极坐标系下扇形截面杆扭转问题的有限差分法

导出了极坐标系下扇形截面杆扭转问题的差分格式；结合Mathcad编程用逐次超松弛迭代法求出了扭转应力函数差分值；采用复化二维辛普生求积公式计算了抗扭刚度；求出了相应的扭转应力的差分解；给出了张角为2π时裂纹尖端附近的第三型应力强度因子．     

扇形截面杆扭转的应力分析

柱体扭转的基本方程为非齐次偏微分方程,在极坐标系下,利用分离变量法及傅立叶级数展开法,求出了扭转应力函数,进一步即可计算出扇形截面杆在外力偶作用下,扭转角和横截面上剪应力的精确解答。这种方法为精确解法,在各种机械及其他工程设备中,对受扭转作用的扇形截面杆设计,有一定实用价值。     

不可压缩平面问题的位移-压力混合重心插值配点法

引入人工压力变量,将弹性本构方程以应力、应变和压力表达,建立求解不可压缩平面弹性问题的位移-压力方程和不可压缩条件方程的耦合偏微分方程组。利用张量积型重心Lagrange插值近似二元函数,得到计算插值节点处偏导数的偏微分矩阵。采用配点法离散不可压缩弹性控制方程,利用偏微分矩阵直接离散弹性力学控制方程为矩阵形式方程组。利用插值公式离散位移和应力边界条件,将离散边界条件与离散控制方程组合为新的方程组,得到求解弹性问题的过约束线性代数方程组;利用最小二乘法求解线性方程组,得到弹性力学问题位移数值解。数值算例验证了所提方法的数值计算精度为10-14~10-10。     

含锥形变截面压杆稳定计算的传递矩阵法

本文从压杆的微分方程出发，推得了锥形压杆的传递矩阵，用传递矩阵法对复杂的变截面压杆进行了一系列的稳定计算。计算结果表明，本文的公式正确。该方法具有精确度高、力学概念清晰及简便易行的优点，适于工程技术人员在微机上应用     

有穿透裂纹的圆柱体扭转应力及应力强度因子

本文对内部有纵向穿透的直线裂纹圆柱体扭转应力及应力强度因子,用线弹性理论复变函数方法进行了讨论.籍助于解析开拓,引进了一解析函数M(Z_1),把在有限的二连通域内求复扭曲函数F(Z)的问题,化归为在无限的单连通城内求解析函数M(Z_1)的问题.而后者很易于应用的方法采用保角映射求得解答.于是马上就可得到复扭曲函数.在此基础上,本文用级数形式给出了位移分量(?)、切应力分量的一般表达式,还给出了抗扭刚度和扭转应力强度因子的一般表达式.最后对αh 小的裂纹体扭转应力强度因子进行了分析和计算,指出了扭转应力强度因子随裂纹中心位置的变化规律,并说明其计算过程是比较简单的.     

梯形应力脉冲在弹性杆中的传播过程和几何弥散

在应力波传播过程中,几何弥散效应往往难以避免.对应力波在弹性杆中传播的几何弥散效应进行解析分析,对于基础波动问题研究以及材料动态力学行为表征等课题,显得至关重要.本文简单说明了弹性杆中考虑横向惯性修正的一维 Rayleigh-Love应力波理论,概述了其波动控制方程的变分法推导过程;针对 Hopkinson杆实验中常用的梯形应力加载脉冲,建立了相应的偏微分方程初边值问题的求解模型,并运用 Laplace变换方法研究了脉冲在杆中传播的几何弥散现象;根据留数定理进行 Laplace反变换,给出了杆中不同位置和时刻的应力波的级数形式解析解,分析了计算项数对结果收敛性的影响;将解析计算结果与采用三维有限元数值模拟的计算结果进行对比,两者吻合程度良好,从而证明 Rayleigh-Love横向惯性修正理论可以有效地表征典型 Hopkinson杆实验中的几何弥散效应.在此基础上围绕梯形加载脉冲的弥散效应进行参数研究,定量描述了传播距离、泊松比、脉冲斜率等参数的影响.本文给出的 Rayleigh-Love杆在梯形加载条件下的解析解,揭示了几何弥散效应的本质规律,可以用于实际实验的弥散修正过程.      

