Complete geometric nonlinear formulation for rigid-flexible coupling dynamics

A complete geometric nonlinear formulation for rigid-flexible coupling dynamics of a flexible beam undergoing large overall motion was proposed based on virtual work principle, in which all the high-order terms related to coupling deformation were included in dynamic equations. Simulation examples of the flexible beam with prescribed rotation and free rotation were investigated. Numerical results show that the use of the first-order approximation coupling (FOAC) model may lead to a significant error when the flexible beam experiences large deformation or large deformation velocity. However, the correct solutions can always be obtained by using the present complete model. The difference in essence between this model and the FOAC model is revealed. These coupling high-order terms, which are ignored in FOAC model, have a remarkable effect on the dynamic behavior of the flexible body. Therefore, these terms should be included for the rigid-flexible dynamic modeling and analysis of flexible body undergoing motions with high speed.

Rigid-Flexible Coupling Dynamic Analysis of Sub-Launched Vehicle During the Vertical Tube-Exit Stage

During the launching stage,hydrodynamic pressure and adapters' reaction loads can influence the vehicle's rigid motion as well as cause its structural vibration,which is a typical rigid-flexible coupling dynamic problem. This paper presents a 2-D rigid-flexible coupling model to calculate the vehicle's dynamic responses in that period.The vehicle was equivalent to a flexure beam with axial deformation. Hybrid coordinate and modal superposition methods were used to describe its large rigid displacement and small deformation. By the second Lagrange equation,the vehicle centroid's displacements,rotational angle and modal coordinates were chosen as generalized coordinates and then the vehicle 's rigid-flexible coupling dynamic equations were obtained. By numerical simulation,the results of vehicle's motion parameters and transverse internal loads were acquired.The calculation results showed that differences of the vehicle's motion parameters between the rigid-flexible coupling model and the rigid body assumption are noticeable and the peak magnitude of the vehicle's transverse internal loads in the rigid-flexible coupling model is higher remarkably than that in the rigid body assumption.

Monolithic Coupling of the Pressure and Rigid Body Motion Equations in Computational Marine Hydrodynamics

In Fluid Structure Interaction(FSI) problems encountered in marine hydrodynamics, the pressure field and the velocity of the rigid body are tightly coupled. This coupling is traditionally resolved in a partitioned manner by solving the rigid body motion equations once per nonlinear correction loop, updating the position of the body and solving the fluid flow equations in the new configuration. The partitioned approach requires a large number of nonlinear iteration loops per time–step. In order to enhance the coupling, a monolithic approach is proposed in Finite Volume(FV) framework,where the pressure equation and the rigid body motion equations are solved in a single linear system. The coupling is resolved by solving the rigid body motion equations once per linear solver iteration of the pressure equation, where updated pressure field is used to calculate new forces acting on the body, and by introducing the updated rigid body boundary velocity in to the pressure equation. In this paper the monolithic coupling is validated on a simple 2D heave decay case. Additionally, the method is compared to the traditional partitioned approach(i.e. "strongly coupled" approach) in terms of computational efficiency and accuracy. The comparison is performed on a seakeeping case in regular head waves, and it shows that the monolithic approach achieves similar accuracy with fewer nonlinear correctors per time–step. Hence, significant savings in computational time can be achieved while retaining the same level of accuracy.

A Study of Dynamic Analysis Method for the Rigid-Flexible Coupled Bar Linkage System

In order to present a dynamic analysis method for the rigid-flexible coupled bar linkage system(RFCBLS),the flexible element motion equation was gotten by Lagrange Equation and the rigid element motion equation was gotten based on rigid constraint conditions.The multi-body system(MBS) is a complex mechanism and its components have quite different rigidities.If it is considered as a rigid MBS(RMBS) to do its dynamic analysis,elastic deformation's ignorance will lead to inaccurate analysis.If it is considered as a flexible MBS(FMBS) to establish,analyze,and solve the model,quite large system equations make it difficult to solve.The better method is as follows:the complex mechanism system is regarded as a rigid-flexible coupled system(RFCS) to make dynamic characteristic of rigid components be equivalent,system equation is established by FMBS' way,and system equation dimensions are reduced by transition matrices' introduction.A dynamic analysis method for rigid element and flexible element coupling was presented based on the FMBS.The analyzed crank slide-block mechanism results show that the dynamic analysis method for RFCBLS is quick and convenient.

Dual equations and solutions of I-type crack of dynamic problems in piezoelectric materials

This paper firstly works out basic differential equations of piezoelectric materials expressed in terms of potential functions, which are introduced in the very beginning. These equations are primarily solved through Laplace transformation, semiinfinite Fourier sine transformation and cosine transformation. Secondly, dual equations of dynamic cracks problem in 2D piezoelectric materials are established with the help of Fourier reverse transformation and the introduction of boundary conditions. Finally, according to the character of the Bessel function and by making full use of the Abel integral equation and its reverse transform, the dual equations are changed into the second type of Fredholm integral equations. The investigation indicates that the study approach taken is feasible and has potential to be an effective method to do research on issues of this kind.

An analysis on a rigid-flexible coupling system of an oscillating massand 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.

Numerical simulation of aircraft arresting process with an efficient full-scale rigid-flexible coupling dynamic model

The arresting process of carrier-based aircraft is widely recognized as a challenging task,characterized by the highest accident rate among all carrier-based aircraft operations.Dynamic simulation plays a crucial role in assessing the intricate responses of the arresting process,favoring the design of carrier-based aircraft.An efficient and accurate rigid-flexible coupling model for analyzing the dynamic response of the arresting process is proposed.By combining the dynamic characteristics of airframe,landing gear,arresting hook and arresting gear system,the rigid-flexible coupling dynamic model is established to reflect the relative motion of the coupling parts and arresting load.The dynamic model is verified through simulations of landing gear landing drops and by comparing the arresting simulation results with corresponding data in the US military standard.Additionally,simulations of the arresting process under different off-center distance and aircraft yaw angle are conducted to obtain the dynamic response of the aircraft during the arresting process.The result indicates that the rigid-flexible coupling dynamic model proposed is effective for analyzing the arresting dynamics response of carrier-based aircraft.The axial force of the arresting cable on both sides of the hook engagement point,pitch and yaw angle of aircraft are inconsistent under yaw and off-center arresting.The analysis method and obtained results provide valuable references for assessing the dynamic responses of carrier-based aircraft during arresting process and offer valuable in-sights in the design of carrier-based aircraft.

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface

The dissipative equilibrium dynamics studies the law of fluid motion under constraints in the contact interface of the coupling system. It needs to examine how con- straints act upon the fluid movement, while the fluid movement reacts to the constraint field. It also needs to examine the coupling fluid field and media within the contact in- terface, and to use the multi-scale analysis to solve the regular and singular perturbation problems in micro-phenomena of laboratories and macro-phenomena of nature. This pa- per describes the field affected by the gravity constraints. Applying the multi-scale anal- ysis to the complex Fourier harmonic analysis, scale changes, and the introduction of new parameters, the complex three-dimensional coupling dynamic equations are transformed into a boundary layer problem in the one-dimensional complex space. Asymptotic analy- sis is carried out for inter and outer solutions to the perturbation characteristic function of the boundary layer equations in multi-field coupling. Examples are given for disturbance analysis in the flow field, showing the turning point from the index oscillation solution to the algebraic solution. With further analysis and calculation on nonlinear eigenfunctions of the contact interface dynamic problems by the eigenvalue relation, an asymptotic per- turbation solution is obtained. Finally, a boundary layer solution to multi-field coupling problems in the contact interface is obtained by asymptotic estimates of eigenvalues for the G-N mode in the large flow limit. Characteristic parameters in the final form of the eigenvalue relation are key factors of the dissipative dynamics in the contact interface.

BSDES IN GAMES,COUPLED WITH THE VALUE FUNCTIONS.ASSOCIATED NONLOCAL BELLMAN-ISAACS EQUATIONS

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.

Novel exact solutions of coupled nonlinear Schro¨dinger equations with time–space modulation

We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.

