一种滚珠丝杠扭转振动模态的测量与分析方法

针对滚珠丝杠副驱动机构的扭转振动模态难以通过测量多点激励-响应的方式进行测量的问题,提出了一种新的测量与分析方法：在丝杠尾端安装高分辨率旋转编码器,利用高速计数卡分别测量该编码器及伺服电机码盘的角加速度信号,并进行频域分析;根据所建立的滚珠丝杠副驱动机构有限元-集中参数模型,通过数值计算并结合实验测量结果获得其扭转振动固有频率和振型。实例研究表明,所提出的测量和分析方法可精确获得驱动机构扭转振动模态的前两阶固有频率和振型。     

Free vibration analysis of isotropic and orthotropic triangular plates

A highly accurate and computationally efficient numerical method is developed for the flexural vibration of isotropic and orthotropic triangular plates. A set of two-dimensional orthogonal plate functions is used as an admissible displacement function in the Rayleigh-Ritz method to obtain the natural frequencies and mode shapes for the plates. From a prescribed starting function satisfying the boundary conditions, the higher terms in the orthogonal plate functions are constructed using the Gram-Schmidt orthogonalization process. Natural frequencies and mode shapes are obtained for three triangular plates with different support conditions. The obtained numerical results are presented, and the isotropic case is verified with other numerical methods in the literature.     

Non-linear free vibrations of Kelvin–Voigt visco-elastic beams

A full visco-elastic non-linear beam with cubic non-linearities is considered, and the governing equations of motion of the system for large amplitude vibrations are derived. By using the method of multiple scales, the non-linear mode shapes and natural frequencies of the beam are then analytically formulated. The resulting formulations for amplitude, non-linear natural frequencies and mode shapes can be used for any type of boundary conditions. Next, method of Galerkin is used to separate the time and space variables. The equations of motion show the presence of a non-linear damping term in addition to the ones with non-linear inertia and geometry. As it is known, the presence of non-linear inertia and the geometric terms make the non-linear natural frequencies to be dependent on constant amplitude of vibration. But, when damping non-linearities are present, it is seen that the amplitude is exponentially time-dependent, and so, the non-linear natural frequencies will be logarithmically time-dependent. Additionally, it is shown that the mode shapes will be dependent on the third power of time-dependent amplitude. The analytical results are applied to hinged–hinged and hinged–clamped boundary conditions and the results are compared with numerical simulations. The results match very closely for both cases specially for the case of hinged–hinged beam.     

Scaling-factor estimation using an optimized mass-change strategy

In operational modal analysis, only unscaled mode shapes can be obtained. The mass-change method is in many cases the simplest way to estimate the scaling factors, which involves repeated modal testing after changing the mass at different points of the structure where the mode shapes are known. With this method, the scaling factors are determined using the natural frequencies and mode shapes of both the modified and the unmodified structure. However, the uncertainty on the scaling-factor estimation depends on the modal-analysis accuracy and the mass-change strategy (number, magnitude, and location of the masses) used to modify the dynamic behavior of the structure. In this paper, a procedure to optimize the mass-change strategy is proposed, which uses the modal parameters (natural frequencies and mode shapes) of the original structure as the basic information.     

Erratum to “Scaling-factor estimation using an optimized mass-change strategy” [Mech. Syst. Signal Process. 24(5) (2010) 1260–1273]

In operational modal analysis, only unscaled mode shapes can be obtained. The mass-change method is in many cases the simplest way to estimate the scaling factors, which involves repeated modal testing after changing the mass at different points of the structure where the mode shapes are known. With this method, the scaling factors are determined using the natural frequencies and mode shapes of both the modified and the unmodified structure. However, the uncertainty on the scaling-factor estimation depends on the modal-analysis accuracy and the mass-change strategy (number, magnitude, and location of the masses) used to modify the dynamic behavior of the structure. In this paper, a procedure to optimize the mass-change strategy is proposed, which uses the modal parameters (natural frequencies and mode shapes) of the original structure as the basic information.     

