing high precision measurements. The inner ring is composed of a single measurement period that functions as a coarse measurement component, thereby providing an absolute position measurement. High measurement accuracy is obtained by adopting single-row excitation electrodes with induction electrodes that adopt a differential sensing structure to eliminate common-mode interference. In addition to the double-ring structure, the compact size of the proposed sensor is facilitated by integrating the electrodes of the stator and rotor and the peripheral circuits required for their respective signal processing using printed circuit board technology. A prototype of the proposed sensor is fabricated with outer and inner radii, and aggregate thickness dimensions of 30 mm, 7 mm, and less than 10 mm, respectively. The sensor design is optimized accordingly by increasing the radial interval between the outer and inner rings and adding shielding. Finally, the experimental results show that the optimized sensor design achieves a precision of $12^{\prime \prime }$ and resolution of $0.31^{\prime \prime }$ over the full measurement range of 0° to 360°. As such, the proposed sensor design is demonstrated to realize the small mechanical dimensions suffered by absolute angular displacement sensors. Accordingly, the relatively low cost and low power consumption of the proposed sensor have excellent potential for a wide variety of applications.

Experimental and numerical analyses of a rotary motor using shape memory alloy mini springs

Abstract: The rotation mechanism based on an eccentric, combined with large deformations of SMA spring type actuators, allowed the design of a compact device with continuous rotation. The proposed rotary motor is driven by NiTi shape memory alloy (SMA) springs. The springs are driven by an electric current using the Joule effect as a physical principle. In this case, the motor can rotate in both directions, by only inverting the drive sequence. When driven, SMA springs combine the superelastic effect (SE) and the shape memory effect (SME), and can suffer deformations of up to 600% of their initial length. To define the design parameters, an electro-thermomechanical characterization of the SMA springs was performed, in addition to antagonistic tests to evaluate the generation of work after thermal heating. An experimental set-up measured the angular displacement, force and torque responses generated by the motor. For the numerical simulations, three different models were tested in order to define which of them best represents the behavior of force and deflection of the actuators. The most appropriate model was selected for the analyses of the static responses of the motor in rotation. The operation of the prototype was demonstrated for different driving modes, presenting results of movement, force, torque and temperature of the actuators. Numerical simulations presented a maximum error of 5.13% when compared to the experiment. The contribution of this work are numerical simulations of a rotating motor, correlated with experimental measurements. It is demonstrated that the proposed motor is in a prominent position regarding its torque/volume and torque/mass ratios when compared to other motors of the same class.

An Angular Displacement Sensor Based on Microstrip Wideband Impedance Transformer With Quasi-Chebyshev Frequency Response

Abstract: In this paper, the design methodology of a microstrip angular displacement sensor based on an impedance transformer with a quasi-Chebyshev frequency response is proposed. This sensor with impedance transforming function can realize the angular displacement detection by bandwidth sensing, and matching sensing simultaneously. The insertion loss in stopband achieves 24.9 dB to 32.6 dB with an average bandwidth selectivity of 2.25 MHz/degree of the sensor centered at 2.4 GHz with two poles of 1.71 GHz and 2.81 GHz. Meanwhile, the average matching level of return loss for displacement sensitivity are 0.18 dB/degree and 0.15 dB/degree, respectively. It is the first time to apply an impedance transformer into the angular displacement sensor design for both bandwidth and matching level sensing among reviewed literatures. In this design, the load-to-source impedance transformation is designed for 250-to- $50~\Omega $ , and by adding a rotated stub on the proposed microstrip transformer, the sensing function is developed with angular displacement.

A new joint friction model for parameter identification and sensor-less hand guiding in industrial robots

