Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base

The non-linear integro-differential equations of motion for a slender cantilever beam subject to axial narrow-band random excitation are investigated. The method of multiple scales is used to determine a uniform first-order expansion of the solution of equations. According to solvability conditions, the non-linear modulation equations for the principal parametric resonance are obtained. Firstly, The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved, in which, the modified Bessel function of the first kind is introduced. Results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response and stabilizes the system for a lower acceleration oscillating amplitude but intensifies the instability of the trivial response for a higher one. Secondly, the first and second order non-trivial steady state response of the system is obtained by perturbation method and the corresponding amplitude–frequency curves are calculated when the bandwidth is very small. Results show that the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the hardening type for the first mode, whereas for the second mode the effective non-linearity of whether the amplitude expectation of the first order steady state response or the amplitude expectation of the second order steady state response is of the softening type. Finally, the stochastic jump and bifurcation is investigated for the first and second modal parametric principal resonance. The basic jump phenomena indicate that, under the conditions of system parameters with a smaller bandwidth, the most probable motion is around the non-trivial branch of the amplitude response curve, whereas with a higher bandwidth, the most probable motion is around the trivial one of the amplitude response curve. However, the stochastic jump is sometimes more sensitive to the change of the bandwidth, in other words, a small change of bandwidth may induce a series of stochastic jump and bifurcation.     

EXPERIMENTAL RESEARCHES ON MAGNETO-THERMO-MECHANICAL CHARACTERIZATION OF TERFENOL-D

The coupling effects of axial pre-stress,temperature and magnetic field on magne- tostrictive strain and magnetization as well as Young's modulus of a Terfenol-D (Tb_(0.3)Dy_(0.7)Fe_(1.93)) rod are tested to give a good understanding of magneto-thermal-mecha-nical characteristics of giant magnetostrictive materials.Results show that magneto-thermo-mechanical coupling of gi- ant magnetostrictive materials is very strong;and the influences of pre-stress and temperature on magnetostrictive strain and Young's modulus vary with the intensity of magnetic field.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

不确定边界条件下悬臂梁裂纹参数识别方法研究

基于Bernoulli-Euler梁振动理论,以等效弹簧来模拟裂纹引起的局部软化效应和由非完全固支边界条件引起的转角效应。推导了悬臂梁在不确定边界条件下确定其振动频率的特征方程,直接利用该特征方程,提出一种有效估计裂纹参数的优化方法,通过计算测量频率和理论频率之间的误差目标函数最小化即可识别裂纹参数-裂纹位置和深度。最后,应用两个实例-理想固支边界条件下和非完全固支边界条件下的悬臂梁实验来说明本文方法的有效性。实验结果表明:只需梁结构前三阶频率即可识别裂纹位置和深度。对于理想边界条件下的裂纹参数识别,在测量频率存在小误差情况下,该方法仍能给出比较满意的结果,对于非完全固支边界条件下的裂纹参数识别,利用本文方法能得到比Narkis的方法更精确的裂纹位置识别结果。同时本文方法还能给出比较满意的裂纹深度识别结果。     

A new technique for large deflection analysis of non-prismatic cantilever beams

This paper studies the very large deflection behavior of prismatic and non-prismatic cantilever beams subjected to various types of loadings. The formulation is based on representing the angle of rotation of the beam by a polynomial on the position variable along the deflected beam axis. The coefficients of the polynomial are obtained by minimizing the integral of the residual error of the governing differential equation and by applying the beam’s boundary conditions. Several numerical examples are presented covering prismatic and non-prismatic cantilever beams subjected to uniform, non-uniform distributed loads and tip concentrated loadings in vertical and horizontal directions. The loads considered in this study are restricted to the non-follower type loads. Cases with different loadings and geometries are compared with MSC/NASTRAN computer package. However, for some very large deflection case, the MSC/NASTRAN failed to predict the deflected shape due to divergence problems.     

空间悬臂梁的动力学分析与迭代求解

本文从柔性体的真-伪混合坐标形式的拉格朗日方程出发,导出了空间双柔性连杆机械臂系统的真-伪混合坐标形式的动力学方程.由于采用伪坐标和矩阵运算,避免了繁冗的对方位参数的求导过程,使方程结构紧凑,便于建立递推形式,并且在选择不同参数描述整体刚性运动情况下,方程形式保持不变,因而具有一般性.     

矩形悬臂薄板的解析解

首先把弹性薄板弯曲问题的控制方程表示成为Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出矩形悬臂薄板的解析解。由于在求解过程中不需要事先人为地选取挠度函数,而是从薄板弯曲的基本方程出发,直接利用数学的方法求出可以满足其边界条件的这类问题的解析解,使得问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文所采用的方法以及所推导出的公式的正确性。     

SIMPLIFIED CALCULATION METHOD FOR DEALING WITH ADDED WATER MASS FOR IMMERSED CANTILEVER CYLINDER 1)

采用解析方法研究了部分浸入水中的圆形悬臂柱的附连水质量分布,给出了无量纲的质量分布系数,通过对质量分布系数的数值拟合分析,附连水质量效应可最终表达为结构的等效质量密度,在振动计算中只需将浸入水中的悬臂柱密度用等效密度替代即可。采用试验方法测量了部分浸入水中的悬臂柱湿频率,简化附连水质量方法计算的湿频率结果与试验测量值吻合很好,表明本文的附连水质量简化公式可很好地描述附连水的质量效应。     

超磁致伸缩换能器系统动力学特性分析

根据第一压磁理论建立超磁致伸缩换能器系统含有平方、立方项非线性动力学模型,利用非线性振动近似解析方法对换能器系统进行分析,揭示了换能器系统非线性特性。利用基于MAT-LAB/Simulink的系统仿真理论,建立换能器系统动态仿真模型,发现换能器系统在一定参数下存在着混沌现象。通过数值模拟的方法对换能器系统做深入的理论分析,得出系统在不同参数下存在着倍周期、倒倍周期分岔等复杂非线性现象。本文的分析结果表明,超磁致伸缩材料的弹性模量、压磁系数、相对磁导率和系统阻尼系数等参数是超磁致伸缩换能器系统动力学特性的主要影响因素。     

DEFLECTION OF FOUR TYPES OF CANTILEVER BEAMS WITH VARIABLE CROSS SECTION UNDER LATERAL TRIANGULAR DISTRIBUTED LOADS1)

研究了圆柱、圆锥、抛物型和双曲型回转变截面悬臂梁在侧向三角形分布载荷下的挠度。基于四类回转悬臂梁的惯性矩沿长度方向的分布规律,得到其任意侧向分布载荷下的挠曲线方程。基于三角形分布载荷下的挠曲线方程,得到其端部挠度值。在等长度和等体积假设下,通过比较端部挠度值找到四类悬臂梁中挠度最小者。研究表明三角形分布载荷下,特征参数在特定范围内,母线为双曲线的悬臂梁挠度最小。