Rigid Body Research Articles

Active interlocking metasurfaces enabled by shape memory alloys

Interlocking metasurfaces (ILMs) are a newly developed joining technology that relies on arrays of interlocking features that transmit force and constrain motion between adjoining bodies in one or more directions. This study explores harnessing the shape memory effect (SME) in Nickel-Titanium shape memory alloys (NiTi SMAs) in structures fabricated using additive manufacturing (AM) to advance the development of active ILMs by creating unit cells that open or close at specific temperatures. The study encompasses designing and fabricating two distinct interlocking array configurations using near-equiatomic NiTi powder and the laser powder bed fusion (L-PBF) AM technique, following a previously developed AM process optimization framework to manufacture defect-free parts. To guide the design process, finite element analysis (FEA) was employed to predict strain values during engage-disengage cycles. The martensitic transformation characteristics of the ILMs were characterized. Thermomechanical testing revealed that the ILMs demonstrate high locking force once engaged, coupled with complete shape recovery and good cyclic stability. Digital image correlation (DIC) was also employed to validate the FEA predictions during the engage-disengage cycles. The results indicate that NiTi SMA-based ILMs can be designed and fabricated into complex shapes using L-PBF. By leveraging the SME, the functionality of an ILM can be improved upon. The combination of computational modeling, additive manufacturing, and thermomechanical and physical property characterization provides a framework for designing future ILMs out of active materials.

Data-driven sensor fault diagnosis for vibration-based structural health monitoring under ambient excitation

In vibration-based structural health monitoring (SHM), the early detection of sensor faults is key to preventing false alarms and misleading conclusions on the condition of monitored structures. Since sensor networks are exposed to hostile environments, they are prone to unexpected errors that might influence the quality of measured data. This paper proposes a novel method for detecting and isolating faulty sensors from vibration response data by establishing an overdetermined system between the measured signals and the actual motion. The method assumes a rigid body motion of the monitored system, describable by a limited number of degrees of freedom (DOFs), to define the overdetermined relation between the sensor outputs and the system’s DOFs. The concept is later extended to systems not governed by rigid body motions by considering their vibration mode shapes. The robustness of the proposed methodology is demonstrated using vibration response data from an experimental monitoring campaign.

Dynamics simulation-based packing of irregular 3D objects

The 3D packing problem has a wide range of applications. However, the complex geometry of irregular objects leads to a sharp increase in the number of placement combinations, making it a challenging problem. In this paper, we propose a packing pipeline based on rigid body dynamics simulation to deal with two types of 3D packing problems. One is the variant bin packing problem, which involves placing more objects into a container of given dimensions to maximize space utilization. The other is the open dimension problem, where the goal is to minimize the container that can accommodate all objects. We first use heuristic placement strategies and a fast collision detection algorithm to efficiently obtain initial packing results. Then, we simulate the shaking of the container according to the dynamic principle. Combined with the vacant space filling operation, shaking the container drives the movement of objects in the container to make the arrangement of objects more compact. For the open dimension packing, the container height is optimized by adjusting the constraints of simulation in the basic pipeline. Experimental results show that our method has advantages over existing methods in both speed and packing density.

Angular velocity observer design using vector measurements with application to attitude tracking

AbstractThis article addresses the design of angular velocity observer for rigid bodies using vector measurements directly. The main contributions of our observer compared with the classical estimation algorithms lie in that it does not need to reconstruct the attitude information from the measurements nor be implemented with velocity sensors. Meanwhile, it provides three‐degrees‐of‐freedom (3DOF) full state estimates. To avoid the unstable critical points for the attitude estimation errors on SO(3), the synergistic potential functions based hybrid method is adopted. Afterwards, an output feedback attitude tracking controller with a proportional‐derivative (PD) plus feedforward form is built based on the estimates from the observer, which can also be implemented using vector measurements directly. Rigorous analysis of the stability of the observer and the combined observer‐controller are presented in detail, with a separation property shown between the observer and the controller. Simulations are conducted to illustrate the effectiveness of the proposed observer and the combined attitude tracking law.