Abstract: The purpose of this paper is to propose a new joint friction model, which can accurately model the real friction, especially in cases with sudden changes in the motion direction. The identification and sensor-less control algorithm are investigated to verify the validity of this model.,The proposed friction model is nonlinear and it considers the angular displacement and angular velocity of the joint as a secondary compensation for identification. In the present study, the authors design a pipeline – including a manually designed excitation trajectory, a weighted least squares algorithm for identifying the dynamic parameters and a hand guiding controller for the arm’s direct teaching.,Compared with the conventional joint friction model, the proposed method can effectively predict friction factors during the dynamic motion of the arm. Then friction parameters are quantitatively obtained and compared with the proposed friction model and the conventional friction model indirectly. It is found that the average root mean square error of predicted six joints in the proposed method decreases by more than 54%. The arm’s force control with the full torque using the estimated dynamic parameters is qualitatively studied. It is concluded that a light-weight industrial robot can be dragged smoothly by the hand guiding.,In the present study, a systematic pipeline is proposed for identifying and controlling an industrial arm. The whole procedure has been verified in a commercial six DOF industrial arm. Based on the conducted experiment, it is found that the proposed approach is more accurate in comparison with conventional methods. A hand-guiding demo also illustrates that the proposed approach can provide the industrial arm with the full torque compensation. This essential functionality is widely required in many industrial arms such as kinaesthetic teaching.,First, a new friction model is proposed. Based on this model, identifying the dynamic parameter is carried out to obtain a set of model parameters of an industrial arm. Finally, a smooth hand guiding control is demonstrated based on the proposed dynamic model.

Motion Estimation for a Compact Electrostatic Microscanner via Shared Driving and Sensing Electrodes in Endomicroscopy

Abstract: In this article, we present a method to estimate high-frequency rotary motion of a highly compact electrostatic microscanner using the same electrodes for both actuation and sensing. The accuracy of estimated rotary motion is critical for reducing blur and distortion in image reconstruction applications with the microscanner given its changing dynamics due to perturbations such as temperature. To overcome the limitation that no dedicated sensing electrodes are available in the proposed applications due to size constraints, the method adopts electromechanical amplitude modulation to separate motion signal from parasitic capacitance feedthrough, and a novel nonlinear measurement model is derived to characterize the relationship between large out-of-plane angular motion and circuit output. To estimate motion, an extended Kalman filter and an unscented Kalman filter are implemented, incorporating a process model based on the microscanner's parametric resonant dynamics and the measurement model. Experimental results show that compared to estimation without using the measurement model, our method can improve the rotary motion estimation accuracy of the microscanner significantly, with a reduction of root-mean-square error (RMSE) in phase shift of 86.1%, and a reduction of RMSE in angular position error of 78.5%.

Diffusion of Anisotropic Particles in Random Energy Landscapes—An Experimental Study

Abstract: If a colloidal particle is exposed to an external field, its Brownian motion is modified. In the case of an anisotropic particle, the external potential might not only affect its translation but also its rotation. We experimentally investigate the dynamics of a trimer, which consists of three spherical particles, within a random potential energy landscape. This energy landscape has energy values drawn from a Gamma distribution, a spatial correlation length similar to the particle size and is realised by a random light field, that is a laser speckle pattern. The particle translation and rotation are quantified by the mean squared (angular) displacement, the van Hove function and other observable quantities. The translation shows an intermediate subdiffusive regime and a long-time diffusion that slows down upon increasing the modulation of the potential. In contrast, the mean squared angular displacement exhibits only small deviations from a linear time dependence but a more detailed analysis reveals discrete angular jumps reflecting the symmetry of the trimer. A coupling between the translation and rotation is observed and found to depend on the length scale.

Establishing metrics and control laws for the learning process: ball and beam balancing.

Abstract: Understanding how dexterity improves with practice is a fundamental challenge of motor control and neurorehabilitation. Here we investigate a ball and beam implementation of a dexterity puzzle in which subjects stabilize a ball at the mid-point of a beam by manipulating the angular position of the beam. Stabilizability analysis of different biomechanical models for the ball and beam task with time-delayed proportional-derivative feedback identified the angular position of the beam as the manipulated variable. Consequently, we monitored the changes in the dynamics with learning by measuring changes in the control parameters. Two types of stable motion are possible: node type (nonoscillatory) and spiral type (oscillatory). Both types of motion are observed experimentally and correspond to well-defined regions in the parameter space of the control gains. With practice the control gains for each subject move close to or on the portion of the boundary which separates the node-type and spiral-type solutions and which is associated with the rightmost characteristic exponent of smallest real part. These observations suggest that with learning the control gains for ball and beam balancing change in such a way that minimizes overshoot and the settling time. This study provides an example of how mathematical analysis together with careful experimental observations can shed light onto the early stages of skill acquisition. Since the difficulty of this task depends on the length of the beam, ball and beam balancing tasks may be useful for the rehabilitation of children with dyspraxia and those recovering from a stroke.

