Rigid-flexible Coupling System Research Articles

Dynamic response of high-speed railway vehicle and welded turnout on large-span bridges based on rigid-flexible coupling system

Dynamic response of high-speed railway vehicle and welded turnout on large-span bridges based on rigid-flexible coupling system

End jitter suppression using FONFTSMC for rigid-flexible coupling systems of PMSpM based on NDO

A permanent magnet spherical motor (PMSpM) with three degrees of freedom rotation characteristics in the rigid rotor has broad application prospects. But the adding flexible material will inevitably produce end jitter issue in the process of the rigid-flexible coupling system (RFCs) movement. To solve the above problem, a fractional order non-singular fast terminal sliding mode control method with a disturbance observer was proposed for the rigid-flexible coupling system of a permanent magnet spherical motor (RFCs-PMSpM). Firstly, a dynamic model of RFCs-PMSpM was established by using the Hamilton principle and Euler–Bernoulli beam theory. Secondly, the influences on the end jitter were discussed from the perspectives of load mass, material parameters, and driving torque of the flexible shaft. The sensitivity analysis is carried out, and the key factors affecting the jitter are obtained. Then, a fractional order non-singular fast terminal sliding mode controller based on a nonlinear disturbance observer (NDO-FONFTSMC) is proposed. The stability of the closed-loop control system was proved by the Lyapunov method. Finally, the effectiveness of the proposed jitter suppression strategy was validated and compared with existing approaches, which also provides an important reference for the future application of RFCs-PMSpM in high-precision industry.

Rigid-flexible coupling dynamic modelling and dynamic accuracy analysis of planar cam four-bar mechanism with multiple clearance joints

Combination mechanism of conjugate cam and four-bar is a complex type of multi-body system, in which the coupling problems such as cam profile law, flexible deformation of connecting rods, and multiple joint clearance collision make it difficult to model and optimise its dynamics. Just as the beating-up mechanism of high-speed rapier looms for multi-layer weaving is a representative conjugate cam four-bar mechanism. In response to the current development of textile machinery in the direction of high-speed and precision, and in order to improve the work efficiency and operational accuracy of the conjugate cam four-bar beating-up mechanism, the following work is done for the purpose of taking into account the multi-joint clearance and the flexible rod based on the mechanism. Firstly, based on Kane's equation, a multi-body dynamic model with multiple clearances is established combined with clearance collision theory. Secondly, a dynamic model of a rigid-flexible coupling system with multiple revolute clearances is established using finite element method combined with modal synthesis technology. Subsequently, the variables separation method and mode superposition method are used to solve the dynamic equations of the coupled system. Finally, numerical examples are used to analyse the effects of clearance quantity, clearance size, mode truncation order, cam speed, and component material properties on the dynamic response and accuracy of the system. The results show that, in the rotational speed range of 600–800 rpm, the dynamic performance of the system is little affected by the speed; Through comparative analysis of multiple materials, it is found that selecting carbon fibre composite materials has the smallest impact on the motion accuracy of the system. In this paper, the clearance collision model, finite element model and rigid-flexible coupling dynamic model are integrated, which are applied to the cam-linkage combined multi-body mechanism, and the dynamic problems of the actual machine are analysed and solved.

Recursive dynamic modeling for attitude control of fully flexible spacecraft with central body and appendages

A spacecraft composed of a central body and flexible appendages is a typical rigid-flexible coupling system. Current modeling methods generally regard the central body as rigid. As the central body becomes larger and more flexible, the elastic deformation of the central body may not be ignored. In this study, considering the flexibility effect of the central body and based on the recursive kinematics, a complete coupled dynamic model of the orbit, attitude, and elastic deformation of a fully flexible spacecraft is established. Compared with traditional methods, the model has smaller generalized coordinate dimension and can more clearly reveal the kinematic relationship between the appendages and the central body. By comparing the simulation results of typical working conditions with the ADAMS software, the correctness of the built model is verified. The PD controller for the attitude of the spacecraft is used, and three models of different rigid/flexible settings are compared in detail. The dynamic response of the system in uncontrolled and controlled states are discussed. Numerical simulation results show that the flexibility of the central body poses a certain influence on the attitude response of the spacecraft, the flexible vibration of the appendages, and the attitude control accuracy.

