Towards linear modal analysis for an L-shaped beam: Equations of motion

We consider an L-shaped beam structure and derive all the equations of motion considering also the rotary inertia terms. We show that the equations are decoupled in two motions, namely the in-plane bending and out-of-plane bending with torsion. In neglecting the rotary inertia terms the torsional equation for the secondary beam is fully decoupled from the other equations for out-of-plane motion. A numerical modal analysis was undertaken for two models of the L-shaped beam, considering two different orientations of the secondary beam, and it was shown that the mode shapes can be grouped into these two motions: in-plane bending and out-of-plane motion. We compared the theoretical natural frequencies of the secondary beam in torsion with finite element results which showed some disagreement, and also it was shown that the torsional mode shapes of the secondary beam are coupled with the other out-of-plane motions. These findings confirm that it is necessary to take rotary inertia terms into account for out-of-plane bending. This work is essential in order to perform accurate linear modal analysis on the L-shaped beam structure.     

Analysis of Coupled Vibration Characteristics of Wind Turbine Blade Based on Green's Functions

This paper presents the analysis of dynamic characteristics of horizontal axis wind turbine blade, where the mode coupling among axial extension, flap vibration(out-of-plane bending), lead/lag vibration(in-plane bending) and torsion is emphasized. By using the Bernoulli-Euler beam to describe the slender blade which is mounted on rigid hub and subjected to unsteady aerodynamic force, the governing equation and characteristic equation of the coupled vibration of the blade are obtained. Due to the combined influences of mode coupling, centrifugal effect, and the non-uniform distribution of mass and stiffness, the explicit solution of characteristic equation is impossible to obtain. An equivalent transformation based on Green's functions is taken for the characteristic equation, and then a system of integrodifferential equations is derived. The numerical difference methods are adopted to solve the integrodifferential equations to get natural frequencies and mode shapes. The influences of mode coupling, centrifugal effect, and rotational speed on natural frequencies and mode shapes are analyzed. Results show that:(1) the influence of bending-torsion coupling on natural frequency is tiny;(2) rotation has dramatic influence on bending frequency but little influence on torsion frequency;(3) the influence of bending-bending coupling on dynamic characteristics is notable at high rotational speed;(4) the effect of rotational speed on bending mode is tiny.     

Analytical and experimental modal analysis of nonuniformly ring-stiffened cylindrical shells

In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e., rings with nonuniform stiffener eccentricity and unequal stiffener spacing, is investigated using analytical and experimental methods. The Ritz method is applied in analytical solution, while stiffeners are treated as discrete elements. The polynomial functions are used for Ritz functions. The effects of nonuniformity of stiffener distribution on natural frequencies are considered for free–free boundary conditions. Results show that, at constant stiffener mass, significant increments in natural frequencies can be achieved using nonuniform stiffener distribution. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes, and damping in each mode. Analytical results are compared with experimental ones, showing good agreement. Because of insufficient experimental modal data for nonuniform stiffener distribution, the results of modal testing obtained in this study could be a useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.     

Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated.     

Mode identification of a rotating disk

A modal testing method permitting identification of the natural frequencies, the number of nodal diameters and wave motions in a rotating disk is presented in this paper. This method is applicable at arbitrary rotation speed without requiring a priori information about the vibration modes of the stationary disk. The influence of disk rotation speed on the prediction of mode shapes with this method is shown, and experimental predictions of modal parameters are presented for both axisymmetric and asymmetric disks.     