复弯矩表示的非圆截面杆平衡的Schrdinger方程及其半解析解

弹性细杆的平衡和稳定性问题的研究在工程和分子生物学中有重要的应用背景。利用文中提出的复柔度概念,建立了用复弯矩表示的非圆截面杆平衡的Schrdinger方程。借助复曲率概念,导出以杆的曲率、挠率和截面相对Frenet坐标系的扭角为未知变量的2阶常微分方程,此方程与传统使用的Kirchhoff方程等价。文献中仅适用于圆截面杆平衡问题的Schrdinger方程为本文导出方程的特例。对于准对称截面杆,用小参数法分别建立了零次和一次近似方程,其中零次近似方程存在解析解。对于截面的主轴坐标轴与中心线的Frenet坐标轴重合的无扭转杆特殊情形,Schrdinger方程转化为Duffing方程,应用数值方法作出了Duffing杆变形后的三维几何图形。     

梯形应力脉冲在弹性杆中的传播过程和几何弥散

在应力波传播过程中,几何弥散效应往往难以避免.对应力波在弹性杆中传播的几何弥散效应进行解析分析,对于基础波动问题研究以及材料动态力学行为表征等课题,显得至关重要.本文简单说明了弹性杆中考虑横向惯性修正的一维Rayleigh-Love应力波理论,概述了其波动控制方程的变分法推导过程;针对Hopkinson杆实验中常用的梯形应力加载脉冲,建立了相应的偏微分方程初边值问题的求解模型,并运用Laplace变换方法研究了脉冲在杆中传播的几何弥散现象;根据留数定理进行Laplace反变换,给出了杆中不同位置和时刻的应力波的级数形式解析解,分析了计算项数对结果收敛性的影响;将解析计算结果与采用三维有限元数值模拟的计算结果进行对比,两者吻合程度良好,从而证明Rayleigh-Love横向惯性修正理论可以有效地表征典型Hopkinson杆实验中的几何弥散效应.在此基础上围绕梯形加载脉冲的弥散效应进行参数研究,定量描述了传播距离、泊松比、脉冲斜率等参数的影响.本文给出的Rayleigh-Love杆在梯形加载条件下的解析解,揭示了几何弥散效应的本质规律,可以用于实际实验的弥散修正过程.     

复弯矩表示的非圆截面杆平衡的Schrǒdinger方程及其半解析解

弹性细杆的平衡和稳定性问题的研究在工程和分子生物学中有重要的应用背景。利用文中提出的复柔度概念，建立了用复弯矩表示的非圆截面杆平衡的Schrǒdinger方程。借助复曲率概念，导出以杆的曲率、挠率和截面相对Frenet坐标系的扭角为未知变量的2阶常微分方程，此方程与传统使用的Kirchhoff方程等价。文献中仅适用于圆截面杆平衡问题的Schrǒdinger方程为本文导出方程的特例。对于准对称截面杆，用小参数法分别建立了零次和一次近似方程，其中零次近似方程存在解析解。对于截面的主轴坐标轴与中心线的Frenet坐标轴重合的无扭转杆特殊情形，Schrǒdinger方程转化为Duffing方程，应用数值方法作出了Duffing杆变形后的三维几何图形。     

弹性基础上无缝轨道应力分析的半解析法

从钢轨应力分析的要求出发,提出了弹性基础上开口厚壁杆的半解析计算方法.轨道截面上沿纵向的正应力分为弯曲正应力和约束扭转正应力,弯曲正应力可以根据弹性基础梁的弯曲理论求得,而约束扭转正应力将采用本文的半解析方法.把钢轨的横截面离散为有限单元,将位移(z方向)解表示为横截面上一个离散的数值函数(称为拟扇性坐标ω(x,y))与长度方向上的解析函数相乘的形式.用最小势能原理求解横截面上拟扇性坐标ω的有限元解和长度方向上解析函数表达式.以75kg钢轨为算例,计算了ω、-((e)ω)/((e)x) y和x-((e)ω)/((e)y)的结果,通过它们可以进一步计算钢轨中的约束扭转正应力和截面上的剪应力.     

Deformations of helical spring and cuboid into hollow cylinders

We study finite inhomogeneous deformations of a helical spring with a rectangular cross-section and a long cuboid. Two surfaces of the spring or the cuboid are joined to obtain a hollow cylinder. When body forces are absent the equilibrium equations reduce to ordinary differential equations. The stress-strain states are the same in each cross-section. The proposed deformations correspond to an inflation, an extension and a torsion of the obtained hollow cylinders. If the obtained cylinders are free of external applied loads, then they have residual stresses.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

THIRD-ORDER GENERALISED BEAM THEORY AND CALCULATION METHOD

一阶广义梁理论描述通过运用加入弯曲、扭转和畸变函数的普通非耦合微分方程组解决棱柱状结构行为.二阶广义梁理论,是添加上偏离力效果的微分方程. 通过引入纵向膜弯矩和膜剪应变虚功到广义梁理论系统当中,完全展开的三阶广义梁方程组将以一串大型离散迭代函数且能转化为可用于数值分析的若干切线刚度矩阵形式出现. 通过膜应力派生出三阶分项^ijrkv_σ和^ijrkv_τ并结合先进数值技术寻求全解,三阶广义梁理论提供了一种严谨和高效的数值工具用于调查薄壁结构后屈曲大变形行为.     