Theoretical research on dynamic modeling of rigid–flexible coupling system with double joint folded wing

For meeting the requirements of tactical missiles seeking miniaturized launch devices for storage, transportation, and launch, a tube-launched missile wing is adopted, which folds before launch and quickly unfolds after launch. As a structure installed on the missile body to generate the required aerodynamic force for manipulating the missile, the tube-launched missile wing can effectively stabilize the missile’s flight attitude. At present, most research on the unfolding mechanism of missile folding wings is focused on one-time folding. When the wingspan is large, multiple folding is required to meet the launch requirements of modern tube-launched missiles. Therefore, this article designs a dual-joint folding wing deployment mechanism and studies the rigid–flexible coupling dynamic modeling and related technologies of folding wings based on this structure. Based on the inertial coupling between large-scale rigid body motion and structural flexible deformation, the folding wing breaks through the element convergence of the model and achieves the applicability of the structural model through zero-order approximation model analysis and other technologies. Simulation results show that the hybrid coordinate method can fully and accurately display the vibration information of flexible folding wings. At different speeds, the first-order coupling model is more advanced than the zero-order coupling model. In addition, increasing rotational speed, increasing wing thickness, and reducing wing span length can effectively increase the fundamental frequency of wing flutter. The structural design of folding wings has shown important reference significance.

Dynamic Modeling and Robust Control for Slant-Axis Telescope

The slant-axis telescope has a novel structure, and its unique structural design is more suitable for extreme environment such as the South Pole. However, there is a lack of research on the dynamic modeling and the controller design of slant-axis telescopes at home and abroad. This paper proposes a dynamic modeling and robust control method for slant-axis telescope. Firstly, the dynamical analysis of the slant-axis telescope is carried out. The 2-degree-of-freedom rigid body model of the telescope is established by the Lagrange method. Next, combining the flexibility of the driver system and disturbance, a mathematical model of the rigid-flexible coupling system of the slant-axis telescope is completed. Then, according to the established mathematical model, a sliding mode controller based on the disturbance observer is designed to suppress the disturbance, and realize the robust control of the slant-axis telescope. Finally, the simulation results show that the disturbance observer based sliding mode controller gets better dynamic performance and anti-disturbance characteristics than the traditional Proportion-Integration-Differentiation (PID) controller in the case of considering nonlinear external interference of the model.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

Establishment of rigid-flexible coupling model of leaf spring detector detection arm and the optimization of its accuracy

A pipeline through-diameter detector is vital in pipeline detection. Still, the commonly used through-diameter detector is a generally complex structure, sizeable structural size, not suitable for small pipe diameters, significant deformation of the pipeline, as well as high-precision detection requirements (Zhang et al., 2019). In this paper, the rigid-flexible coupling system and accuracy optimization of the detection arm of the leaf spring detector is proposed to improve the detection accuracy under the rigid-flexible coupling state. Under the condition of rigid-flexible coupling, the process and stage of over-convex defects are studied, the effects of different motion speeds, different preload forces, and different defect L/D ratios on the detection accuracy are analyzed, and the fitting values of the defects of the leaf spring detector detecting arm are obtained under the motion condition, which needs to be corrected for the right values to reduce the error and improve the detection accuracy under the condition of rigid-flexible coupling.

Rigid–Flexible Coupled System Attitude–Orbit Integration Fixed-Time Control

A diffractive imaging system consisting of two satellites is analyzed in view of dynamics. The mathematical model of rigid and flexion couples is studied to describe the relative motion of diffractive satellites and imaging satellites. Based on an integrated dynamics model with dual quaternion, a fixed-time non-singular terminal sliding mode controller is designed to meet the requirements of Earth observation. Finally, introducing the non-singular terminal sliding mode as the control group, a comparative simulation of relative motion and control is implemented to verify the controller and dynamics model.

