Data fusion of acceleration and angular velocity for improved model updating

In the traditional finite element (FE) model updating, translational responses, such as acceleration, have generally been employed to identify the structural properties. However, the boundary conditions of a structure are associated with both translational and rotational DOFs. Thus, the combinational measurement of translational and rotational responses (e.g., angular velocity) would increase accuracy of FE model updating of structures, especially in identifying their boundary conditions. This paper proposes data fusion of translational and rotational responses for improved system identification using FE model updating technique. In the proposed method, the accelerometers and gyroscopes are installed in between and near the supports of a structure, respectively, and FE model updating is carried out using the natural frequencies, the translational mode shapes obtained from accelerations, and the rotational mode shapes obtained from angular velocities. Numerical and experimental verifications are carried out on simply-supported beam structures. The verifications show that the proposed FE model updating strategy based on the data fusion results in more accurate assessment of both structural properties and boundary conditions than the traditional FE model updating using translational responses only.     

Stiffness Analysis of Corrugated Flexure Beam Used in Compliant Mechanisms

Conventional flexible joints generally have limited range of motion and high stress concentration. To overcome these shortcomings, corrugated flexure beam(CF beam) is designed because of its large flex...     

任意边界条件的弹性长方体封闭腔辐射声场分析

在弹性板边界施加假想的弹簧系统模拟板的不同边界条件,利用汉密尔顿函数和瑞利—李兹方法,建立了由1块四边弹性支承的弹性板及5块刚性板构成的封闭矩形腔外的辐射声场模型,推导了腔体外辐射声场的解析解。该模型考虑了任意边界条件下的弹性板与腔体内声场之间的耦合。将仿真计算结果与实验结果进行比较,验证了理论推导的正确性。算例表明:低频段时支承板的线弹簧的刚度变化对辐射声场的影响较旋转弹簧的大。     

钳夹车双层并联过约束机构力学解算与均载性分析

为研究钳夹车双层并联过约束起升机构力传递与均载特性,基于Timoshenko梁理论建立了起升机构钳形梁刚度模型,通过联立该双层并联过约束机构弹性变形协调方程与静力平衡方程,对起升机构进行了力学解算,得到了机构分支承载力与机构刚度的映射关系。在此基础上,定义了不均衡载荷系数,分析得到了机构刚度变化及其载运偏载量对起升机构均载性能的影响规律。基于动力学仿真软件RecurDyn建立了双层并联过约束起升机构的刚柔耦合仿真模型,仿真分析结果验证了理论建模分析的正确性。     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.     

计入齿圈柔性的直齿行星传动动力学建模

为揭示齿圈柔性对传动系统动态性能的影响,在系杆随动坐标系下建立计入齿圈柔性的直齿行星传动精细化动力学模型,建模中计入时变啮合刚度、支承刚度、陀螺效应和齿圈柔性等诸多影响因素。采用有限段单元的离散化建模方法,将连续体的柔性齿圈离散成由等效虚拟弹簧连接的刚性齿圈段,并推导齿圈段与行星轮的啮合判据。通过分析各构件间的相对位移关系及其受力,最终建立系统的运动微分方程。以三行星轮的NGW型直齿行星传动为例对其进行固有特性分析,得出系统的低阶固有频率分布及相应的振动模式,并进一步分析齿圈柔性和安装方式对系统固有特性的影响。研究结果表明,直齿行星传动的振型可归结为中心构件(太阳轮、系杆)扭转振动和中心构件平移振动两种模式,两种模式下齿圈和行星轮均做复杂平面振动;齿圈的柔性会降低系统的低阶固有频率,且其影响程度与齿圈安装方式有关,齿圈完全固定时其影响最小,径向浮动或周向转动时影响较明显。     

Dynamic analysis of rotating damped beams with an elastically restrained root

The vibration of a rotating damped blade with an elastically restrained root is investigated. The effects of viscous damping and the translational and rotational damping at the root of blade are considered. The flow-indued force and moment between the tip of a blade and the casing are simulated by using the time-dependent boundary conditions. A simple and efficient algorithm for deriving the semi-analytical steady state solution of the general system is proposed. The governing equation is divided into two coupled real differential equations. The two coupled equations are uncoupled into an eighth-order characteristic differential equation. The eight corresponding boundary conditions are obtained. The eight linearly independent homogenous semi-analytical solutions of the eighth-order characteristic differential equation are derived. If the coefficients of the uncoupled governing differential equation are constant, the exact fundamental solutions are obtained. The exact steady state solution is obtained by using Green's function in terms of the eight homogenous solutions. Moreover, the influence of the translational, rotational and viscous damping constants on the frequency response curves of a rotating beam are investigated. The opposite influence of the transverse viscous damping constant and the root damping constants on the frequency of resonance is revealed.     

基于曲率模态与柔度曲率的损伤识别

以含两条裂纹的两端固定梁为例,采用曲率模态差和模态柔度曲率差来检测结构的损伤。首先将梁的裂纹模拟为无质量的等效扭转弹簧,推导了裂纹梁的特征微分方程,利用边界条件和裂纹位置的连续性条件推导得到该裂纹梁的振形函数解析表达式。然后用中心差分法分别求解裂纹梁损伤前后的曲率模态值和模态柔度曲率值,利用其差值确定梁的损伤位置,进而确定损伤程度。最后讨论了曲率模态和柔度曲率对结构损伤识别的敏感性。     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

考虑边界条件的弹性长方体封闭结构腔辐射声场分析

在弹性板边界施加假想的弹簧系统模拟板的不同边界条件,利用汉密尔顿函数和瑞利-李兹方法,建立由2块四边弹性支承的弹性板及4块刚性板构成的封闭矩形腔外的辐射声场模型,推导腔体外辐射声场的解析解.该模型考虑两弹性板之间以及弹性板与腔体内声场之间的耦合.算例表明,低频段时两弹性板之间为弱耦合,辐射声场主要由受到外激励的结构所决定;支承板的线弹簧的刚度变化对辐射声场的影响较旋转弹簧大.同时,运用有限元法计算腔体内的声模态和弹性板的振动模态及其耦合系数,结合边界元法计算出腔体外的声压响应,并将此数值仿真结果与前面导出的解析结果进行比较,理论推导的正确性得到验证.     

利用局部振动频率识别框架结构构件刚度参数

鉴于依据结构整体动力反应很难准确识别结构局部构件的物理参数,提出了结构局部激振检测方法,既对结构局部进行激振,依据结构局部激振反应提取其固有频率,再利用此固有频率运用方法识别结构局部构件的物理参数。本文将框架结构中的梁柱简化为两端有弹性约束、具有等刚度和等分布质量的杆件模型,推导了这一模型的频率特征方程,并进一步分析了两端弹性约束对其振动固有频率的影响。仿真算例表明：在固有频率误差较小的情况下,本文方法可准确地给出构件抗弯刚度和边界条件。     

I-beam Crack Identification Based on Study of Local Flexibility due to Crack

Local flexibility of crack plays an important role in crack identification of structures.Analytical methods on local flexibility in a cracked beam with simple geometric crossing sections,such as rectangle,circle,have been made,but there are some difficulties in calculating local flexibility in a cracked beam with complex crossing section,such as pipe and I-beam.In this paper,an analytical method to calculate the local flexibility and rotational spring stiffness due to crack in I-beam is proposed.The local flexibility with respect to various crack depths can be calculated by dividing a cracked I-beam into a series of thin rectangles.The forward and inverse problems in crack detection of I-beam are studied.The forward problem comprises the construction of crack model exclusively for crack section and the construction of a numerically I-beam model to gain crack detection database.The inverse problem consists of the measurement of modal parameters and the detection of crack parameters.Two experiments including measurement of rotational spring stiffness and prediction of cracks in I-beam are conducted.Experimental results based on the current methods indicate that relative error of crack location is less than 3%,while the error of crack depth identification is less than 6%.Crack identification of I-beam is expected to contribute to the development of automated crack detection techniques for railway lines and building skeletons.     

A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect

A single-walled nanotube structure embedded in an elastic matrix is simulated by the nonlocal Euler-Bernoulli, Timoshenko, and higher order beams. The beams are assumed to be elastically supported and attached to continuous lateral and rotational springs to take into account the effects of the surrounding matrix. The discrete equations of motion associated with free transverse vibration of each model are established in the context of the nonlocal continuum mechanics of Eringen using Hamilton's principle and an efficient meshless method. The effects of slenderness ratio of the nanotube, small scale effect parameter, initial axial force and the stiffness of the surrounding matrix on the natural frequencies of various beam models are investigated for different boundary conditions. The capabilities of the proposed nonlocal beam models in capturing the natural frequencies of the nanotube are also addressed.     

The effect of root flexibility on the torsional vibration of uniform section blades

Turbomachinery blades are mounted rather loosely in a disc using different root fixing devices. This results in the root offering stiffnesses against bending as well as torsional motions. The effect of root flexibility on the bending vibrations has been studied considerably in the past; the present paper deals with its effect on the torsional vibrations. Solution of the governing differential equation with appropriate boundary conditions is obtained. The proper non-dimensional root flexibility and frequency parameters are derived. The effect on the first three torsional natural frequencies and the corresponding mode shapes is studied for the complete spectrum of root flexibility ranging from a clamped to a free root. An empirical formula is suggested for the value of the correction factor by which the fixed root frequency may be multiplied to obtain the flexible root frequency.     