Multi-Sensor Orientation Tracking for a Façade-Cleaning Robot.

Abstract: Glass-facade-cleaning robots are an emerging class of service robots. This kind of cleaning robot is designed to operate on vertical surfaces, for which tracking the position and orientation becomes more challenging. In this article, we have presented a glass-facade-cleaning robot, Mantis v2, who can shift from one window panel to another like any other in the market. Due to the complexity of the panel shifting, we proposed and evaluated different methods for estimating its orientation using different kinds of sensors working together on the Robot Operating System (ROS). For this application, we used an onboard Inertial Measurement Unit (IMU), wheel encoders, a beacon-based system, Time-of-Flight (ToF) range sensors, and an external vision sensor (camera) for angular position estimation of the Mantis v2 robot. The external camera is used to monitor the robot's operation and to track the coordinates of two colored markers attached along the longitudinal axis of the robot to estimate its orientation angle. ToF lidar sensors are attached on both sides of the robot to detect the window frame. ToF sensors are used for calculating the distance to the window frame; differences between beam readings are used to calculate the orientation angle of the robot. Differential drive wheel encoder data are used to estimate the robot's heading angle on a 2D facade surface. An integrated heading angle estimation is also provided by using simple fusion techniques, i.e., a complementary filter (CF) and 1D Kalman filter (KF) utilizing the IMU sensor's raw data. The heading angle information provided by different sensory systems is then evaluated in static and dynamic tests against an off-the-shelf attitude and heading reference system (AHRS). It is observed that ToF sensors work effectively from 0 to 30 degrees, beacons have a delay up to five seconds, and the odometry error increases according to the navigation distance due to slippage and/or sliding on the glass. Among all tested orientation sensors and methods, the vision sensor scheme proved to be better, with an orientation angle error of less than 0.8 degrees for this application. The experimental results demonstrate the efficacy of our proposed techniques in this orientation tracking, which has never applied in this specific application of cleaning robots.

Angular measurement of high precision reducer for industrial robot

Abstract: High precision reducer is the core component of industrial robot. Its performance test is the key to improve product quality and ensure the accuracy of robot operation. One of the basic geometric parameters in performance test is the angle, which refers to the angular displacement of the input or output of the tested reducer. Based on a vertical instrument, angular measurement systems are designed in this article. In order to ensure the precision of results, the angle encoder is as close as possible to the measured object, which shortens the measurement chain and reduces the number of error sources. A cylindrical and disk structure improves the structural stiffness, ensures regular deformation patterns, and minimizes the angular errors caused by weak stiffness supports in the conventional instrument. Another system consisting of an autocollimator and a regular polyhedron, whose output is taken as angular standard, calibrates absolute circular gratings in the instrument. The model based on back propagation neural network can compensate the errors over the whole circumference, and finally get the angular displacement of input and output of the tested reducer in the drive process accurately. After compensation, the angular measurement errors of the instrument are within ±2 arc seconds.

Angular displacement estimation enhanced by squeezing and parametric amplification

Abstract: We theoretically study angular displacement estimation based on a modified Mach-Zehnder interferometer (MZI), in which two optical parametric amplifiers (PAs) are introduced into two arms of the standard MZI, respectively. The employment of PAs can both squeeze the shot noise and amplify the photon number inside the interferometer. When the unknown angular displacements are introduced to both arms, we derive the multiparameter quantum Cramer-Rao bound (QCRB) using the quantum Fisher information matrix approach, and the bound of angular displacement difference between the two arms is compared with the sensitivity of angular displacement using the intensity detection. On the other hand, in the case where the unknown angular displacement is in only one arm, we give the sensitivity of angular displacement using the method of homodyne detection. It can surpass the standard quantum limit (SQL) and approach the single parameter QCRB. Finally, the effect of photon losses on sensitivity is discussed.

