Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells北大核心CSCD

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

A Spatial Geometric Nonlinearity Spline Beam Element With Nodal Parameters Containing Strains北大核心CSCD

Many slender rods in engineering can be modeled as Euler-Bernoulli beams. For the analysis of their dynamic behaviors, it is necessary to establish the dynamic models for the flexible multi-body systems. Geometric nonlinear elements with absolute nodal coordinates help solve a large number of dynamic problems of flexible beams, but they still face such problems as shear locking, nodal stress discontinuity and low computation efficiency. Based on the theory of large deformation beams’ virtual power equations, the functional formulas between displacements and rotation angles at the nodes were established, which can satisfy the deformation coupling relationships. The generalized strains to describe geometric nonlinear effects in this case were derived. Some parameters of boundary nodes were replaced by axial strains and sectional curvatures to obtain a more accurate and concise constraint method for applying external forces. To improve the numerical efficiency and stability of the system’s motion equations, a model-smoothing method was used to filter high frequencies out of the model. The numerical examples verify the rationality and effectiveness of the proposed element. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

NUMERICAL STUDIES OF 2D FREE SURFACE WAVES WITH FIXED BOTTOM

AbstractThe motion of surface waves under the effect of bottom is a very interesting and challenging phenomenon in the nature, we use boundary integral method to compute and analyze this problem. In the linear analysis, the linearized equations have bounded error increase under some compatible conditions. This contributes to the cancellation of instable Kelvin-Helmholtz terms. Under the effect of bottom, the existence of equations is hard to determine, but given some limitations it proves true. These limitations are that the swing of interfaces should be small enough, and the distance between surface and bottom should be large enough. In order to maintain the stability of computation, some compatible relationship must be satisfied like that of [5]. In the numerical examples, the simulation of standing waves and breaking waves are calculated. And in the case of shallow bottom, we found that the behavior of waves are rather singular.     

ASYMPTOTIC ANALYSIS OF DYNAMIC PROBLEMS FOR LINEARLY ELASTIC SHELLS-JUSTIFICATION OF EQUATIONS FOR DYNAMIC FLEXURAL SHELLS

61. IntroductionIn this paper) or,P,a,T,' take their values in the set {1, 2}; i, i, k, l,' take their valuesin the set {1, 2, 3}.In [1] under appropriate conditions on the body force density and the middle surfaceof elastic shells, starting from the three-dimensional dynamic equations of elastic shells wehave given the justification of two-dimensional dynamic equations of membrane shells. Inthis paper, under different assumptions on the body force density and the middle surfaceof elastic…     

Thermal Buckling Analysis of FGM Sandwich Circular Plates Under Transverse Nonuniform Temperature Field Actions北大核心CSCD

Based on the von Kármán geometric nonlinear plate theory, the displacement⁃type geometric nonlinear governing equations for FGM sandwich circular plates under transverse nonlinear temperature field actions were derived. With the immovable clamped boundary condition, the analytical formula for dimensional critical buckling temperature differences of the system was obtained from the solution of the linear eigenvalue problem. Moreover, the 2⁃point boundary value problem of ordinary differential equations was solved with the shooting method. The effects of geometric parameters, constituent material properties, gradient indexes, temperature field parameters and layer⁃thickness ratios on the critical buckling temperature differences, the thermal postbuckling equilibrium paths, and the buckling equilibrium configurations of FGM sandwich circular plates, were investigated. The results show that, with the increases of the thickness⁃radius ratio, the relative thickness of the FGM layer and the gradient index, the FGM sandwich circular plate's critical buckling temperature difference will increase monotonically. Given a fixed radius and a fixed total thickness, the postbuckling deformation of the FGM sandwich circular plate will decrease significantly with the relative thickness of the FGM layer. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

响应变量随机缺失下变系数模型的变量选择

In this paper,we present a variable selection procedure by combining basis function approximations with penalized estimating equations for varying-coefficient models with missing response at random.With appropriate selection of the tuning parameters,we establish the consistency of the variable selection procedure and the optimal convergence rate of the regularized estimators.A simulation study is undertaken to assess the finite sample performance of the proposed variable selection procedure.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.     

