Analytical models of mesh stiffness for cracked spur gears considering gear body deflection and dynamic simulation

Meccanica - Accurate mesh stiffness models for cracked spur gears are critical to obtain gearbox failure characteristics. However, the influence of crack on gear body deflection is often ignored by...     

Dynamic features of planetary gear set with tooth plastic inclination deformation due to tooth root crack

Gear tooth root crack, as one of the popular gear tooth failures, is always caused by the dynamic load or excessive load applied to the tooth. It will devastate the working performance of the gear system, by problems such as vibration and noise, or even lead to a broken tooth, which will stop the normal working process of the gear system. It has attracted wide attention from researchers. However, the previous studies focused their concentration only on the mesh stiffness reduction due to tooth root crack, while the tooth plastic inclination due to tooth bending damages like gear tooth root crack is seldom considered. In this paper, a tooth plastic inclination model for spur gear with tooth root crack is developed by regarding the cracked tooth as a cantilever beam. It influences not only the displacement excitation but also the mesh stiffness and load-sharing factor among tooth pairs in mesh. The simulation results obtained by incorporating the tooth plastic inclination deformation model together with the tooth root crack model into a 21-Degree-of-Freedom planetary gear dynamic model indicate that the tooth plastic inclination has a significant effect on the performance of the gear system rather than the mesh stiffness reduction due to tooth root crack.     

Local stresses and strains in axially cracked cylindrical shells

A perturbation solution is obtained for the local stress-strain fields in an axially cracked cylindrical shell. The tenth-order differential equations are used that take into account the transverse shear deformation. The perturbation of a curvature parameter, λ, is employed, where . The stress intensity factors for finite size cylindrical shells subjected to bending and internal pressure are evaluated. Sufficient accuracy can be obtained without using fine mesh sizes in regions near the crack tip. Also analyzed are the influence of cylinder diameter and shearing stiffness on bulging.     

Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.     

裂纹转子的振动响应研究

本文研究了裂纹转子的振动响应。文章首先通过应力强度因子积分得到含裂纹轴单元的刚度矩阵；建立了裂纹转子的运动微分方程。进而研究了裂纹转子的振动响应，得出了裂纹转子的振动响应随裂纹位置和深度的变化关系。为工程上早期诊断微小裂纹提供了理论根据。     

Stiffness reduction of cracked general symmetric laminates using a variational approach

In this paper, stiffness reduction of general symmetric laminates containing a uniform distribution of matrix cracks in a single orientation is analyzed. An admissible stress field is considered, which satisfies equilibrium and all the boundary and continuity conditions. This stress field has been used in conjunction with the principle of minimum complementary energy to get the effective stiffness matrix of a cracked general symmetric laminate. Natural boundary conditions have been derived from the variational principle to overcome the limitations of the existing variational methods on the analysis of general symmetric laminates. Therefore, the capability of analyzing cracked symmetric laminates using the variational approach has been enhanced significantly. It has been shown that the method provides a rigorous lower bound for the stiffness matrix of a cracked laminate, which is very important for practical applications. Results derived from the developed method for the properties of the cracked laminates showed an excellent agreement with experimental data and with those obtained from McCartney’s stress transfer model. The differences of the developed model with McCartney’s model are discussed in detail. It can be emphasized that the current approach is simpler than McCartney’s model, which needs an averaging procedure to obtain the governing equations. Moreover, it has been shown that the existing variational models are special cases of the current formulation.     