A DYNAMIC MODEL FOR ROCKET LAUNCHER WITHCOUPLED RIGID AND FLEXIBLE MOTION

The dynamics of a coupled rigid-flexible rocket launcher is reported. The coupled rigid-flexible rocket launcher is divided into two subsystems, one is a system of rigid bodies, the other a flexible launch tube which can undergo large overall motions spatially. First, the mathematical models for these two subsystems were established respectively. Then the dynamic model for the whole system was obtained by considering the coupling effect between these two subsystems. The approach, which divides a complex system into several simple subsystems first and then obtains the dynamic model for the whole system via combining the existing dynamic models for simple subsystems, can make the modeling procedure efficient and convenient.

Dynamic response analysis of a moored crane-ship with a flexible boom

The dynamic response of moored crane-ship is studied. Governing equations for the dynamic response of a crane-ship coupled with the pendulum motion of the payload are derived based on Lagrange’s equations. The boom is modeled based on finite element method, while the payload is modeled as a planar pendulum of point mass. The dynamic response was studied using numerical method. The calculation results show that the large-amplitude responses occur at wave periods near the natural period of the payload. Load swing angle is smaller for crane-ship with flexible boom, in comparison with rigid boom. The ship surge mo- tions have large vibrations for crane-ship with flexible boom, which were not observed for a rigid boom. The analysis identifies the significance of key parameters and reveals how the system design can be adjusted to avoid critical conditions.

Rigid-flexible coupling dynamic modeling and vibration control for flexible spacecraft based on its global analytical modes

In this study, we used global analytical modeswfny(GAMs) to develop a rigid-flexible dynamic modeling approach for spacecraft with large flexible appendages(SwLFA). This approach enables the convenient and precise calculation of the natural characteristics for designing an attitude control law for the spacecraft while simultaneously suppressing the active vibration of its flexible appendages. We simplify the flexible spacecraft as a rigid-flexible coupling hub-beam system with tip mass and derive the system's governing equations of motion based on Hamilton's principle. By solving the linearized form of those equations with their associated boundary conditions, we obtain the frequencies as well as the corresponding GAMs of flexible spacecraft,which we use to discretize the equations of motion. Using this approach, we performed numerical simulations to investigate the system's global modes and assess the performance of the controller based on the GAM model. The results reveal that the GAM model can be used to directly calculate the exact global modes of SwLFAs and that the controller based on the discrete GAM model can achieve a control-index for a SwLFA in a shorter time with less input energy than other methods.

Impacts of flexible obstructive working environment on dynamic performances of inspection robot for power transmission line

The rigid-flexible coupling dynamic modeling and simulation of an inspection robot were conducted to study the influences of the flexible obstructive working environment i.e. overhead transmission line on the robot＇s dynamic performance. First, considering the structure of the obstacles and symmetrical mechanism of the robot prototype, four basic subactions were abstracted to fulfill full-path kinematic tasks. Then, a multi-rigid-body dynamic model of the robot was built with Lagrange equation, whil^e a multi-flexible-body dynamic model of a span of lin~ was obtained by combining finite element method （FEM）, modal synthesis method and Lagrange equation. The two subsystem models were coupled under rolling along no-obstacle segment and overcoming obstacle poses, and these simulations of three subactions along different spans of line were performed in ADMAS. The simulation results, including the coupling vibration parameters and driving moment of joint motors, show the dynamic performances of the robot along ftexibile obstructive working path： in flexible obstructive working environment, the robot can fulfill the preset motion goals; it responses slower in more flexible path; the fluctuation of robot as well as driving moment of the corresponding joint in startup and brake region is greater than that in rigid environment; the fluctuation amplitude increases with increasing working environment flexibility.

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method(NS-PIM),for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection.This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations,but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method(FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.

Macroscopic damping model for structural dynamics with random polycrystalline configurations

In this paper the macroscopic damping model for dynamical behavior of the structures with random polycrystalline configurations at micro-nano scales is established. First, the global motion equation of a crystal is decomposed into a set of motion equations with independent single degree of freedom （SDOF） along normal discrete modes, and then damping behavior is introduced into each SDOF motion. Through the interpolation of discrete modes, the continuous representation of damping effects for the crystal is obtained. Second, from energy conservation law the expression of the damping coefficient is derived, and the approximate formula of damping coefficient is given. Next, the continuous damping coefficient for polycrystalline cluster is expressed, the continuous dynamical equation with damping term is obtained, and then the concrete damping coefficients for a polycrystalline Cu sample are shown. Finally, by using statistical two-scale homogenization method, the macroscopic homogenized dynamical equation containing damping term for the structures with random polycrystalline configurations at micro-nano scales is set up.