An image encryption algorithm based on compound homogeneous hyper-chaotic system

A new dynamic model of a rotating flexible beam with a concentrated mass located in arbitrary position is derived based on the absolute nodal coordinate formulation, and its modal characteristics are investigated in this paper. To consider the concentrated mass at an arbitrary location of the beam, a Dirac’s delta function is used to express the mass per unit length of the beam. Based on the proposed dynamic model, the frequency analysis is performed. The nonlinear equation is transformed into the linear one via employing the linear perturbation analysis method. The stiffness matrix of static equilibrium of the system under the deformed condition is obtained, in which the effect of coupling between the longitudinal deformation and transversal deformation is included. This means even if only the chordwise bending equation is solved, the longitudinal vibration effect can be still considered. As we know, once the longitudinal deformation is large, it will significantly affect the chordwise bending vibration. So the proposed model in this paper is more accurate than the traditional dynamic models which are usually lack of the coupling terms between the longitudinal deformation and transversal deformation. In fact, the traditional dynamic models for the chordwise vibration analysis in the existing literature are usually linear due to neglecting the coupling terms, and consequently, they are only suitable for the modal characteristic analysis of a beam under small deformations. In order to get some general conclusions of the natural frequencies and mode shapes, the equation which governs the chordwise bending vibration of the rotating beam is transformed into a dimensionless form. The dynamic model presented in this paper is nonlinear and can be conveniently used to analyze the modal characteristics of a rotating flexible beam with large deformations. To demonstrate the power of the new dynamic model presented in this paper, the dynamic simulations involving the comparisons between the different frequencies obtained using the model proposed in this paper and the models in the existing literature and the investigating in frequency veering and mode shift phenomena are given. The simulation results show that the angular velocity of the flexible beam will give rise to the phenomena of the natural frequency loci veering and the associated mode shift which is verified in the previous studies. In addition, the phenomena of the natural frequency loci veering rather than crossing can be observed due to the changing of the magnitude of the concentrated mass or of the location of the concentrated mass which are found for the first time. Furthermore, there is an interesting phenomenon that the natural frequency loci will veer more than once due to different types of mode coupling between the bending and stretching vibrations of the rotating beam. At the same time, the mode shift phenomenon will occur correspondingly. Additionally, the characteristics of the vibration nodes are also investigated in this paper.     

An exact analytical approach for in-plane and out-of-plane free vibration analysis of thick laminated transversely isotropic plates

In this article, the governing equations of motion of thick laminated transversely isotropic plates are derived based on Reddy’s third-order shear deformation theory. These equations are exactly converted to four uncoupled equations to study the in-plane and out-of-plane free vibrations of thick laminated plates without any usage of approximate methods. Based on the present analytical approach, exact Levy-type solutions are obtained for thick laminated transversely isotropic plates and, for some boundary conditions, the exact characteristic equations hitherto not reported in the literature are given. Also, the in-plane and out-of-plane deformed mode shapes are plotted for different boundary conditions. The present solutions can accurately predict both the in-plane and out-of-plane natural frequencies and mode shapes of thick laminated transversely isotropic plates.     

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 1: linear modal properties

A geometrically exact mechanical model for the overall dynamics of elastic isotropic rotating blades is proposed. The mechanical formulation is based on the special Cosserat theory of rods which includes all geometric terms in the kinematics and in the balance laws without any restriction on the geometry of deformation besides the enforcement of the local rigidity of the blade cross sections. All apparent forces acting on the blade moving in a rotating frame are accounted for in exact form. The role of internal kinematic constraints such as the unshearability of the slender blades is discussed. The Taylor expansion of the governing equations obtained via an Updated Lagrangian formulation is then employed to obtain the linearized perturbed form about the prestressed configuration under the centrifugal forces. By applying the Galerkin approach to the linearized equations of motion, the linear eigenvalue problem is solved to yield the frequencies and mode shapes. In particular, the natural frequencies of unshearable blades including coupling between flapping, lagging, axial and torsional components are investigated. The angular speeds at which internal resonances may arise due to specific ratios between the frequencies of different modes are determined thus shedding light onto the overall modal couplings in rotating beam structures depending on the angular speed regime. The companion paper (part?2) discusses the nonlinear modes of vibration away from internal resonances.     

Determination of mode shapes using wavelet transform of free vibration data

In this article, a new method is proposed to determine the mode shapes of linear dynamic systems with proportional viscous damping excited by an impact force. The time signals of responses and a priori knowledge of the natural frequencies are required in this method. The method is particularly suitable for the wavelet techniques which can precisely estimate the natural frequencies. A previously proposed method based on a modified Morlet wavelet function with an adjusting parameter is used to identify the natural frequencies and damping ratios of system, and the mode shapes are estimated using the proposed method in this work. It is shown that the extracted mode shapes are not scaled. Therefore, mass change method is used for scaling the mode shapes. Moreover, the effect of noise on the extracted modal parameters is investigated. The validity of method is demonstrated using numerical and experimental case studies.     

Experimental modal analysis of a synthetic composite femur

In this paper we describe the experimental characterization of the modal parameters of a synthetic composite femur model widely used in biomechanical research studies. The objective of the experimental procedure was to identify the natural frequencies and mode shapes of an unconstrained (free-free) femur. The experimental data were compared with the same obtained in an analog study performed with a fresh cadaveric femur bone. Other objective of the study was to investigate modal analysis as a technique to validate a finite element model of a composite femur with isotropic material properties.     

DYNAMIC CHARACTERISTICS OF LAG VIBRATION OF A WIND TURBINE BLADE

Lagwise dynamic characteristics of a wind turbine blade subjected to unsteady aero- dynamic loads are studied in this paper. The partial differential equations governing the coupled longitudinal-transverse vibration of the blade with large bending deflection are obtained by ap- plying Hamilton＇s principle. The modal problem of the coupled vibration is handled by using the method of numerical integration of Green＇s function. Influences of the rotating speed, the pitch angle, the setting angle, and the aerodynamic loads on natural frequencies are discussed. Results show that： （I） Lagwise natural frequencies ascend with the increase of rotating speed; effects of the rotating speed on low-frequencies are dramatic while these effects on high-frequencies become less. （2） Influences of the pitch angle on natural frequencies are little; in the range of the normal rotating speed, the first frequency ascends with the increase of the absolute value of the pitch angle, while it is contrary to the second and third frequencies. （3） Effects of the setting angle on natural frequencies depend on the rotating speed; influences are not significant at low speed, while they are dramatic on the first frequency at high speed. （4） Effects of the aerodynamic loads on natural frequencies are very little; frequencies derived from the model considering aerodynamic loads are smaller than those from the model neglecting aerodynamic loads; relative errors of the results corresponding to two models ascend with the increase of the absolute value of the setting angle.     

弹性连接旋转柔性梁动力学分析

采用Chebyshev 谱方法对考虑根部连接弹性的平面内旋转柔性梁动力学特性进行研究. 基于Gauss-Lobatto 节点与Chebyshev 多项式方法对柔性梁变形场进行离散,通过投影矩阵法施加固定及弹性连接边界条件. 利用Chebyshev 谱方法获得了系统固有频率和模态振型数值解,通过与有限元方法及加权残余法的比较,验证了方法的有效性. 分析了弹性连接刚度、角速度比率、系统径长比及梁的长细比等参数对系统固有频率及模态振型的影响. 研究发现:由于系统弯曲模态、拉伸模态的频率随各参数的变化规律不一致,将出现频率转向与振型转换现象;随着弹性连接刚度、角速度比率及系统径长比的增大,低阶弯曲模态频率增大并超过高阶拉伸模态频率,随着梁的长细比的增大,低阶拉伸模态频率增大并超过高阶弯曲模态频率.      

Experimental full field investigations of resonant vibrations for piezoceramic plates by an optical interferometry method

The experimental measurement of resonant frequencies for piezoelectric material is generally performed by impedance analysis. In this paper we employ an optical interferometry method, called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), to investigate the vibration characteristics of piezoceramic plates. This method demonstrates its advantages of combining noise reduction, like the subtraction method, and high fringe sensitivity, like the time-averaged method. As compared with the film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Based on the fact that clear fringe patterns measured by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and corresponding mode shapes are obtained experimentally at the same time. Excellent quality for the interferometric fringe patterns of the mode shapes is demonstrated. We find from experimental results that the out-of-plane vibration modes (type A) with lower resonant frequencies cannot be measured by impedance analysis and only the in-plane vibration modes (type B) will be shown. However, both the out-of-plane (bending) and in-plane (extension) vibration modes of piezoceramic plates are obtained by the AF-ESPI method. Finally, numberical finite element calculations are also performed, and the results are compared with the experimental measurements. Excellent agreement for the resonant frequencies and mode shapes are obtained from both results.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Vibrations of segmented shells

An analytical and experimental investigation was performed to determine the natural frequencies and mode shapes of a cone-cylinder segmented shell. The finite-element technique was used to predict the natural frequencies and mode shapes of a clamped segmented shell. In the experimental phase of the program, the shell was excited by an electromagnet and the natural frequencies were determined with the aid of a microphone. Holographic interferometry was used to identify the mode shapes for each resonant frequency. The analytical and experimental results were in good agreement with one another.     

Cyclo-symmetric modal synthesis of rotating bladed disc system

The natural mode analysis of bladed disc system at an arbitrary constant rotational speed is investigated in this paper and the large deformation effect caused by the pretorsion of the blades is considered. The computational method of group theory is employed in substructure technique and equilibrium position and stress distribution of the bladed disc system in centrifugal field are obtained through simultaneous iteration, furthermore, the initial stress stiffness matrix, the large deformation stiffness matrix and the corresponding rotating natural frequencies and modal shapes are determined. Because the large deformation effect of the blades is considered and all the interface degrees of freedom between the substructures can be eliminated in the present method, the computation work can be cut down remarkably under the precondition that the computational accuracy is ensured. This shows that, in general, the substructure simultaneous iteration method presented in this paper has a good convergency, thus greatly reducing the computation work.     