A review of space tether in new applications

Gear eccentricities are one of the practical types of the manufacturing errors that affect the dynamic performance of a planetary gear train (PGT). Previous research about the effects of the gear eccentricities is abundant, and many of them focus on the parallel shaft gear set. However, almost none of them have considered the influence of the gear eccentricities on the mesh stiffness. In fact, the existence of the gear eccentricities can change the center distance and the mesh positions of a meshing gear pair, which will directly affect the mesh stiffness. Situation can be even more complex for the PGT with either sun gear eccentricities or planet gear eccentricities or both of them. Based on that, a new dynamic model of a PGT with gear eccentricities is established. The planar motions of the PGT and the mesh stiffness are integrated and solved simultaneously where the mesh stiffness is determined by the actual mesh positions of the meshing gear pair. The mesh stiffness is calculated by the energy potential method. The time-varying center distance caused by the gear eccentricities is also considered, which can result in the change of line of action, pressure angle, contact ratio and mesh positions. The influence of gear eccentricities on the dynamic performance of a 4-planet PGT is studied. Some useful results are derived at last.     

A three-dimensional edge-crack finite element for fracture mechanics applications

A three-dimensional extension of a previously published two-dimensional cracked finite element [Potirniche, G.P., Hearndon, J., Daniewicz, S.R., Parker, D., Cuevas, P., Wang, P.T., Horstemeyer, M.F., 2008. A two-dimensional damaged finite element for fracture applications. Engineering Fracture Mechanics 17(13), 3895–3908] is presented in this paper. The new element has an embedded edge crack, and was developed to model damage in three-dimensional structures using the finite element method. The element simulates the presence of a crack without physically inserting it in the three-dimensional finite element mesh. The method involves the derivation of a modified stiffness matrix that accounts for the change in the element flexibility due to the crack presence. The cracked element was analytically formulated and implemented in the finite element code ABAQUS Standard as a User-defined Element (UEL) subroutine. Tests of various cracked beam configurations were used to estimate the accuracy of the element by comparing two models: one with a UEL and another with an embedded edge crack. Beam deflections and natural frequencies were analyzed and compared for the two models. The results indicate that the new element has a good potential in modeling cracks in three-dimensional parts. Moreover, the method using this UEL computes the global response of damaged structures, in which cracks can be placed at various locations and in an unlimited number.     

基于均匀荷载面曲率的简支裂纹梁损伤识别

为了采用模态参数对结构裂纹进行定位与定量,基于集中柔度模型,采用无质量的扭转弹簧模拟裂纹,建立简支裂纹梁的振动微分方程。针对现有柔度曲率指标仅能判断裂纹的大致范围,基于线性插值理论,建立裂纹位置与相邻测点均匀荷载面曲率差的关系,提出裂纹进一步定位公式,实现裂纹位置的精确定位。针对现有大多数损伤识别方法无法实现裂纹的损伤定量,基于位移曲率与结构刚度和弯矩的关系,理论推导了均匀荷载面曲率的结构刚度损伤程度识别方法,基于弹簧串联原理和线刚度思想,首次提出串联等效线刚度模型,建立裂纹深度与均匀荷载面曲率的关系,实现裂纹深度的定量。通过简支裂纹梁数值算例,考虑多裂纹的损伤情况,验证了新方法对裂纹定位与定量的有效性。     

Monitoring structural damage of components using an effective modulus approach

This paper describes a novel nondestructive damage detection method that was developed to study the influence of a crack on the dynamic properties of a cantilever beam subjected to bending. Experimental measurements of transfer functions for the cracked cantilever beam revealed a change in the natural frequency with increasing crack length. A finite element model of a cracked element was created to compute the influence of severity and location of damage on the structural stiffness. The proposed model is based on the response of the cracked beam element under a static load. The change in beam deflection as a result of the crack is used to calculate the reduction in the global component stiffness. The reduction of the beam stiffness is then used to determine its dynamic response employing a modal analysis computational model. Euler–Bernoulli and Timoshenko beam theories are used to quantify the elastic stiffness matrix of a finite element. The transfer functions from both theories compare well with the experimental results. The experimental and computational natural frequencies decreased with increasing crack length. Furthermore the Euler–Bernoulli and Timoshenko beam theories resulted in approximately the same decrease in the natural frequency with increasing crack length as experimentally measured.     