Dynamic Analysis on the Main Transmission Device of Wooden Ice Cream Sticks Branding Machine

In order to reduce the labor intensity,improve the production efficiency and enhance the equipment stability and the branding accuracy of the pattern,we have completed a double-row high-efficiency wooden ice cream stick branding machine structural design.The rigid-flexible coupling dynamics model is established and the movement and stress of the first-stage chain drive are calculated and analyzed.The comparison of the theoretical calculation results shows that the dynamic modeling and the structural design of the equipment are reasonable and the result of dynamic calculation also provides the basis of load data for dynamic strength calculation of structural components.

Application of Lagrange＇s equation to rigid-elastic coupling dynamics

A consistent focus in theoretical mechanics has been on how to apply Lagrange's equation to continuum mechanics.This paper uses the concept of a variational derivative and its laws of operation to investigate the derivation of Lagrange's equation,which is then applied to nonlinear elasto-dynamics.In accordance with the work-energy principle and the energy conservation law,kinetic and potential energies are proposed for rigid-elastic coupling dynamics,whose governing equation is established by manipulating Lagrange's equation.In addition,case studies are used to demonstrate the application of the proposed method to spacecraft dynamics.

Parameter optimization for torsion spring of deployable solar array system with multiple clearance joints considering rigid–flexible coupling dynamics

In this paper,four novel evaluation indices and corresponding hierarchical optimization strategies are proposed for a deployable solar array system considering panel flexibility and joint clearance.The deployable solar array model consists of a rigid main-body,two panels and four key mechanisms,containing torsion spring mechanism,closed cable loop mechanism,latch mechanism and attitude adjustment mechanism.Rigid and flexible components are established by Nodal Coordinate Formulation and Absolute Nodal Coordinate Formulation,respectively.The clearance joint model is described by nonlinear contact force model and amendatory Coulomb friction model.The latch time,stabilization time,maximum contact force and impulse sum of the contact force of the solar array system are selected as the four novel evaluation indices to represent the complex dynamic responses of a deployable solar array with clearance joints instead of the lock torque widely used in conventional works.To eliminate the gross errors caused by the nonlinear and nonsmooth mechanical properties,a hierarchical optimization strategy based on an adaptive simulated annealing algorithm and a nondominated sorting genetic algorithm is adopted for the solar array system with clearance joints.Results indicate that the effects of panel flexibility on the evaluation index responses and design optimization of the solar array system cannot be neglected.Besides,increasing the weight factor of the stabilization time index of the rigid system may compensate for the differences in optimal results of the rigid–flexible coupling system.That may provide some references for optimization design of deployable space mechanisms considering clearance joints.

Magnetic-structure coupling dynamic model of a ferromagnetic plate parallel moving in air-gap magnetic field

Aiming at the air-gap magnetic field excited by wall armatures,Laplace’s partial differential equation of air-gap magnetic potential is achieved by means of the electromagnetic field theory.According to the magnetic boundary conditions and the method of separation of variables,the magnetic potential of the air-gap magnetic field is obtained.Based on the magnetization force model and Lorentz force of ferromagnetic thin-walled structures,and introducing the electromagnetic constitutive relations and boundary conditions,the calculation model of electromagnetic force of the soft ferromagnetic thin plate moving in air-gap magnetic field is established.Considering geometric nonlinearity,expressions of strain energy and kinetic energy of the elastic thin plate and the work of forces are given,respectively.The magnetic-structure coupling nonlinear vibration equations of ferromagnetic thin plate parallel moving in the air-gap magnetic field excited by armatures are obtained by using the Hamilton principle,which can be of the characterization of the system dynamics model with electro-magneto-velocity-mechanical interaction.Through numerical examples,primary resonance characteristics of the strip thin plate under the action of air-gap magnetic force are obtained.The results show that the two stable amplitude values will increase as amplitude of magnetic potential increases and thickness of air-gap decreases,and the amplitude’s multi-valued region will change due to the varieties of magnetic potential,air-gap and velocity.The model established in this paper is a theoretical reference for investigation on the multi-field coupling dynamic behaviors of structures moving in complex electromagnetic fields.