三阶广义梁理论及其计算方法

一阶广义梁理论描述通过运用加入弯曲、扭转和畸变函数的普通非耦合微分方程组解决棱柱状结构行为.二阶广义梁理论,是添加上偏离力效果的微分方程. 通过引入纵向膜弯矩和膜剪应变虚功到广义梁理论系统当中,完全展开的三阶广义梁方程组将以一串大型离散迭代函数且能转化为可用于数值分析的若干切线刚度矩阵形式出现. 通过膜应力派生出三阶分项ijrkv_σ和ijrkv_τ并结合先进数值技术寻求全解,三阶广义梁理论提供了一种严谨和高效的数值工具用于调查薄壁结构后屈曲大变形行为.      

Non-linear response of a magneto-elastic translating beam with prismatic joint for higher resonance conditions

The non-linear response of a magneto-elastic translating beam having prismatic joint for higher resonance conditions is studied. A periodically varying transverse magnetic field is applied to the system. Two frequencies of prismatic motion and oscillating transverse magnetic field are implemented to the system. The method of multiple scales as one of the perturbation techniques is used to derive two first order ordinary differential equations that govern the time variation of the amplitude and phase of the response. Then a stability analysis is conducted for subharmonic resonance and simultaneous resonance conditions. A parametric study is performed to investigate the effect of magnetic field strength, amplitude of prismatic motion, damping and payload mass on the frequency response curves for both the resonance conditions. The catastrophic failure of the system may occur due to the presence of saddle-node and pitchfork bifurcations. The results obtained by method of multiple scales are compared with those obtained by numerically integrating the reduced equations and are found to be in good agreement. The developed results can be applied to control the vibration of a beam with prismatic joint subjected to magnetic field for third order subharmonic resonance and simultaneous resonance conditions.     

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介绍了求解复杂截面闭口薄壁杆件扭转问题的网络理论解法,以普朗特应力函 数解法为基础,通过与电路问题比拟,借鉴了电路理论中的网络理论解法. 本文方法对解决复 杂截面闭口薄壁杆件扭转的工程问题有一定的应用价值. 在教学上,可以起到拓宽学生思路 的作用.     

Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars

The Saint-Venant torsion of linearly elastic anisotropic cylindrical bars with solid and hollow cross-section is treated. The shear flexibility moduli of the non-homogeneous bar are given functions of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsion problem of non-homogeneous anisotropic bar is expressed in terms of the torsion and Prandtl's stress functions of the corresponding homogeneous anisotropic bar having the same cross-section as the non-homogeneous bar.     

变截面梁动力响应的频域传递矩阵计算方法

为提高变截面梁振动分析的计算效率,提出了基于频域传递矩阵法的动力计算算法．首先,选择线速度、角速度、弯矩和剪力作为求解变量,通过Laplace变换将变截面梁的动力响应时域偏微分方程转换为频域常微分方程;然后,通过求解频域方程并结合协调和边界条件建立变截面梁的频域传递矩阵;通过构造傅里叶级数展开形式的时域响应函数,对变截面梁传递矩阵方法求解的频响函数进行Laplace逆变换,建立了变截面梁的固有特性计算和时域瞬态响应计算方法,最后,借助数值仿真软件,开发了变截面梁动力响应分析的计算程序．完成对算例的仿真计算和分析,并与有限元计算结果进行对比,数值结果验证了该方法的正确性和有效性．     

Lateral buckling of beams with shear deformations – A hyperelastic formulation

The equilibrium and buckling equations are derived for the lateral buckling of a prismatic straight beam. A consistent finite strain constitutive law is used, which is based on a hyperelastic model for an isotropic material. The kinematics of the cross-sectional deformations are based on a Timoshenko type beam displacement of the cross-sectional plane using Euler angles and two shear finite rotations coupled with warping taken normal to the displaced plane. Also derived are the second order approximations to the displacements, curvatures, twist and internal actions. The constitutive relationships for the internal actions reveal new coupling terms between the bending moments, torsion and bimoment, which are functions of the cross-sectional warping and shear deformations. New Wagner type nonlinear torsion terms are derived which are functions of the warping of the cross-sectional plane, and are coupled to the twisting and shear deformations of the cross-section. Solutions are determined for the lateral buckling of a prismatic monosymmetric beam under pure bending and the flexural–torsional buckling under axial compression. For the flexural–torsional buckling problem it is found that the Euler type column buckling formula is consistent with Haringx’s column buckling formula while the torsional buckling formula is different to conventional equations. The second variation of the total potential is also derived. The effects of shear deformations are explored by examining the non-dimensional lateral buckling equation for a simply supported beam.