Semiempirical identification of nonlinear dynamics of a two-degree-of-freedom real torsion pendulum with a nonuniform planar stick–slip friction and elastic barriers

Abstract: The purpose of this study is to identify the nonlinear dynamics of the double torsion pendulum with planar friction and elastic barriers. The original experimental stand consists of a disk-shaped body that rotates freely on top of a forced column with a system of barriers limiting the torsional vibrations of the pendulum bodies that create an nonuniform planar rotational friction contact. Two beam springs form soft barriers modeled by Voigt elements that limit the angular displacement of one of the pendulum bodies—the disk, while the second limiting system, made of a much more rigid barrier, limits the movement of the pendulum’s second body. The dynamic behavior of the asymmetrical system of two degrees of freedom with discontinuities is identified with the use of the described strategy, numerical solutions of the derived mathematical model and the Nelder–Mead simplex algorithm. The actual measurement series and numerical solutions show a good similarity of the dynamical reaction of the mechanical system and its virtual analog.

A Singularity-free Steering Law of Roof Array of Control Moment Gyros for Agile Spacecraft Maneuver

Abstract: In this paper, a singularity-free steering law for single gimbal control moment gyros (CMGs) is addressed for agile spacecraft. The geometrical array considered particularly in this work is a roof array due to the simplicity of singularity envelope. A feasible angular momentum chart which can provide a singularity-free bound is employed. The chart allows a guaranteed maximum torque output and angular momentum at any time without concerning the geometrical singularity of the array. Furthermore, a new deterministic allocation algorithm, called half-leading steering logic, of gimbal angular rates, is also suggested instead of the well-known pseudo-inverse technique to meet control torque commands required and to keep away from the singularity. It is noted that a momentum vector recovery to the initial direction is also an important task for the CMG array to overcome the singularity and for the reliable operation of CMGs. An optimization technique is addressed to restore the gimbal vectors back to their original angular position after the attitude reorientation mission. The techniques proposed are demonstrated using illustrative numerical simulations.

Dynamics of two freely rotating dipoles

Abstract: The equations of motion for two spherical dipoles moving freely in a plane are obtained. Special consideration is given to when the two spheres are in contact. Investigations of equilibria, small-amplitude motion, and large-amplitude motion reveal that possible motions are exclusively quasi-periodic. Two distinct modes are identified, one of which is isomorphic with the simple pendulum, complete with a regime where it ceases to be periodic, and the angular displacement grows continuously at high energy.

A New Butterfly-Inspired Compliant Joint with 3-DOF In-plane Motion

Abstract: This paper proposes a new butterfly-inspired compliant joint by combining a butterfly’s profile and a foldable mechanism. The joint can achieve three degrees of freedom in its fabrication plane, including two translations in the x- and y-axes and a rotary motion around the z-axis. Equivalent spring stiffness of the joint is easy to be adjusted by adopting a foldable mechanism, while a butterfly’s profile gives better compliance. Closed-form model of the joint is established to calculate stiffness, displacement, and rotational angle. Performances and effectiveness of the proposed joint are verified by comparing with other joints. A prototype is fabricated, and experiments are conducted. The results show a good agreement between the analytical model, simulation, and experiment. Compared with a conventional rectangular flexure joint in terms of bending displacement and compressive displacement, the results found that performances of the joint are greatly improved. Bending displacement and angular displacement of the joint are increased by up to 14.6% and 12.6%, respectively. The proposed joint can be considered as a potential candidate for a precise positioning system.

Smart-Device-Based Radar System Performing Angular Estimation Using Machine Learning

Abstract: Techniques and apparatuses are described that implement a smart-device-based radar system capable of performing angular estimation using machine learning. In particular, a radar system (102) includes an angle-estimation module (504) that employs machine learning to estimate an angular position of one or more objects (e.g., users). By analyzing an irregular shape of the radar system (102)'s spatial response across a wide field of view, the angle-estimation module (504) can resolve angular ambiguities that may be present based on the angle to the object or based on a design of the radar system (102) to correctly identify the angular position of the object. Using machine-learning techniques, the radar system (102) can achieve a high probability of detection and a low false-alarm rate for a variety of different antenna element spacings and frequencies.

Equilibrium analysis of an immersed rigid leaflet by the virtual element method

Abstract: We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees $1$ and $2$ combined with a bisection algorithm. We present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology.