Analytical modeling of the flexible rim of space antenna reflectors

This paper presents a geometrically nonlinear analytical model of the flexible cylindrical rim of a deployable precision large space antenna reflectors made of shape-memory polymer composites. A nonlinear boundary-value problem for the rim in the deformed (folded) configuration is formulated and exact analytical solutions in elliptic functions and integrals describing the deformation modes of the rim are obtained. Exact analytical solutions based on the geometrically nonlinear model are obtained and can be used to determine preliminary geometric dimensions and optimal shape of the flexible rim along with the estimation of the accumulated energy.     

Numerical investigation of structural geometric nonlinearity effect in high-aspect-ratio wing using CFD/CSD coupled approach

An efficient and robust fluid–structure coupled methodology has been developed to investigate the linear and non-linear static aeroelastic behavior of flexible high-aspect-ratio wing. A three-dimensional open source finite element solver has been loosely coupled with an in-house Reynolds-averaged Navier–Stokes solver, designed for hybrid-unstructured meshes, to perform aero-structural coupled simulations. For volume mesh deformation and two-way data interpolation over non-matching grids interface, a radial basis function methodology combined with a data reduction algorithm has been used. This technique is efficient in handling large deflections and provides high-quality deformed meshes. Structural geometric nonlinearity has been considered to predict the deformations in the vertical and torsional directions caused by gravitational and aerodynamic loading. A multi-material finite element model has been generated to match the experimental configuration. Computational aeroelastic simulations were performed on an experimental high-aspect-ratio aeroelastic wing model with a slender body at the tip to get non-linear static deflections, twist and structure natural frequencies. The effect of the geometric nonlinearity is significant for large deformation analysis and has been highlighted in the predicted maximum tip deflection and twist. Good qualitative and quantitative agreement has been achieved between the predicted results and the available experimental data.     

DYNAMIC ANALYSIS OF FLEXIBLE BODY WITH DEFINITE MOVING ATTITUTE

The nonlinear dynamic control equation of a flexible multi-body system with definite moving attitude is discussed.The motion of the aircraft in space is regarded as known and the influence of the flexible structural members in the aircraft on the motion and attitude of the aircraft is analyzed.By means of a hypothetical mode,the defor- mation of flexible members is regarded as composed of the line element vibration in the axial direction of rectangular coordinates in space.According to Kane’s method in dy- namics,a dynamic equation is established,which contains the structural stiffness matrix that represents the elastic deformation and the geometric stiffness matrix that represents the nonlinear deformation of the deformed body.Through simplification the dynamic equation of the influence of the planar flexible body with a windsurfboard structure on the spacecraft motion is obtained.The numerical solution for this kind of equation can be realized by a computer.     

Dynamic event-triggered and asynchronous sliding mode control for T-S fuzzy Markov jump systems

Nonlinear Dynamics - In this paper, the motion of a smart rigid-flexible satellite considering large deflections for its flexible appendages in general planar motion is modeled. Also, the satellite...     

一种正交各向异性板的等效各向同性板计算法

上、下表面平整的正交各向异性板的有效抗扭刚度是2个主方向抗弯刚度的几何平均值. 根据这一特性、从正交各向异性板挠曲面的偏微分方程出发,保持一个主方向尺寸不变,另一主方向的尺寸做线性缩放,保持弹性模量与第一主方向相同,泊松比取2个主方向泊松比的几何平均值,将这类正交各向异性板等效为一块各向同性板;通过分析得到：各向同性板任意点的挠度就是原正交各向异性板对应点的挠度,各对应点内力之间存在简单的比例关系,该文的方法将为工程技术人员提供方便,同时可以更加直观的认识正交各向异性板的受力.     

Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.     

Dynamic modeling of a revolute-prismatic flexible robot arm fabricated from advanced composite materials

A dynamic model for a two degree-of-freedom planar robot arm is derived in this study. The links of the arm, connected to prismatic and revolute joints, are considered to be flexible. They are assumed to be fabricated from either aluminum or laminated composite materials. The model is derived based on the Timoshenko beam theory in order to account for the rotary inertia and shear deformation. These effects are significant in modeling flexible links connected to prismatic joints. The deflections of the links are approximated by using a shear-deformable beam finite element. Hamilton's principle is implemented to derive the equations describing the combined rigid and flexible motions of the arm. The resulting equations are coupled and highly nonlinear. In view of the large number of equations involved and their geometric nonlinearity (topological and quadratic), the solution of the equations of motion is obtained numerically by using a stiff integrator.The digital simulation studies examine the interaction between the flexible and the rigid body motions of the robot arm, investigate the improvement in the accuracy of the model by considering the flexibility of all rather than some of the links of the arm, assess the significance of the rotary inertia and shear deformation, and illustrate the advantages of using advanced composites in the structural design of robotic manipulators.     

Isogeometric Analysis of the Nonlinear Deformation of Planar Flexible Beams with Snap-back

Based on the continuity of the derivatives of the Non-Uniform Rational B-Splines(NURBS) curve and the Jaumann strain measure, the present paper adopted the position coordinates of the control points as the degrees of freedom and developed a planar rotation-free Euler-Bernoulli beam element for isogeometric analysis, where the derivatives of the field variables with respect to the arc-length were expressed as the sum of the weighted sum of the position coordinates of the control points, and the NURBS basis functions were used as the weight functions. Furthermore, the concept of bending strip was used to involve the rigid connection between multiple patches. Several typical examples with geometric nonlinearities were used to demonstrate the accuracy and effectiveness of the proposed algorithm. The presented formulation fully accounts for the geometric nonlinearities and can be used to study the snap-through and snap-back phenomena of flexible beams.     

基于PETSc的非线性逆向运动并行计算方法研究

在考虑几何非线性的有限元分析中,初始构型和变形构型是严格区分的,并且变形后的构型对结构性能和功能的实现往往具有重要意义.传统的非线性有限元分析主要面向变形前的初始构型为导向的设计问题,而对于变形后构型为导向的设计问题则有较大局限性.针对此问题,引入非线性逆向运动分析方法,为了保证大变形非线性分析迭代的收敛性和计算效率,基于PETSc函数库建立了并行分析框架,并对并行框架中的模块划分、数据并行存储以及通信锁死和负载平衡等进行了详细阐述.在算例部分,首先通过正向运动和逆向运动分析结果对比,阐述了两种分析方法的不同以及逆向运动分析方法对变形前构型求解的准确性;其次,采用不同MPI进程数对并行分析程序的效率进行了测试.结果 表明,合理地选择MPI进程数目可显著提高非线性逆向运动分析的效率.     

Study of static and dynamic stability of flexible rods in a geometrically nonlinear statement

We study static and dynamic stability problems for a thin flexible rod subjected to axial compression with the geometric nonlinearity explicitly taken into account. In the case of static action of a force, the critical load and the bending shapes of the rod were determined by Euler. Lavrent’ev and Ishlinsky discovered that, in the case of rod dynamic loading significantly greater than the Euler static critical load, there arise buckling modes with a large number of waves in the longitudinal direction. Lavrent’ev and Ishlinsky referred to the first loading threshold discovered by Euler as the static threshold, and the subsequent ones were called dynamic thresholds; they can be attained under impact loading if the pulse growth time is less than the system relaxation time. Later, the buckling mechanism in this case and the arising parametric resonance were studied in detail by Academician Morozov and his colleagues.In this paper, we complete and develop the approach to studying dynamic rod systems suggested by Morozov; in particular, we construct exact and approximate analytic solutions by using a system of special functions generalizing the Jacobi elliptic functions. We obtain approximate analytic solutions of the nonlinear dynamic problem of flexible rod deformation under longitudinal loading with regard to the boundary conditions and show that the analytic solution of static rod system stability problems in a geometrically nonlinear statement permits exactly determining all possible shapes of the bent rod and the complete system of buckling thresholds. The study of approximate analytic solutions of dynamic problems of nonlinear vibrations of rod systems loaded by lumped forces after buckling in the deformed state allows one to determine the vibration frequencies and then the parametric resonance thresholds.     

Dynamics of flexible multibody systems with tree topologies

The dynamic equations of flexible multibody systems with tree topological configuration are derived by using the Jourdain's principle. The independent joint coordinates are introduced to describe the large displacements of the bodies, and the modal coordinates are used to describe small deformations of flexible bodies based on the consistent mass finite element method and normal vibration mode analysis. The minimum differential equations are developed, which are compatible with the equations of multi-rigid body systems or structural dynamics. The stiff problem in the numerical integration is thus alleviated effectively. The method used in this paper can be extended to deal with systems with other topological configurations. Finally, the validity and feasibility of the presented mathematical model are demonstrated by a numerical example of a manipulator with two elastic links. The project supported by National Natural Science Foundation of China     

A nonlinear finite-element approach for kineto-static analysis of multibody systems

Methods that treat rigid/flexible multibody systems undergoing large motion as well as deformations are often accompanied with inefficiencies and instabilities in the numerical solution due to the large number of state variables, differences in the magnitudes of the rigid and flexible body coordinates, and the time dependencies of the mass and stiffness matrices. The kineto-static methodology of this paper treats a multibody mechanical system to consist of two collections of bulky (rigid) bodies and relatively flexible ones. A mixed boundary condition nonlinear finite element problem is then formulated at each time step whose known quantities are the displacements of the nodes at the boundary of rigid and flexible bodies and its unknowns are the deformed shape of the entire structure and the loads (forces and moments) at the boundary. Partitioning techniques are used to solve the systems of equations for the unknowns, and the numerical solution of the rigid multibody system governing equations of motion is carried out. The methodology is very much suitable in modelling and predicting the impact responses of multibody system since both nonlinear and large gross motion as well as deformations are encountered. Therefore, it has been adopted for the studies of the dynamic responses of ground vehicle or aircraft occupants in different crash scenarios. The kineto-static methodology is used to determine the large motion of the rigid segments of the occupant such as the limbs and the small deformations of the flexible bodies such as the spinal column. One of the most dangerous modes of injury is the amount of compressive load that the spine experiences. Based on the developed method, a mathematical model of the occupant with a nonlinear finite element model of the lumbar spine is developed for a Hybrid II (Part 572) anthropomorphic test dummy. The lumbar spine model is then incorporated into a gross motion occupant model. The analytical results are correlated with the experimental results from the impact sled test of the dummy/seat/restraint system. With this extended occupant model containing the lumbar spine, the gross motion of occupant segments, including displacements, velocities and accelerations as well as spinal axial loads, bending moments, shear forces, internal forces, nodal forces, and deformation time histories are evaluated. This detailed information helps in assessing the level of spinal injury, determining mechanisms of spinal injury, and designing better occupant safety devices.     

Nonlinear Random Response of Cylindrical Panels to Acoustic Excitations Using Finite Element Modal Method

This paper investigates large amplitude multi-mode free vibration andrandom response of thin cylindrical panels of rectangular planform usinga finite element modal formulation. A thin laminated composite doublycurved element is developed. The system equation in structural nodal DOFis transformed into the modal coordinates by the using the modes of theunderlying linear system. The nonlinear stiffness matrices are alsotransformed into nonlinear modal stiffness matrices. Numericalintegration is employed to determine free vibration and random response.Single-mode free vibration results are compared with existing classicalanalytical solutions to validate the nonlinear modal formulation.Nonlinear random analysis results for cylindrical panels have shown thatthe root mean square of panel deflections could be larger than thoseobtained using the linear structure theory. Time histories, probabilitydistribution functions, power spectral densities, and phase plane plotsare also presented.     

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation. The project supported by the National Natural Science Foundation of China (10472066, 10372057). The English text was polished by Yunming Chen.     

Characterization of the Mechanics of Compliant Wing Designs for Flapping-Wing Miniature Air Vehicles

Flapping-wing miniature air vehicles (MAVs) offer multiple performance benefits relative to fixed-wing and rotary-wing MAVs. This investigation focused on the problem of designing compliant wings for a flapping-wing MAV where only the spar configuration was varied to achieve improved performance. Because the computational tools needed for identifying the optimal spar configuration for highly compliant wing designs have yet to be developed, a new experimental methodology was developed to explore the effects of spar configuration on the wing performance. This technique optically characterized the wing deformations associated with a given spar configuration and used a customized test stand for measuring lift and thrust loads on the wings during flapping. This revealed that spar configurations achieving large and stable deformed volume during the flapping cycle provided the best combination of lift and thrust. The approach also included a sensitivity and reproducibility analysis on potential spar configurations. Results indicated that the wing shape and corresponding lift and thrust performance were very sensitive to slight changes in volume and 3-D shape associated with slight variations in the spar locations.     

Flexible Multibody Simulation and Choice of Shape Functions

The approach most widely used for the modelling of flexible bodies in multibody systems has been called the floating frame of reference formulation. In this methodology the flexible body motion is subdivided into a reference motion and deformation. The displacement field due to deformation is approximated by the Ritz method as a product of known shape functions and unknown coordinates depending on time only. The shape functions may be obtained using finite-element-models of flexible bodies in multibody systems, resulting in a detailed system representation and a high number of system equations. The number of system equations of such a nodal approach can be reduced considerably using a modal representation of deformation. This modal approach, however, leads to the fundamental problem of selecting the shape functions.The floating frame of reference formulation is reviewed here using a generic flexible body model, from which the various body models used in multibody simulations may be derived by formulation of specific constraint equations. Special attention is given in this investigation to the following subjects: The separation of flexible body motion into a reference motion and deformation requires the definition of a body reference frame, which in turn affects the choice of shape functions. Some alternatives will be outlined together with their advantages and disadvantages. Assuming the body deformation to be small, the system equations can be linearized. This may require considering geometric stiffening terms. The problem of how to compute these terms has been solved in literature on the instability of structures under critical loads. For finite element models the geometric stiffening terms are obtained from the tangential stiffness matrix. The generality of the flexible body model allows the definition of an object oriented data base to describe the system bodies. Such a data base includes a general interface between multibody- and finite-element-codes. By combining eigenfunctions and static deformation modes to represent body deformation one obtains a set of so-called quasi-comparison functions. When selected properly these functions can be shown to improve the representation of stresses significantly.     

An index 0 differential-algebraic equation formulation for multibody dynamics: Nonholonomic constraints

A method is presented for formulating and numerically integrating index 0 differential-algebraic equations of motion for multibody systems with holonomic and nonholonomic constraints. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation. Orthogonal dependent coordinates and velocities are used to enforce position, velocity, and acceleration constraints to within specified error tolerances. Explicit and implicit numerical integration algorithms are presented and used in solution of three examples: one planar and two spatial. Numerical results verify that accurate results are obtained, satisfying all three forms of kinematic constraint to within error tolerances embedded in the formulation.     

Wave propagation in heterogeneous anisotropic plates involving large deflection

Equations of motion for laminated plates of composite materials are derived for motions of large amplitudes as well as for incremental deformations superposed on large deflections. Free waves of large and small amplitudes propagating, respectively, in initially flat and deformed plates are investigated. By using a special substitution for the extensional displacement components the governing equations appear to be linear and consequently the analysis of free wave propagation greatly simplified. It is found that a train of free wave consists of a finite number of simple harmonic waves.     

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].One of the methods for solving the equations with rapidly oscillating coefficients is the asymptotic method of small geometric parameter. The standard procedure of this method usually includes preparatory stages. At the first stage, as a rule, a rectangular periodicity cell is distinguished to be a typical element. At the second stage, the scale of global coordinates is changed so that the rectangular structure periodicity cells became square cells of size l × l. The third stage consists in passing to the dimensionless global coordinates relative to the plate characteristic dimension L. As a result, the dependence between the new local variables and the new scaled dimensionless variables is such that the factor 1/α, where α=l/L ? 1 is a small geometric parameter, appears in differentiating any function of the local coordinate with respect to the global coordinate. After this, the solution of the problem in new coordinates is sought as an asymptotic expansion in the small geometric parameter [1], [4–10].We note that, in the small geometric parameter method, the asymptotic series simultaneously have the form of expansions in the gradients of functions depending only on the global coordinates. This averaging procedure can be applied to linear and nonlinear boundary value problems for differential equations with variable periodic coefficients for which the periodicity cell can be affinely transformed into the periodicity cube. In the case of an arbitrary dependence of the coefficients on the coordinates (including periodic dependence), another averaging technique can be used in linear problems. This technique is based on the possibility of the integral representation of the solution of the original problem for the linear equation with variables coefficients in terms of the solution of the same problem for an equation of the same type but with constant coefficients [11–13]. The integral representation implies that the solution of the original problem can be represented in the form of the series in the gradients of the solution of the problem for the equation with constant coefficients [13].The aim of the present paper is to develop methods for calculating effective characteristics of corrugated plates. To this end, we first write out the equilibrium equations for a flexible anisotropic plate, which is inhomogeneous in the thickness direction and in the horizontal projection, with an initial deflection. We write these equations in matrix form, which allows one to significantly reduce the length of the expressions and to simplify further calculations. After this, we average the initial matrix equations with variable coefficients. The averaging procedure implies the statement of problems such that, after solving them, we can calculate the desired effective characteristics. By way of example, we consider the case of a corrugated plate made of a homogeneous isotropic material whose corrugations are hexagonal in the horizontal projection. In this case, we obtain approximate expressions for the components of the effective tensors of flexural rigidity and longitudinal compliance and expressions for the effective plate thickness.