Dynamic and structural analyses of floating dock operations considering dock-vessel coupling loads

Floating docks are important for increasing the capacity of land-based shipyards and serve as essential platforms for vessel construction, maintenance, and repair. As the size of modern vessels increases, ensuring the structural safety of the floating docks becomes challenging. To enhance the operational safety of the floating docks, this study aims to develop an in-house code for a comprehensive global response assessment of a full-scale floating dock. The code is developed under a quasi-static assumption and enables dynamic and structural response assessments for the floating dock and docked vessel during the vessel-docking operations. The floating dock and docked vessel are represented as rigid bodies with six degrees of freedom and subjected to hydrostatic, hydrodynamic, mooring, and contact loads. The piping network of the floating dock's ballast water system is modelled to calculate the flow rates out of all ballast tanks. A modified proportional controller (P-controller) is employed to achieve automatic ballast control, regulating opening angles of ballast tank valves to minimize roll and pitch motions of the floating dock and docked vessel. For structural response assessment, a bending model is proposed to evaluate the bending deformation of the floating dock based on the load distribution along the dock's length. Results show that the present code can successfully simulate the vessel-docking process. The automatic ballast control strategy effectively minimizes roll and pitch for both the floating dock and docked vessel, ensuring stable control of the entire system. Minimal discrepancies are observed in the dynamic responses and bending of the floating dock between one- and two-way couplings. The longitudinal shift of the docked vessel's centre of gravity (CoG) increases the relative motions between the docked vessel and the floating dock, especially in surge, sway and pitch. A significant shift distance might cause the mooring rope breakage. Positioning the vessel's CoG closer to the dock's CoG proves safer for vessel-docking operations. In general, the developed code can support the decision-making for dock masters and optimize the docking plan.

Design and Implementation of Fabric Wrinkle Detection System Based on YOLOv5 Algorithm

Background Nowadays, robots have been widely used in handling rigid objects, but research on deformable objects like fabrics is still in its early stages. This is because fabrics possess infinite degrees of freedom and their state modeling is highly complex, making robot manipulation of fabrics challenging due to the occurrence of wrinkles and deformations during the operation. The detection and recognition of fabric deformations such as wrinkles and fabric manipulation features like corners are of great significance in enhancing a robot's capability to handle deformable objects. Methods In response to the issue of fabric wrinkles in various scenarios, we propose a real-time fabric wrinkle and corner detection system based on the YOLOv5 detection algorithm. Additionally, we implement a fabric flattening operation on a hardware platform using the detected wrinkle and corner information. Results We collected and created a dataset of fabric deformation features and trained a detection model, achieving a detection accuracy of over 90%. The model was deployed in the fabric wrinkle detection system, using a heuristic operation strategy of flattening the fabric from the four corners. As a result, the robot successfully performed the flattening operation on wrinkled fabric. Conclusions The application of the YOLOv5 algorithm enables effective detection of fabric wrinkles and corner points. Based on the detection information and using the quadrilateral flattening operation method, the robotic system achieves fabric flattening operations.

Research on Attitude Analysis of Hydraulic Support Based on Column Length

The existing hydraulic support attitude monitoring and control methods integrate multiple sensing technologies, but it is difficult to apply and promote due to unclear monitoring mechanisms, complicated types, and a large number of sensors. Aiming at the problem of attitude monitoring and control of hydraulic support, firstly, the kinematics mathematical model of hydraulic support is established by the key marker point method and the two-dimensional bar system model, and the forward and inverse solution algorithm of hydraulic support attitude based on the double-drive length of front and rear columns is proposed. Secondly, the rigid body dynamics simulation model of hydraulic support is constructed to verify the theoretical analytical model and prove the correctness of the algorithm. The rigid body simulation model can be used as the attitude monitoring system of hydraulic support. Thirdly, considering the influence of material elastic deformation, a rigid-flexible coupling dynamic simulation model of hydraulic support is established to explore the influence of elastic deformation of components on the attitude of hydraulic support and its causes. On this basis, the rigid-flexible coupling dynamic simulation model of the hydraulic support with clearance is established by replacing the revolute pair in the simulation model with the solid axis pin connection unit. The influence of the axis pin connection clearance on the attitude of the hydraulic support and its causes are explored, and the actual attitude and change characteristics of the hydraulic support under simulated real conditions are obtained. Finally, considering the force compression of the hydraulic cylinder of the column, the rigid-flexible and machine-liquid coupling dynamic simulation model of the hydraulic support with clearance is constructed.