Effect of transmission error on the dynamic behaviour of gearbox housing

The dynamic response of gearbox remains a paramount concern because of noise generation. This work is concerned with numerical simulation of the overall dynamic behaviour of a parallel helical gear transmission. A dynamic sub-structuring method using different types of substructure (carrying and slave) is made to determine the natural frequencies and the corresponding mode shapes. The structure to be investigated is subdivided into components or sub-structures, which are then analyzed independently for natural frequencies and mode shapes. A numerical model taking into account the elastic coupling between the various components of a gearbox was developed. It allows studying and analyzing the dynamic behaviour of elastic housing in the presence of gear process. The static transmission error is introduced as a vibratory excitation source and it is represented by time-varying mesh stiffness. The discretization of the housing deformation energy and the kinetic energy expressions using plate finite elements leads to constructions of the stiffness and the mass matrixes. In dynamic analyses, time-discretization based on the Newmark method is used. The different equations governing movement of gearbox are established in a truncated modal base deduced from the average characteristics of the structure. A gearbox example is presented, and analyzed. A presentation and discussion of the numerical results was emphasized. The numerical results allow us to conclude on the dominant phenomena of the overall dynamic behaviour of the gear transmission.     

Exact analysis of resonance frequency and mode shapes of isotropic and laminated composite cylindrical shells; Part I: analytical studies

In order to study the free vibration of simply supported circular cylindrical shells, an exact analytical procedure is developed and discussed in detail. Part I presents a general approach for exact analysis of natural frequencies and mode shapes of circular cylindrical shells. The validity of the exact technique is verified using four different shell theories 1) Soedel, 2) Flugge, 3) Morley-Koiter and 4) Donnell. The exact procedure is compared favorably with experimental results and those obtained using a numerical finite element method. A literature review reveals that beam functions are used extensively as an approximation for simply supported boundary conditions. The accuracy of the resonance frequencies obtained using the approximate method are also investigated by comparing results with those of the exact analysis. Part II presents effects of different parameters on mode shapes and natural frequencies of circular cylindrical shells.     

An assumed mode method and finite element method investigation of the coupled vibration in a flexible-disk rotor system with lacing wires

The Assumed mode method (AMM) and Finite element method (FEM) were used. Their results were compared to investigate the coupled shaft-torsion, disk-transverse, and blade-bending vibrations in a flexible-disk rotor system. The blades were grouped with a spring. The flexible-disk rotor system was divided into three modes of coupled vibrations: Shaft-disk-blade, disk-blade, and blade-blade. Two new modes of coupled vibrations were introduced, namely, lacing wires-blade and lacing wires-disk-blade. The patterns of change of the natural frequencies and mode shapes of the system were discussed. The results showed the following: first, mode shapes and natural frequencies varied, and the results of the AMM and FEM differed; second, numerical calculation results showed three influencing factors on natural frequencies, namely, the lacing wire constant, the lacing wire location, and the flexible disk; lastly, the flexible disk could affect the stability of the system as reflected in the effect of the rotational speed.     

Four-beam model for vibration analysis of a cantilever beam with an embedded horizontal crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton's principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.     

基于瑞利-里兹法的低翅片管固有频率数值解

针对低翅片管几何形状复杂,质量和刚度分布不均匀,振动特征值不能有封闭解的问题,采用瑞利-里兹法对低翅片管的固有频率进行了数值研究,获得了低翅片管弯曲振动固有频率的数值计算模型。对于一定规格的低翅片管,应用该数值计算模型可求得其各阶固有频率的数值解。采用共振法测试了低翅片管的固有频率,前三阶固有频率的数值解与实测值相比,误差范围为-3.28%--3.51%。该数值计算模型可用于低翅片管固有频率的求解。     

Vibration analysis of cantilever plates undergoing translationally accelerated motion

This paper presents a modeling method for the vibration analysis of translationally accelerated cantilever plates. An accurate dynamic modeling method, which was introduced in the previous study, is employed to obtain the equations of motion for the vibration analysis. Dimensionless parameters are identified to generalize the conclusions from numerical results. The effects of the dimensionless parameters on the natural frequencies and mode shapes are investigated. Particularly, the magnitude of critical acceleration which causes the dynamic buckling of the structure is calculated. Incidentally, the natural frequency loci veering phenomena are observed and discussed.     

Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures

A spline finite strip capability is described for predicting the buckling stresses and natural frequencies of vibration of prismatic plate structures which may be of composite laminated construction with arbitrary lay-ups. The plate structures may have general boundary conditions. The capability embraces analyses based on the use of first-order shear deformation plate theory and of classical plate theory, and utilizes substructuring procedures which include the use of superstrips. The theoretical development is not detailed since the present paper reports a very direct extension of a theoretical study developed for the analysis of single plates in an earlier paper in this Journal. A considerable range of buckling and vibration applications is documented and comparison of spline finite strip numerical values of buckling stresses and frequencies is made with results generated using the semi-analytical finite strip method and, in some cases, the finite element method. Buckled and vibrational mode shapes are presented for some applications.     

Free vibration analysis of multi-directional functionally graded circular and annular plates

This paper addresses the free vibration of multi-directional functionally graded circular and annular plates using a semianalytical/ numerical method, called state space-based differential quadrature method. Three-dimensional elasticity equations are derived for multi-directional functionally graded plates and a solution is given by the semi-analytical/numerical method. This method gives an analytical solution along the thickness direction, using a state space method and a numerical solution using differential quadrature method. Some numerical examples are presented to show the accuracy and convergence of the method. The most of simulations of the present study have been validated by the existing literature. The non-dimensional frequencies and corresponding displacements mode shapes are obtained. Then the influences of thickness ratio and graded indexes are demonstrated on the non-dimensional natural frequencies.     

On the equivalent mass-spring parameters and assumed mode of a cantilevered beam with a tip mass

In the Rayleigh-Ritz approach to the mathematical model of a cantilevered beam with a tip mass, the proper selection of basis functions is critical in representing the original system by an equivalent mass-spring system. Although the fundamental bending mode shape of a beam varies for a different tip mass magnitude, the numerical values of 33/140 and 3 have been conventionally employed as those of the normalized dimensionless equivalent mass and spring constants, respectively, which correspondingly yield errors in its calculated natural frequencies. This work firstly proposes a method to evaluate more accurate values of the equivalent mass and spring for a wide range of the tip mass-to-beam mass ratio by direct use of a fundamental mode, and then proposes a new basis function as a linear combination of two polynomials, which represent static deflection shapes of a beam under a tip force and a uniformly distributed force, respectively, yielding natural frequencies fairly close to those by the continuous beam equation.     

Dynamic and acoustic characteristics of bell type structure using finite element method

Dynamic and acoustic characteristics of the bell type structure were analyzed numerically. The finite element method with 3-D general shell element was used to identify the natural frequencies and mode shapes of the structure. Mode shapes and stress distributions of a transient dynamic analysis were effectively displayed by using computer graphic technique. The results of this numerical study were in good agreement with those obtained from the experimental test of fast Fourier transform analyer. Vibrational modes, which effect the acoustic characteristics of the typical bell-type structure were found to be the first flexural mode (4-0 mode) and the second flexural mode (6-0 mode). Asymmetric effects by Dangjwas and acoustic holes gave rise to beat frequencies, and the Dangjwa was found to be most effective. When the impact load was applied to the bell, the stress concentration occured at the rim part of the bell. It was found that the bell type structure should be designed thickly at the rim part in order to prevent failure from impact loads.     