Spherical Joint Actuator with Backstepping Sliding Mode Control for Robotic Manipulator Rigid-Flexible Coupling System

At present, the output shaft of a permanent magnet spherical actuator (PMSA) is a rigid structure, which has the problems of small load weight ratio and high energy consumption. It is difficult to meet the application requirements of lightweight, high speed, and low energy consumption. In this paper, a PMSA is proposed as the driving motor for the rigid-flexible coupling system (R-FCS) of the robotic manipulator, and a flexible mechanical arm is used to replace the rigid output shaft of the PMSA to connect the hand claw to form a spherical joint. Firstly, the partial differential dynamics model of the robotic manipulator in the coordinate system is established based on the Hamilton method, and the R-FCS is constructed according to the momentum conservation equation combined with the rigid axis dynamics model of the PMSA. Then, a backstepping sliding mode controller (BSMC) is designed to form a closed-loop control system by selecting an appropriate nonlinear gain function to estimate the unknown disturbances of the system, and the stability of the controller is guaranteed by the Lyapunov theory. Finally, the simulation and experimental results verify that the R-FCS can achieve better trajectory tracking and provide an effective driving method for robotic manipulator spherical joint.

Dual-Quaternion-Based Dynamics Modeling for a Rigid–Flexible Coupling Satellite

Dual quaternions can describe both three-dimensional translation and three-dimensional rotation and have the advantages of a simple form, accurate modeling, and precise physical meaning in terms of dynamic modeling. The traditional rigid-body dynamics modeling theory based on dual quaternions is only applicable to the case of modeling for the center of mass. When modeling appendages, the flexible appendages must be symmetrically configured to reduce the modeling complexity. However, in space operations, flexible appendages are usually not symmetrically installed, which significantly limits the applications of dual-quaternion theory. In this paper, we propose a new dual-quaternion derivative rule and a rigid-body dynamic modeling method suitable for any point. The theory can be applied to model flexible appendages with any configuration without changing the mathematical modeling framework. Finally, we demonstrate the integrated dynamic equation of a rigid–flexible coupling system considering the coupling of position and attitude, and we verify the correctness of the model by numerical simulation.

Sensitivity and uncertainty analysis of the nonlinear flight dynamics system of the flexible body

Free flight of a flexible body, such as large slenderness missiles, is a complex process involving rigid-flexible coupling dynamics. The strong flexible effect and the sensitive rotational characteristics caused by large slenderness ratio provide an extreme environment for the rigid-flexible coupled system. To study the performance of nonlinear flight dynamics system of the flexible body under extreme conditions, the trajectory of javelin model with a slenderness ratio of 98.15 flying in vacuum condition is calculated, and the uncertainty and the sensitivity of the coupled system are analyzed for the first time. The Monte Carlo (MC) method and the non-intrusive polynomial chaos (NIPC) method are compared by quantifying the uncertainty in the coupled system with the random moment of inertia of principal axis (Ixx). The uncertainty in the coupled system is quantified using the second-order NIPC method. It is found that the decoupled method is sensitive to the time step due to the unsynchronization of status variables between the rotational equation and structural equation. In contrast, the coupled method is insensitive to the time step. Ixx, stiffness according to the primary modes, and the initial deformation have the largest influence on the uncertainty of the coupled system. The rotational velocity in x direction (ωx) is significantly sensitive to the random inputs. The total uncertainty region of the ωx gradually expands over time.

Dynamic Response Modeling of Mountain Transmission Tower-Line Coupling System under Wind–Ice Load

Transmission lines have the characteristics of being tall tower structures with a large span distribution of transmission lines that are sensitive to external loads such as wind and ice, and belong to strong, nonlinear, complex, rigid-flexible coupling systems. The force process of the tower-line structure is a combination of instantaneous and continuously stressed, so it is not accurate to judge the safety of the transmission line based only on the operation status of the transmission tower or the conductor. In this paper, a finite element model of three towers and two lines with large span and large elevation differences is established by taking into account the tower-line coupling system. From the static point of view, the static axial force of a single tower and the contribution rate of wind and ice load are analyzed, and the ultimate bearing capacity of a tension-type electric tower is obtained by considering the bending effect and critical initial defects. From the perspective of transient dynamic response, the displacement of the tower-line coupling system under wind–ice load is calculated, and the force characteristics and force transmission process of the straight tower under wind–ice load are observed. Multiple comparison models are set up to compare and analyze the sway and tension under large span and large elevation differences, and the maximum icing thickness of each group model is obtained by repeated trials. The experimental results show that under the tower-line coupling system, the contribution of wind load to the axial force of the main material is 72.92%, and the contribution of wind–ice load to the axial force of main material is 27.6%. The maximum increase tension under transient ice-off effect is 59.58%, the ultimate force of the tension tower is 545.5 kN, and the maximum icing thickness of the transmission line under large span and large elevation differences is 28.7 cm, which is slightly larger than the design icing thickness. In conclusion, this paper can provide reference for the construction of mountain transmission towers, power safety inspection, and line health status assessment.