Cultural relics fragment assembly based on fracture surface contour features

Addressing the issue of registration deviations caused by local fragment loss and shape diversity during the process of cultural relics fragment assembly, this paper proposes a method for assembling cultural relics fragments based on the contour features of fracture surfaces. Initially, the curvature of discrete point clouds is calculated using the quartic surface method and the Weingarten matrix, and curvature discontinuity points are identified using a proportional method as a candidate feature point set. Subsequently, the angle difference between a point and its neighboring points’ directed segments is defined as a neighborhood feature parameter, which further optimizes the feature point set and selects the optimal matching set. Then the 4-points congruent sets (4PCS) algorithm is utilized for the initial registration of the fragments, and the unit quaternion method is employed to calculate the rigid body transformation matrix during the iteration process. Finally, corresponding point pairs are determined by the intersection of a point’s normal with a set of points, improving the iterative closest point (ICP) algorithm to achieve precise assembly of the fragments. The experimental results show that, compared to other assembly algorithms, the proposed method is more accurate in feature point extraction, robust, and effectively enhances the accuracy and efficiency of fragment assembly.

Study on Fault Diagnosis of Single-Group Springs of Mining Vibrating Screen

In the mining vibrating screen’s working process, the support spring often fails due to alternating loads and a poor working environment. The six-degrees-of-freedom simplified rigid body dynamic model is established to obtain the vibration response of the mining vibrating screen with a faulty single-group spring and is verified using the ABAQUS (2020) simulation in this paper. Unlike previous studies focusing on multiple-group spring failures or using lower-degree-of-freedom models, this study addresses the common single-group spring failure with the comprehensive model capturing complex spatial motion. This study also reveals that the vibration angle can be used to diagnose the failure location, while the vibration displacement can be used to evaluate the health state of single-group springs. Based on these rules, the novel method for locating a faulty spring group and new evaluation formulas for their health states are validated by the simulations. The results indicate that the failure location is accurately diagnosed, and the errors of the health state evaluation are minimal. The evaluated values are 66.12% and 85.56% when the health states of the spring groups at the feed side and the out-feed side are 65% and 85%, respectively. Therefore, this diagnostic model provides significant theoretical assistance for diagnosing single-group spring failure in the mining vibrating screen, enhancing maintenance strategies and operational reliability.

High delta angle after reverse total shoulder arthroplasty increases stresses of the acromion: biomechanical study of different implant positions

BackgroundAcromial fractures after Reverse Total Shoulder Arthroplasty (RTSA) are a common complication. Nevertheless, only a few studies have identified risk factors for acromial fractures after RTSA. High delta angle (combination of inferiorization and medialization of the center of rotation) after RTSA was identified as a risk factor in recent studies. The aim of this study was the biomechanical exploration of different delta angles and implant configurations with regard to the acromial stress. MethodsIn a rigid body model of the upper extremity muscle, forces of the deltoid muscle were calculated before and after implanting RTSA in different arm and implant positions. The deltoid muscle was divided into an anterior, middle, and posterior part. Implant positions of the glenoid components were changed in the medialization, lateralization and inferiorization of the center of rotation (COR) as well as lateralization of the humeral component. Further, in a finite element model of the upper extremity, the stresses of the acromion in the same implant design configurations were measured. ResultsDifferences in acromial stress between different delta angle model configurations were observed. Lateralization (5 mm, 10 mm) of the glenosphere reduced maximal acromial stress by 21% (1.5 MPa) and 31% (1.3 MPa), respectively. Inferiorization (5 mm, 10 mm) of the glenosphere increased maximal acromial stress by 5% (2.0 MPa) and 15% (2.2MPa), respectively. Changes in positioning the humeral component was found to have the highest impact in this model configuration. A 10 mm lateralized humeral component reduced acromial stress by 37% (1.2 MPa) while in the 6 mm medialized configuration, an increase in acromial stress by 83% (3.48 MPa) was observed. There was a high correlation between delta angle and acromial stress (R-squared = 0.967). ConclusionImplant design configuration has an impact on the acromial stress. High delta angles correlate with an increase in acromial stress. Both lateralization of the COR and the humerus decreased the acromial stress in our study. The lateralization of the humerus has the highest impact in influencing acromial stress. Due to contrary results in the current literature, further studies with focus on the acromial stress influenced by different anatomical variants of the shoulder and the acromion are needed before a clinical recommendation can be made.