THE CONFOCAL INVOLUTIVE SYSTEM AND THE INTEGRABILITY OF THE NONLINEARIZED LAX SYSTEMS FOR AKNS HIERARCHY

The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.     

Three-Level Safety Evaluation of Suspended Sections of Underwater Buried Pipelines北大核心CSCD

The uneven riverbed, and the impact and scour actions by water flow, make the underwater buried pipeline vulnerable to exposure and suspension, and endanger the pipeline operation safety. To investigate the mechanical properties and failure behaviors of the suspended pipeline section under water impact, according to the failure mechanism of the pipeline, the statics and dynamics analyses of the pipeline were carried out, and the graded safety evaluation technique for the buried pipeline with suspended sections was presented. First, a “static strength safety evaluation under static loads” (level 1) was conducted according to the mechanical features and stress states of the pipeline’s suspended section. Second, a “resonance safety evaluation under dynamic loads” (level 2) was conducted based on the correlation between the natural vibration frequencies of the suspended pipeline and the vortex emission frequencies of water flow. Finally, the periodical change process of the pipeline’s alternating stress was studied to solve the fatigue damage and fatigue life of the pipeline, and the “fatigue strength safety evaluation under dynamic loads” (level 3) was performed. Thus, a 3-level safety assessment procedure for pipelines with suspended sections was proposed. The stabilizing measures for pipelines of poor safety were suggested, and through an example, a specific calculation process was provided. The work serves as a theoretical guide for the safety evaluation of the suspended sections of underwater buried pipelines. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

混合型分段连续微分方程的数值稳定性

This paper deals with the stability analysis of the Euler-Maclaurin method for differential equations with piecewise constant arguments of mixed type. The expression of analytical solution is derived and the stability regions of the analytical solution are given. The necessary and sufficient conditions under which the numerical solution is asymptotically stable are discussed. The conditions under which the analytical stability region is contained in the numerical stability region are obtained and some numerical examples are given.     

Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection

This paper presents the conditions that can possibly lead to chaotic motion and bifurcation behavior for a simply-supported large deflection thermo-elastic circular plate with variable thickness by utilizing the criteria of fractal dimensions, maximum Lyapunov exponents and bifurcation diagrams. The governing partial differential equation of the simply supported thermo-elastic circular plate with variable thickness is first derived by means of Galerkin method. Several different features including Fourier spectra, phase plot, Poincar’e map and bifurcation diagrams are numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitations of lateral loads and thermal loads. Numerical examples are presented to verify the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. Numerical modeling results indicate that large deflection motion of a thermo-elastic circular plate with variable thickness possesses chaotic motions and bifurcation motion under different lateral loads and thermal loads. The simulation results also indicate that the periodic motion of a circular plate can be obtained for the convex or the concave circular plate. The dynamic motion of the circular plate is periodic for the cases including (1) the lateral loading frequency is within a specific range, (2) thermal and lateral loadings are operated in a specific range and (3) the thickness parameter is less than a specific critical value for the convex circular plate or greater than a specific critical value for the concave circular plate. The modeling results show that the proposed method can be employed to predict the non-linear dynamics of any large deflection circular plate with variable thickness.     

Modelling of dynamic interaction between structure and trolley for mega container cranes

This article deals with the analysis of trolley impact on the dynamic behaviour of the flexible structure of the mega quayside container crane (QCC) boom, identified as the most relevant structural part. It develops a modelling method for the dynamic response of the large flexible structure of the QCC boom under a moving trolley. By using FEM the original structure of the whole crane structure is reduced to an equivalent model of the boom. The boom is in this way modelled as a system with distributed parameters, comprising reduced stiffnesses and lumped masses from other parts of the upper structure. The article looks at the moving mass approach to achieve the desired performance of the QCC. Differential equations of the mathematical model are obtained by using Lagrange's equations and the assumed mode method. The continuum is discretized by a finite number of admissible functions. Deterministic simulation gives the dynamic response of the boom for quay-to-ship container transfer. Results are obtained for the boom deflection and bending moment values, as well as for the dynamic amplification factor of deflection.     