Detection of crack location using cracked beam element method for structural analysis

The local flexibility introduced by cracks changes the dynamic behavior of the structure and, by examining this change, crack position and magnitude can be identified. In order to model the structure for FEM analysis, a special finite element for a cracked Timoshenko beam is developed. Shape functions for rotational and translational displacements are used to obtain the consistent mass matrix for the cracked beam element. Effect of the crack on the stiffness matrix and consistent mass matrix is investigated. Proposed is a procedure for identifying cracks in structures using modal test data.     

考虑界面滑移效应的组合裂纹梁弯曲解析解

将上部子梁的裂纹等效为线性扭转弹簧,考虑组合梁连接面的滑移位移,建立了以组合裂纹梁挠度和滑移位移为基本未知量的组合裂纹梁弯曲变形一维数学模型．利用Laplace变换及其逆变换,给出了组合裂纹梁弯曲变形一维数学模型的解析通解．在此基础上,研究了均布载荷作用下简支组合裂纹梁的弯曲变形问题,数值分析了连接面剪切刚度、裂纹深度、数目和位置等参数对组合裂纹梁弯曲变形的影响,结果表明：在裂纹处,组合裂纹梁挠度曲线存在尖点,而横截面转角曲线存在跳跃,且随着裂纹数目和深度的增加,挠度和横截面转角跳跃值增大;随着连接面剪切刚度的增加,挠度和横截面转角减小,并最终趋于定值．并且,随着组合梁跨高比的增加,连接面剪切刚度对梁挠度影响逐渐减弱．     

CONSTITUTIVE RELATION AND STIFFNESS DEGRADATION OF CRACKED ANISOTROPIC COMPOSITE LAMINATES(II)─DETERMINATION OF RESOLVED STIFFNESS AND INVESTIGATION OF STIFFNESS DEGRADATION FOR CRACKED LAMINATES

I.IntroductionTheconstitutiverelationsofacracked(omm.),laminatearedevelopedin(l)ofthispaper.Forthecrackedanisotropiclaminateswithsymmetricconstrainedcracking,themethodofstiffnesspartionisusedtosolvethecoupled'problemandtheeffectivestiffnessesofthelaminatesareexpressedasfunctionsofcrackdensity.Duringthestiffnesspartition,thein-planestiffnessmatrixesandtheout-of-planestiffnessmatrixesforthecrackedplyandtheconstraintplyshouldberesolvedintotwoparts.Fortheconstraintply(2),allcouplingstiffnesseleme…     

Stability analysis of the whirl motion of a breathing cracked rotor with asymmetric rotational damping

The stability of the whirl motion of a breathing cracked rotor with the distinction of stationary damping and the asymmetric rotational damping is studied. By Lagrange’s principal, the motion equations are formed in rotational frame such that the multi-asymmetric system, i.e., asymmetric rotational damping and asymmetric time-periodic varying stiffness, is simplified to be a system with anisotropic damping and anisotropic time-periodical varying stiffness in rotational operation. Based on the multiple scales solution of the simplified whirling equation in moving frame, root locus method for stability analysis is proposed. Different from the former stability estimation method, the corresponding Campbell diagram, decay rate plot, and root locus plot of the fifth-order approach are derived to prove the effects of both crack depth and damping effects. The numerical results of the instabilizing effects of the crack depth are well agreeing with the previous studies. In addition, the destabilizing influence of the rotational damping on the breathing cracked rotor is presented for the first time.     

CONSTITUTIVE RELATION AND STIFFNESS DEGRADATION OF CRACKED ANISOTROPIC COMPOSITE LAMINATES(II)─DETERMINATION OF RESOLVED STIFFNESS AND INVESTIGATION OF STIFFNESS DEGRADATION

The study on properly degradation of damaged,.onlj,osile laminates is extendedto anisotropic laminates with matrix cracking. In (II) of the paper, a solution forpartitioned stiffness is given to complete the constitutive relations developed in (I) Thestiffness degradation in (θ_m/90_n)_s cracked laminates is calculated and the results arediscussed.     

Structural analysis of gradient elastic components

The present study investigates the size effects in the problems of cantilever beam bending and cracked bar tension within the gradient elasticity framework. Analytical solutions for metrics that characterize both the normalized stiffness and toughness are derived. It is found that the gradient elastic beam exhibits a significantly stiffer but also more brittle response, while the gradient cracked bar exhibits considerable toughening. These results compare well with respective finite element computations.     

Formulation and implementation of a singular anisotropic finite element

The formulation and implementation of a singular finite element for analyzing homogeneous anistropic materials is presented in this paper. Lekhnitskii's stress function method is used to formulate the boundary value problem with the stress function expressed as a Laurent series. The development of the element stiffness matrix and the method of integrating the element to conventional displacement based finite element programs is shown. The stiffness matrix generation is based on a least squates collocation technique to satisfy displacement continuity boundary conditions at the element interface. Implementation of the element is demonstrated for cracked anisotropic materials subjected to inplane loading. Center cracked, on and off-axis coupons under tensile loading are analyzed using the element. It is shown that the stress distributions and intensity factors compare well with those obtained using other methods.     

A NEW APPROACH TO FREE VIBRATION ANALYSIS OF A BEAM WITH A BREATHING CRACK BASED ON MECHANICAL ENERGY BALANCE METHOD

In this paper, a new approach to free vibration analysis of a cracked cantilever beam is proposed. By considering the effect of opening and closing the crack during the beam vibration, it is modeled as a fatigue crack. Also, local stiffness changes at the crack location are considered to be a nonlinear amplitude-dependent function and it is assumed that during one half a cycle, the frequencies and mode shapes of the beam vary continuously with time. In addition, by using the experimental tests, it is shown that the local stiffness at the crack location varies continuously between the two extreme values corresponding to the fully closed and the fully open cases of the crack. Then, by using the mechanical energy balance the dynamic response of the cracked beam is obtained at every time instant. The results show that for a specific crack depth, by approaching the crack location to the fixed end of the beam, more reduction in the fundamental frequency occurs. Furthermore, for a specific crack location, the fundamental frequency diminishes and the nonlinearity of the system increases by increasing the crack depth. In order to validate the results, the variations of the fundamental frequency ratio against the crack location are compared with experimental results.     

基于模板的子结构多分辨率拓扑优化

多分辨率拓扑优化(multi-resolution topology optimization, MTOP)方法将有限元网格和密度网格解耦, 采用较粗的网格(超单元)进行有限元分析, 从而大大降低了拓扑优化过程中的结构分析成本. 但MTOP方法每次迭代都需要根据超单元内的平均密度计算有限元单刚, 不仅精度不够且在过滤半径较小的情况容易出现棋盘格现象和QR模式. 为解决相应问题, 本文将超单元视为子结构, 通过静态凝聚得到超单元刚度阵, 并进一步根据拓扑优化过程中子结构的密度分布特征组建了其模板库, 从而省去了超单元单刚的重复计算, 显著提高了MTOP方法的分析精度, 有效抑制了数值不稳定现象.      

Effects of a crack on the stability of a non-linear rotor system

The stability of a rotor system presenting a transverse breathing crack is studied by considering the effects of crack depth, crack location and the shaft's rotational speed. The harmonic balance method, in combination with a path-following continuation procedure, is used to calculate the periodic response of a non-linear model of a cracked rotor system. The stability of the rotor's periodic movements is studied in the frequency domain by introducing the effects of a perturbation on the periodic solution for the cracked rotor system.It is shown that the areas of instability increase considerably when the crack deepens, and that the crack's position and depth are the main factors affecting not only the non-linear behaviour of the rotor system but also the different zones of dynamic instability in the periodic solution for the cracked rotor. The effects of some other system parameters (including the disk position and the stiffness of the supports) on the dynamic stability of the non-linear periodic response of the cracked rotor system are also investigated.