Development of Modified Rhombus Amplifier of Positioning Stage Compliant Mechanisms

Abstract This paper presents a modified design of rhombus-type amplifier used in positioning stage compliant mechanism. The new rhombus design is introduced to enhance the performance of rhombus and the results are promising. The new approach of rhombus amplifier is based on mixing the concept of rhombus type with the idea of hinges (reduced section portions) existed in bridge type. The performance of the proposed design is investigated by the pseudo rigid body model and validated by the finite element analysis. The new rhombus design improves displacement amplification ratio by (22.3%) with slightly change in master mode natural frequency (1.3%). The pseudo rigid body model for new rhombus design is consistent with the ANSYS results, and hence can be used in modelling and analysis of the new rhombus design.

Left ventricular deformation in the presence of left ventricular 'rigid body rotation' in healthy adults-three-dimensional speckle-tracking echocardiographic insights from the MAGYAR-Healthy Study.

During the heart cycle, the left ventricle (LV) not only shows a contraction-relaxation pattern, but LV has a rotational mechanics, as well. It is a known fact that certain pathologies may be associated with an absence of LV twist, when LV basal and apical regions rotate in the same clockwise (cw) or counterclockwise (ccw) direction called LV 'rigid body rotation' (LV-RBR), but it can also occur in healthy subjects. The present cohort study aimed to examine LV strains in healthy subjects with LV-RBR versus with normally directed LV rotational mechanics by three-dimensional speckle-tracking echocardiography (3DSTE). The study consisted of 181 healthy individuals, from which 171 cases had normally directed LV rotational mechanics (mean age: 32.5±12.3 years, 79 males) and 10 healthy subject showed LV-RBR (mean age: 35.4±11.3 years, 3 males). Complete two-dimensional (2D) Doppler echocardiography and 3DSTE were performed in all healthy individuals. None of routine 2D Doppler echocardiographic parameters showed differences between the groups examined. There were no subjects with ≥ grade 1 regurgitation on any valves or with significant stenosis on any valves. 3DSTE-derived LV volumes, global and mean segmental strains did not differ between the groups examined. Apical anterior and lateral segments showed reduced segmental LV circumferential strain (CS) (-18.9%±8.5% vs. -26.7%±10.7%, P=0.02; -27.3%±12.6% vs. -34.8%±13.2%, P=0.08, respectively) and LV area strain (AS) (-26.8%±9.8% vs -36.8%±12.0%, P=0.01; -35.7%±13.2% vs. -45.0%±14.6%) in healthy subjects having LV-RBR as compared to cases with normally directed LV rotational mechanics. These abnormalities were present only in subjects having cwLV-RBR. Although global LV deformation is normal in the presence of LV-RBR in healthy adults, reduction of apical anterior and lateral LV-CS (and LV-AS) are present in cases with cwLV-RBR only suggesting segmental deformation abnormalities.

Design and Research of a Strain Elastic Element with a Double-Layer Cross Floating Beam for Strain Gauge Wireless Rotating Dynamometers.

Cutting force is one of the most basic signals that can reflect the information of the cutting process, so it is very necessary to study the strain elastic element of strain gauge wireless rotating dynamometers. This paper proposes a strain elastic element with a double-layer cross floating beam that can be applied to the strain gauge wireless rotating dynamometer, which can simultaneously obtain the four-component cutting force/torque information of FX, FY, FZ, and MZ. Based on the proposed strain elastic element, a compact strain gauge wireless rotating dynamometer is designed, which is composed of a tool holder, upper connection flange, strain elastic element, lower connection flange, tool base, and data acquisition and wireless transmission system. The static model of the double-layer cross floating beam on the strain elastic element is established by the segmented rigid body method, and the relationships between the material, force, structural parameters, and the strain and deformation of the floating beam are obtained. The static model is consistent with the finite element solution, which proves the rationality of the static model. Based on the established static model, the sequential quadratic programming algorithm is used to optimize the structural parameters of the double-layer cross floating beam to maximize the sensitivity of the floating beam. The overall structure of the strain elastic element is analyzed by finite element software, and the strain of the structure under simulation conditions is obtained, which provides a reference for subsequent calibration tests and circuit design. The calibration matrix and dynamic performance of the strain elastic element are obtained by the static calibration test, dynamic calibration test, and cutting test. The results show that the proposed strain elastic element has high sensitivity and low cross-sensitivity error, and can be applied to the strain gauge wireless rotating dynamometer to measure medium- and low-speed cutting forces.