Symmetry breaking of a closed flexible filament behind a rigid plate at low Reynolds numbers

In this paper, the motion modes transition and dynamic performance of the flow past a rigid–flexible coupling system were investigated at low Reynolds numbers. The coupling system consisted of a rigid plate and a trailing closed flexible filament and was simulated numerically using the immersed boundary method. According to whether the filament moves and the symmetry of its movement, six motion modes have been identified for different filament lengths and Reynolds numbers (Re), i.e., the symmetric and stationary mode, the asymmetric and stationary (AS) mode, the regular and unilateral flap (RUF) mode, the transition motion (TM) mode, the symmetric and bilateral flap mode, and the asymmetric and bilateral flap (ABF) mode. Moreover, symmetry breaking occurred in the AS mode, RUF mode, and ABF mode. Drag reduction was found at Re≥40, which was mainly influenced by the pressure behind the filament. However, drag reduction was weakened for the filament with Lr≥2.57 because of the large width suffered to the flow, where Lr is the ratio of the length of the filament to the width of the plate. The time-averaged lift was also found to be directly dependent on the symmetry of the filament. In addition, the motion modes were significantly affected by the bending coefficient of the filament. Symmetry breaking was prevented by the reduction of the bending coefficient before vortex shedding. Moreover, if the filament was symmetric before vortex shedding, the RUF mode and the TM mode did not appear with the increase in Re.

AILC for Rigid-Flexible Coupled Manipulator System in Three-Dimensional Space with Time-Varying Disturbances and Input Constraints

In this paper, an adaptive iterative learning control (AILC) law is developed for two-link rigid-flexible coupled manipulator system in three-dimensional (3D) space with time-varying disturbances and input constraints. Based on the Hamilton’s principle, a dynamic model of a manipulator system is established. The conditional equation that is coupled by ordinary differential equations and partial differential equations is derived. In order to achieve high-precision tracking of the revolving angles and vibration suppression of the elastic part, the iterative learning control law based on the disturbance observer is considered in the process of the design controller. The composite Lyapunov energy function is proposed to prove that the angle errors and elastic deformation can eventually converge to zero with the increase of the number of iterations. Ultimately, the simulation results to rigid-flexible coupled manipulator system are given to prove the convergence of the control objectives under the adaptive iterative learning control law.

An analysis on a rigid-flexible coupling system of an oscillating mass and a rotating disk

A mass-rod-disk system consisting of an oscillating mass attached to a rigid rotating disk by an elastic rod is designed to study rigid-flexible coupling mechanism. Suppose the rod is lightweight and has enough stiffness, the theorems of linear momentum and angular momentum are applied to the mass-rod-disk system based on the kinematic description of the system. With respect to two deflections of the mass and one angular velocity of the system, a group of nonlinear differential equations are established where the tangential inertial force, centrifugal force, Coriolis force as well as the moments of additional inertial forces take important effects on the dynamic response. For the sake of description, these three types of inertial forces mentioned before are referred to as additional inertial forces in this paper. The horizontal deflections of the mass and the angular velocity of the disk rotating about a fixed-axis are numerically solved for the prescribed external torque. The oscillating trajectory of the mass is deeply influenced by the additional inertial forces, meanwhile the dynamic fluctuations of the angular velocity and rotary inertia of the system are strongly affected by the mass oscillation.