Validation of the soft tissue restraints in a force-controlled knee simulator

In vitro testing of total knee replacements (TKRs) is important both at the design stage and after the production of the final components. It can predict long-term in vivo wear of TKRs. The two philosophies for knee testing are to drive the motion by displacement or to drive the motion by force. Both methods have advantages and disadvantages. For force control an accurate simulation of soft tissue restraints is required. This study was devised to assess the accuracy of the soft tissue restraints of the force-controlled Stanmore knee simulator in simulating the restraining forces of the anterior cruciate ligament (ACL) and posterior cruciate ligament (PCL). In order to do this, human cadaver knee joints were subjected to the ISO Standard Walking Cycle. The resulting kinematics were monitored when the soft tissue structures were intact, when the ACL and PCL were resected, and when they were simulated by springs positioned anteriorly and posteriorly. The stiffness of the springs was determined from the literature. Two different stiffnesses of springs were used which were 7.24 N/mm (designated as soft springs) and 33.8 N/mm (designated as hard springs). All the intact knees showed displacements that were within the range of the machine. Cutting the ACL and PCL resulted in anterior and posterior motion and internal external rotation that were significantly greater than the intact knee. Results showed that when the ACL and PCL were cut hard springs positioned anterior and posterior to the knee returned the knee to near normal anterior-posterior (AP) motion. Using hard springs in the posterior position in any condition reduced rotational displacements. Therefore using springs in a force-controlled simulator is a compromise. More accuracy may be obtained using springs that are of intermediate stiffness.     

多交叉曲梁簧片柔性铰链的力学建模与性能分析

针对直梁型多交叉簧片柔性铰链难以兼顾大转角、低刚度的问题,设计了一种多交叉曲梁簧片柔性铰链。采用圆弧曲梁簧片减小变形应力和转动刚度,在纯转矩驱动条件下具有高转动精度和大转角行程。推导了圆弧曲梁簧片的变形应力计算方程,建立了柔性铰链的大变形力学分析模型,并通过转动刚度实验和变形应力仿真计算验证了理论模型的准确性。进一步分析了半径系数、簧片曲率和位形角等设计变量对柔性铰链驱动转矩和变形应力的影响关系,提出了同时减小转动刚度和变形应力的设计方案,确定了使柔性铰链获得零刚度特性和负刚度特性的设计变量组合。所设计的柔性铰链结构及其性能分析结果可为新型大行程柔性铰链的设计提供参考。     

Exact dynamic stiffness matrix of a Timoshenko three-beam system

An exact dynamic stiffness matrix is established for an elastically connected three-beam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the free vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Numerical results for the natural frequencies and mode shapes of the illustrative examples are discussed for 10 interesting boundary conditions and three different stiffness constants of springs.     

压电位移放大机构的力学解析模型及有限元分析

研究了压电位移放大机构的运动学和动力学建模问题。基于能量守恒原理和弹性梁弯曲理论推导了桥式位移放大机构的位移放大比等静力学解析模型;在此基础上,通过拉格朗日方程建立了桥式位移放大机构的固有频率解析模型。通过有限元计算验证和分析了提出的解析模型的可行性和优越性,并与国内外典型的位移放大比数学模型进行了比较。结果表明:由于本文提出的模型考虑了位移放大机构的拉伸和弯曲变形,并且摒弃了国内外普遍采用近似几何关系进行数学推导的思路,因此所建立的位移放大比解析模型精度更高;固有频率解析计算结果与有限元模态分析结果的相对误差约为5%。得到的结果显示:本文给出的建模方法以及位移放大比、固有频率等解析模型可为柔性机构的优化设计和研制提供依据和参考。     

Parametric instability of a Leipholz beam due to distributed frictional axial load

In this paper, the parametric instability of a beam subjected to a moving distributed contact load that also oscillates in the transverse direction of the beam is studied. Due to the presence of friction between the beam and the moving contact surface, the beam is excited by a pulsating, distributed, follower-type axial load, which results in the problem of a Leipholz column subjected to parametric excitation. The study of this prototypical one-dimensional model is intended to provide further understanding of the instability of systems that have similar loading conditions such as automotive disc brake pads rubbed by the oscillatory rotor or beam-like structures. Effects of several parameters such as boundary conditions, coefficient of friction, intermediate supports, and waviness of the contact load on the instability are examined. Results show that the combination resonance of both the sum and difference types exists for the boundary conditions considered in this study. It is also found that the shearing motion of the friction material plays an important role in determining the stability of the Leipholz beam model. Analogous behavior of instability between the present model and the disc brake pad system is also discussed.     

An anti-backlash two-part shaft coupling with interlocking elastically averaged teeth

A new family of compliant and self-locking tapered torsional couplings have been developed as an alternate to spline-type couplings. The couplings use designed compliance to ensure constant contact between mating beam-like teeth. The tip of one tooth is larger, but tapered, so that when it mates at the base of the opposed tooth, it deflects radially. The taper is self-locking to prevent radial deflection under torsional load. This eliminates backlash between parts while maintaining a relatively high torsional stiffness. The large number of mating teeth elastically average errors in the teeth.