Experimental and analytical investigations on flexural performance of corroded RC beams using RBSM-based corrosion model

Residual strength and structural performance of reinforced concrete (RC) structures subjected to rebar corrosion need to be precisely evaluated for an efficient rehabilitation process. However, the performance of deteriorated structures cannot be specified by using design methods since the damaged conditions are different in each type of structures due to corrosion degree, corrosion position, environmental condition. This study first presents an experimental investigation on the flexural performance of RC beam subjected to rebar corrosion. To complement the test data, an analytical corrosion model developed by the authors, which is based on Rigid Body Spring Method (RBSM), was employed. The results obtained from both analytical model and experiment confirmed that the corroded beam appeared to obtain a significantly lower bending stiffness and lower flexural capacity than those of the sound beam. The developed corrosion model can be able to capture the load-displacement curve of experimental results with good agreement.

In Silico Research in Glioma Vaccine Discovery from Isocitrate Dehydrogenase Type 1 (R132H) Epitopes

Glioma is a primary malignant brain tumor, which is often detected using the mutation of isocitrate dehydrogenase type 1 (IDH1) at the R132H position. Several studies have also reported the use of mutated IDH1 (R132H) specific immunogenic epitopes as vaccines against this tumor. Therefore, this study aimed to determine the high-affinity epitopes of IDH1 (R132H) as a plausible candidate of preventive and curative glioma vaccines and to predict the stability of epitope-receptor complex through molecular dynamics simulation. The binding affinity of epitopes for preventing and treating glioma were predicted by docking epitope to major histocompatibility complexes class II (MHC II) and ephrin type-A receptor 3 (EphA3), respectively, using Dock version 6.7. This study used the rigid body docking method, where the samples were treated in their compact state. The highest binding affinity for MHC II was exhibited by epitope 42, as indicated by a grid score of -62.73 kcal/mol. Meanwhile, epitope 54, with a grid score of -55.56 kcal/mol, had the highest binding affinity for the EphA3 receptor. The results showed that the protein conformation in the 42-MHC II epitope complex changed significantly in molecular dynamics simulations using GROMACS version 5.0.6 at 300 K for 25 ns with RMSD > 3 Å, while epitope 54-EphA3 complex was stable from the beginning up to 15.29 ns. Based on these findings, the best candidates for prophylactic and curative glioma vaccination were epitope 42 and 54, respectively.

Design and performance evaluation of a novel parallel kinematic micromanipulator

The targeted placement of selected carbon nanotubes is associated with low throughput rates. Hence, this paper presents a novel stage design for the fully automatic assembly of carbon nanotube field-effect transistors (CNTFETs) via mechanical dry transfer. It constitutes the core module of an assembly machine for the precise deployment of nanotubes onto a prefabricated wafer. The mechanical dry transfer approach allows high level of carbon nanotube selectivity with the aim of enabling a high-volume fabrication of ultra-clean devices. The previous production rate of such ultra-clean devices is less than one per hour, which offers a high potential for improvement. The stage consists of a parallel kinematic mechanism (PKM) with 3 degrees of freedom (X2Y2C1) carrying two additional stacked axes (Z1C2). The PKM is suspended on a vacuum preloaded aerostatic bearing, voice coil motors (VCM) and flexure hinges. Together with a combination of high precision and resolution touch probe measurement systems close to its tool center point (TCP), high accuracy and low settling time can be achieved. Based on realistic manufacturing tolerances, a sensitivity analysis of the mechanism’s pseudo rigid body model (PRBM) suggests a theoretical closed-loop error below 0.15μm over the entire XY workspace of 20mm×2.8mm. Measurements of positioning motions show that the settling time can be decreased by the compensation of friction resistances inside the internal bearings of the VCMs.

Performance Assessment of Object Detection Models Trained with Synthetic Data: A Case Study on Electrical Equipment Detection.