流固冲击下加筋板的非线性动态屈曲

分析了流固冲击下加筋板的非线性弹性动态屈曲．考虑板与筋的膜力,忽略面内位移,运用Hamilton变分原理,得出非线性控制方程,采用双级数形式的挠度假设,由Galerkin方法得到离散方程组,根据Budiansky-Roth(B-R)曲线,判断加筋板的动态屈曲．     

均布载荷下正交异性变厚度圆薄板大挠度问题

本文导出了正交各向异性变厚度圆薄板大挠度问题的基本方程,用修正迭代法求解了正交各向异性变厚度圆薄板在均布载荷下的大挠度问题.作为特例,令ε=0,则由本文结果得到的表达式与J.Nowinski用摄动法得到的正交各向异性等厚度圆薄板大挠度问题的解完全一致.     

Chaotic and bifurcation for a simply-supported thermo-mechanical coupling circular plate with variable thickness

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads , η₁=1.1, β=0.5, and _∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection.     

弹性地基上正交各向异性变厚度圆薄板的大挠度问题

本文推出了均布载荷下弹性基地上的正交各向异性变厚度圆薄板大挠度问题的基本方程。利用修正迭代法获得了该问题的二阶近似解。     

变厚度圆薄板非对称非线性弯曲问题

首先将直角坐标系中的横向变厚度薄板的大挠度方程,转化到极坐标系中的变厚度圆薄板的非对称大挠度方程·　此方程和极坐标系中径向、切向两个平衡方程联立求解·　将物理方程和中面应变非线性变形方程,代入3个平衡方程,可得用3个变形位移表示的3个非对称非线性方程·　用Fourier级数表示的解代入基本方程,获得相应的基本方程·　在周边夹紧边界条件下,用修正迭代法求解·　作为算例,研究了余弦形式载荷作用下的问题,还给出了载荷与挠度的特征曲线,曲线依据变厚度参数变化而变化,其结果和物理概念完全吻合·     

Geometrically nonlinear static and dynamic analysis of functionally graded skew plates

The present paper deals with nonlinear static and dynamic behavior of functionally graded skew plates. The equations of motion are derived using higher order shear deformation theory in conjunction with von-Karman’s nonlinear kinematics. The physical domain is mapped into computational domain using linear mapping and chain rule of differentiation. The spatial and temporal discretization is based on fast converging finite double Chebyshev series and Houbolt’s method. Quadratic extrapolation technique is employed to linearize the governing nonlinear equations. The spatial and temporal convergence and validation studies have been carried out to establish the efficacy of the present solution methodology. In case of dynamic analysis, the results are obtained for uniform step, sine, half sine, triangular and exponential type of loadings. The effect of volume fraction index, skew angle and boundary conditions on nonlinear displacement and moment response are presented.     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

关于圆簿板大挠度问题的正交条件解法

本文重新考察了钱伟长教授求解圆薄板大挠度问题的系统近似法,发现此法实质上可视为奇异摄动理论中的变形参数法.以无量纲中心挠度为小参数,将挠度、中面薄膜力和载荷参数作渐近展开,我们对所得的递推方程给出了正交条件(可解性条件),据此可确定圆薄板的刚度特性.本文指出,利用圆薄板小挠度解和正交条件,可以不经求解方程而导得载荷参数与中心挠度关系的三阶近似以及中心点、边缘处的薄膜力的首项近似.文中对若干特例(均布载荷、复合载荷、各种边界条件)进行了具体计算,所得的结果与钱伟长、叶开沅、黄黔等人在文[1～4]中给出的结果完全相符.