Dynamics of a rigid-flexible coupling system in a uniform flow

Dynamics of two-dimensional flow past a rigid flat plate with a trailing closed flexible filament acting as a deformable afterbody are investigated numerically by an immersed boundary-lattice Boltzmann method for the fluid flow and a finite element method for the filament motion. The effects of Reynolds number ( $Re$ ) and length ratio ( $Lr$ ) on the flow patterns and dynamics of the rigid-flexible coupling system are studied. Based on our numerical results, five typical state modes have been identified in $Lr\unicode{x2013}Re$ plane in terms of the filament shape and corresponding dynamics, i.e. static deformation, micro-vibration, multi-frequency flapping, periodic flapping and chaotic flapping modes, respectively. Benefiting from the passive flow control by using the flexible filament as a deformable afterbody, the coupled system may enjoy a significant drag reduction (up to $22\,\%$ ) compared with bare plate scenarios ( $Lr=1$ ). Maximum drag reduction achieved at $L_{c,{min}} \in [1.8, 2]$ is often accompanied by the onset of the system state transition. The flow characteristic and its relation to the change in hydrodynamic drag are further explored in order to reveal the underlying mechanisms of the counterintuitive dynamical behaviour of the coupled system. The scaling laws for the form drag and the friction drag, which arise from the pressure and viscous effects, respectively, are proposed to estimate the overall drag acting on the system. The results obtained in the present study may shed some light on understanding the dynamical behaviour of rigid-flexible coupling systems.

Optimization of Strip Fertilization Planter for Straw Throwing and Paving

To enhance the operation effect and working performance of our previously developed strip fertilization planter for broken straw back throwing and inter-row laying, and to improve the stability of straw crushing and consistency of straw mulching between rows (broken straw inter-row mulching), the key operation parameters of the planter were optimized in this study. On the basis of determining the transmission route and matching power consumption, the discrete element method was used to establish a mechanical model of straw particles using the EDEM software, which was then imported into the rigid–flexible coupled system of the ‘shredded straw-mechanism’. Quadratic regression orthogonal methods and rotation combination experiments were then designed to carry out a DEM virtual simulation and numerical simulation, and the optimal combination of operating parameters affecting planter working performance was obtained, which was also verified by field tests. The simulation test results showed that the smashing spindle speed (A) had the most significant influence on the coefficient of variation (Y1) of straw crushing, followed by the planter working forward speed (C). The conveying impeller speed (B) had the most significant influence on the coefficient of variation (Y2) of inter-row straw mulching, also followed by (C). The optimal combination of operating parameters after optimization were A = 2060.79 rpm, B = 206.25 rpm, and C = 0.95 m·s−1, and the optimal working performance of the planter was obtained as Y1 = 8.51% and Y2 = 10.34%. The evaluation index results corresponding to the field test were Y1 = 9.35% and Y2 = 10.97%, which met the technical requirements of the relevant operation machinery; the relative errors of the simulation test results were 9.87% and 9.63%, respectively, indicating the effectiveness of the virtual numerical simulation and the rationality of the optimized operation parameters. Our results provide a technical reference for realizing high-quality and smooth no-tillage seeding operations.

Rigid-Flexible Coupling Dynamics Modeling of Spatial Crank-Slider Mechanism Based on Absolute Node Coordinate Formulation

In order to study the influence of compliance parts on spatial multibody systems, a rigid-flexible coupling dynamic equation of a spatial crank-slider mechanism is established based on the finite element method. Specifically, absolute node coordinate formulation (ANCF) is used to formulate a three-dimensional, two-node flexible cable element. The rigid-flexible coupling dynamic equation of the mechanism is derived by the Lagrange multiplier method and solved by the generalized α method and Newton–Raphson iteration method combined. Comparison of the kinematics and dynamics response between rigid-flexible coupling system and pure rigid system implies that the flexible part causes a certain degree of nonlinearity and reduces the reaction forces of joints. The elastic modulus of the flexible part is also important to the dynamics of the rigid-flexible multibody system. With smaller elastic modulus, the motion accuracy and reaction forces become lower.

Tracking and vibration control of a free-floating space manipulator system with flexible links and joints

The problem of modeling and controlling of a free-floating space manipulator with flexures in both links and joints is addressed in this study. A mathematical model of the system is developed by combining Lagrange’s equations and momentum conservation. The finite element method is introduced to discretize multi-links with complex cross-sections. In order to reduce the dimensions and maintain the precision of a rigid-flexible coupled system, an iterated improved reduction system method is adopted. Then, a novel composite control scheme for the reduced system is presented that uses the concept of integral manifolds and singular perturbation theory. Finally, an augmented computed torque controller is applied to the under-actuated slow subsystem to realize trajectory tracking in joint space, while a linear-quadratic controller is designed to damp out the vibration of joints and links. Numerical simulation results verified that the proposed hybrid controller can successfully suppress vibration and track trajectory at the same time.