This paper explores a data augmentation approach for images of rigid bodies, particularly focusing on electrical equipment and analogous industrial objects. By leveraging manufacturer-provided datasheets containing precise equipment dimensions, we employed straightforward algorithms to generate synthetic images, permitting the expansion of the training dataset from a potentially unlimited viewpoint. In scenarios lacking genuine target images, we conducted a case study using two well-known detectors, representing two machine-learning paradigms: the Viola-Jones (VJ) and You Only Look Once (YOLO) detectors, trained exclusively on datasets featuring synthetic images as the positive examples of the target equipment, namely lightning rods and potential transformers. Performances of both detectors were assessed using real images in both visible and infrared spectra. YOLO consistently demonstrates F1 scores below 26% in both spectra, while VJ's scores lie in the interval from 38% to 61%. This performance discrepancy is discussed in view of paradigms' strengths and weaknesses, whereas the relatively high scores of at least one detector are taken as empirical evidence in favor of the proposed data augmentation approach.

ACTION OF A SOLID BODY ON THE INNER SURFACE OF A THICK-WALLED BIMETAL CYLINDER

Purpose. The goal of the work is to obtain an exact solution to the problem of the stress-strain state of a long thick-walled bimetallic cylinder within the framework of the classical theory of elastic materials. Then, using the results obtained with this formulation, it is necessary to propose simpler engineering approaches and explore the possibilities of using asymptotic formulas for express analyzes at the design stage of such structural elements. Research methods. For the main body of the cylinder, the classical equations of the theory of elasticity in displacements are used. For the outer coating (sputtering), the shell theory equations are written based on the Kirchhoff-Love hypotheses. The complex integral Fourier transform and the Failon method are used to approximately find the original stresses and displacements. Asymptotic representations of cylindrical Bessel functions for large values of the argument and representations of improper integrals in the form of combinations of elementary and special tabulated functions are also used. Results. A mathematical model has been constructed to analyze the stress-strain state of a bimetallic cylinder with a thin outer layer of a material different from that of the inner layer. Various boundary conditions are recorded on the inner surface of the cylinder, describing the transmission from a rigid body of either specified forces or specified displacements. For all considered options, using the method of integral transformations, the results were obtained in the form of improper integrals, for the calculation of which a special method was used, aimed at calculating integrals with highly oscillating functions. Examples of specific graphs of changes in the components of the stress-strain state in the cylinder material are given. Depending on the conditions on the inner surface of the cylinder, simpler models are proposed to describe the main body, which are based, depending on the nature of the description of the interaction of the rigid body and the cylinder, on one equation of the theory of elasticity. With this approach, in some important cases it was possible to obtain improper inversion integrals using the asymptotic approach in closed form as a combination of elementary and special tabulated functions. Comparison with the exact approach allowed us to prove the possibility of using approximate models. Scientific novelty. A model of the behavior of a bimetallic cylinder as a body is constructed, the main layer of which is described by the equations of the theory of elasticity, and the theory of shells is used for the outer coating. Various methods of describing the transfer of forces and displacements from a rigid body to the inner surface of the cylinder are considered. The possibility of using the asymptotic approach to obtain relatively simple formulas for carrying out preliminary calculations at the design stage of such structural elements is shown.

Semiglobal Exponential Stability for Dual-Quaternion-Based Rigid-Body Tracking Control

Semi-Global Exponential Stability (SGES) is proven for the combined attitude and position rigid-body motion tracking problem, presenting a significant upgrade compared to previously proposed controllers which can only guarantee asymptotic stability. Dual quaternions are used to jointly represent the rotational and translational tracking error dynamics of the rigid body. A novel nonlinear feedback tracking controller is proposed, and a Lyapunov-based analysis is provided to prove SGES of the closed-loop dynamics. The analysis does not place any restrictions on the reference trajectory or the feedback gains. This stronger SGES result aids in further analyzing the robustness of the rigid-body system by establishing input-to-state stability (ISS) in the presence of time-varying additive and bounded external disturbances. Motivated by the fact that, in many aerospace applications, stringent adherence to safety constraints such as approach path and input constraints is critical for overall mission success, a framework for safe control of spacecraft is presented that combines the proposed feedback controller with control barrier functions. Numerical simulations are provided to verify the SGES and ISS results and also showcase the efficacy of the proposed nonlinear feedback controller in several nontrivial scenarios, including trajectory tracking, transposition and docking, flip maneuvers, collision avoidance, and rendezvous.

On the stability of rotation in an environment with resistance of a free system of two rigid bodies connected by an elastic Hook joint

On the stability of rotation in an environment with resistance of a free system of two rigid bodies connected by an elastic Hook joint