基于键合图的模态分析与应用研究

传统的模态分析方法采用牛顿定律建立运动方程,分析质量矩阵[M]和刚度矩阵[K],求解系统的固有频率和振型的过程十分复杂、繁琐。将键合图理论与方法运用于系统的模态分析,通过建立系统的状态空间方程,可以得到系统的模态。该方法能简便、正确计算系统的模态,获得完整的数据。文中提供的算例进一步证明了该方法的正确性和有效性。     

A two-stage method to identify structural damage sites and extents by using evidence theory and micro-search genetic algorithm

A two-stage method of determining the location and extent of multiple structural damages by using information fusion technique and genetic algorithm is presented in this paper. First the damage detection strategy is to localize the damage sites by using an evidence theory, which can perfectly integrate the damage identification information coming from both natural frequencies and mode shapes. Then, a micro-search genetic algorithm (MSGA) is proposed to determine the damage extent. A cantilever beam is analyzed as a numerical example to compare the performance of the proposed method with the multiple damage location assurance criterions (MDLAC) and the simple genetic algorithms. Simulation results show that identification results of the evidence theory are better than those both of the frequency MDLAC method and the mode shape MDLAC method, and the MSGA is also more accurate and effective than the simple genetic algorithms. Therefore, the two-stage method is very effective for the identification of multiple structural damages.     

改进的特征值灵敏度在结构损伤识别中的应用

传统的特征值灵敏度分析中忽略了振型变化的影响,导致特征值灵敏度难以准确地识别出结构损伤。为提高特征值灵敏度的准确性,利用未损伤结构的振型线性表示损伤情况下结构振型的变化,进而改进了特征值灵敏度的理论分析方法。在不同的损伤工况下,计算数值模型改进前后的特征值灵敏度,验证表明改进后的特征值灵敏度具有更高的精度。以孔洞直径和位置均未知的实际结构为研究对象,利用改进后灵敏度方法准确识别了结构损伤的位置和程度,与结构的孔洞直径和位置实测值相吻合,表明了改进后灵敏度方法具有准确实用性,从而弥补了传统特征值灵敏度法在结构损伤识别中的缺陷。     

Three methods for studying coupled vibration in a multi flexible disk rotor system

This paper improved and developed Shaft-disk-blade (SDB) rotor system based on the previous studies of the authors in the last decade from Yang and Huang (2005) to Chiu et al. (2017). This paper also explored blade-bending, disk-transverse, and shaft-torsion coupling vibration of a multi flexible disk rotor system. Unlike the previous studies of the authors, this paper adopted three methods: (a) Assumed mode method (AMM), (b) Finite element method (FEM), and (c) experimental method. The first approach is the main method, and the two other methods are complementary. Results generated from the three methods were then compared and analyzed. Based on the previous definition of the authors, a flexible disk rotor system displays three types of coupling vibrations: Inter-blade, SDB, and diskblade modes. The system changes the rules of natural frequencies and mode shapes. This paper presents several interesting results. First, the author determined the change rules of the mode shapes and natural frequencies using the AMM, FEM (including three kinds of software), and the experimental method. Second, numerical calculation results also revealed that two phenomena regarding the distance of disk and flexible disk would be affected by the natural frequencies. Third, the experimental results would be explored in this paper. Last, the flexible disk could affect the system instability in the case study of rotation effects.     

大型分体式磁轴承电动机系统定子模态分析

准确分析定子固有模态对低振动噪声电机系统优化设计和控制有十分重要的意义。针对大型分体式磁轴承电动机系统,采用有限元仿真方法分析轴向磁轴承定子绕组、径向磁轴承定子叠片结构、机座螺栓连接以及底座附加质量对大型分体式磁轴承电动机系统定子模态的影响。结果表明:轴向轴承定子绕组对定子固有频率影响较大,宜采用质量计入铁心的方式处理;径向轴承定子叠片结构对定子固有频率的影响较小,可按各向同性处理;机座螺栓连接刚度对机座固有频率有一定影响,但当连接刚度远大于机座刚度时,影响较小;底座附加质量对整机模态频率有一定的影响,但对不同振型的固有频率的影响效果相差较大。计算了整机的模态频率,并与振动试验结果进行了对比验证。