Modal analysis of a spinning disk in a dense fluid as a model for high head hydraulic turbines

In high head Francis turbines and pump-turbines in particular, Rotor Stator Interaction (RSI) is an unavoidable source of excitation that needs to be predicted accurately. Precise knowledge of turbine dynamic characteristics, notably the variation of the rotor natural frequencies with rotation speed and added mass of the surrounding water, is essential to assess potential resonance and resulting amplification of vibrations. In these machines, the disk-like structures of the runner crown and band as well as the head cover and bottom ring give rise to the emergence of diametrical modes and a mode split phenomenon for which no efficient prediction method exists to date. Fully coupled Fluid–Structure Interaction (FSI) methods are too computationally expensive; hence, we seek a simplified modeling tool for the design and the expected-life prediction of these turbines.We present the development of both an analytical modal analysis based on the assumed mode approach and potential flow theory, and a modal force Computational Fluid Dynamics (CFD) approach for rotating disks in dense fluid. Both methods accurately predict the natural frequency split as well as the natural frequency drift within 7.9% of the values measured experimentally. The analytical model explains how mode split and drift are respectively caused by linear and quadratic dependence of the added mass with relative circumferential velocity between flexural waves and fluid rotation.     

转动盘一片系的循环对称模综

本文考察在任意恒定转速下叶片/轮盘系的固有模态分析,计及叶片预扭所引起的大变形效应.把群论算法用于子结构技术,通过联立迭代求出叶片/轮盘系在离心场中的平衡位置和应力分布,进而确定初应力刚度矩阵、大变形刚度矩阵,以及相应的动频模态.由于本方法考虑了叶片大变形的影响,且可消去子结构间的全体内连自由度。从而在确保精度的前提下,显著地压缩了计算的规模.实践证明本文提出的子结构联立迭代法有很好的收敛性,它使计算量有巨大的节省.     

A strain-energy criterion for recognition of identified modes of continuous structural models

In structural modal analysis and modal testing, an important but difficult task is to match the identified natural frequencies and the corresponding modal deflections. This process is called the modal recognition in this paper. There were some treatments towards this problem for the lumped parameter structural models. For the distributed parameter models, however, little research has been reported on the modal recognition problem. In this paper, a strain-energy criterion for modal recognition has been developed. As an example, a distributed parameter model for a two-beam structural system has been formulated, which is expected to simulate the dynamics of a two-arm manipulating system fixed on a shuttle. Transfer matrix method has been used to set up the dynamic equation of the system. The natural frequencies are obtained from the solution of the characteristics equation. Consequently, the mode shape functions are found out analytically.

Strain energy can be viewed as a measure of the structural deformation. When performing modal analysis, we always assume that the structural system is vibrating at a particular natural frequency. The strain energy is, therefore, stored in the deflection caused by such a harmonic motion. The vibration at a particular natural frequency will not produce any strain energy in the other modal components. On the other hand, if a particular mode shape is contributed mostly by the deformation of a specific component of the global structural system, then the great percentage of the total strain energy will be stored in the deformation of that component. Based upon the calculation of the strain energy in the structural components we can find out which component is deformed most and in what motion it is deformed, thereby, the mode shape can be detected. The computer simulation demonstrated that the strain energy indicated an essentially perfect recognition of the identified natural frequencies with the corresponding mode shapes. The creation of the strain-energy criterion consummates the procedure of the distributed parameter modeling, modal identification and parameter estimation.     

Geometrically exact planar beams with initial pre-stress and large curvature: Static configurations,natural frequencies,and mode shapes

Within this paper, an analytical formulation is provided and used to determine the natural frequencies and mode shapes of a planar beam with initial pre-stress and large variable curvature. The static configuration, mode shapes, and natural frequencies of the pre-stressed beam are obtained by using geometrically exact, Euler–Bernoulli beam theory. The beam is assumed to be not shear deformable and inextensible because of its slenderness and uniform, closed cross-section, as well as the boundary conditions under consideration. The static configuration and the modal information are validated with experimental data and compared to results obtained from nonlinear finite-element analysis software. In addition to the modal analysis about general static configurations, special consideration is given to an initially straight beam that is deformed into semi-circular and circular static configurations. For these special circular cases, the partial differential equation of motion is reduced to a sixth-order differential equation with constant coefficients, and solutions of this system are examined. This work can serve as a basis for studying slender structures with large curvatures.