Close Natural Frequencies Research Articles

Modeling and analysis of the in-plane vibration of a complex cable-stayed bridge

The in-plane vibration of a complex cable-stayed bridge that consists of a simply-supported four-cable-stayed deck beam and two rigid towers is studied. The nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of the cables and transverse vibrations of segments of the deck beam, respectively, are derived, along with their boundary and matching conditions. The undamped natural frequencies and mode shapes of the linearized model of the cable-stayed bridge are determined, and orthogonality relations of the mode shapes are established. Numerical analysis of the natural frequencies and mode shapes of the cable-stayed bridge is conducted for various symmetrical and non-symmetrical bridge cases with regards to the sizes of the components of the bridge and the initial sags of the cables. The results show that there are very close natural frequencies when the bridge model is symmetrical and/or partially symmetrical, and the mode shapes tend to be more localized when the bridge model is less symmetrical. The relationships between the natural frequencies and mode shapes of the cable-stayed bridge and those of a single fixed–fixed cable and the single simply-supported deck beam are analyzed. The results, which are validated by commercial finite element software, demonstrate some complex classical resonance behavior of the cable-stayed bridge.

The non-linear oscillations of a rod with close values of the cross-section axial moments of inertia

An analytical solution of the problem of the forced flexural oscillations of a rod with fixed hinged supports is presented. The rod has close natural frequencies of flexural oscillations in two mutually perpendicular planes due to the close values of the principal axial moments of inertia of the cross-section. The geometrical non-linearity, due to the change in the length of the middle line of the rod when it undergoes three-dimensional motion, is taken into account. The oscillations of the rod in the neighbourhood of the principal and first superharmonic resonances are investigated.

Stationary coupled modes at forced oscillations of nonlinear symmetric 2DoF systems having close natural frequencies

In the work cubic symmetrical systems with close eigenfrequencies are considered. Such systems are widely used in various applications and include, in particular, thin-walled beams, rectangular and circular plates and cylindrical shells. Note that circular disks and cylindrical shells always possess the internal 1:1 resonance since for each natural mode there exists so-called conjugate mode, which has equal or close eigenfrequency. Our attention is focused on the primary resonance under harmonic external excitation which for the considered systems coincides, precisely or approximately, with the internal resonance. Coincidence of the external and internal resonances can result in complicated nonlinear behavior, in particular, leads to the appearance of coupled steady-state modes, along with uncoupled modes. The aim of the work is to carry out the detailed analysis of coupled steady-state modes (CSM) in the vicinity of the primary external resonance.

A linear least-squares version of the algorithm of mode isolation for identifying modal properties. Part II: Application and assessment

The latest modifications of the algorithm of mode isolation (AMI) for identification of modal properties from frequency response data are tested with synthetic data derived from an analytical model of an elastic frame in which flexure and torsion are coupled. The parameters of this model are selected to cause the occurrence of localized modal patterns in two modes having close natural frequencies. The response data is contaminated with white noise at a level sufficient to almost mask the two close modes. Results for the real and imaginary part of the eigenvalues are tabulated. The analytical modal patterns of displacement and torsional rotation are depicted graphically, accompanied by the discrete values obtained from AMI. Excellent agreement is found to occur for each mode, other than one of the pair of close modes. The poorer quality of that mode’s identified properties is shown to be a consequence of its localized modal pattern. Results for the eigenvalues obtained by the rational fraction polynomial algorithm, which is an alternative modal identification technique, are found to be substantially less accurate as a consequence of difficulty in the presence of noise.

近接した固有振動数をもつ二自由度非線形系の1/2次分数調波・和差調波振動(機械力学・計測制御IV)

近接した固有振動数をもつ二自由度非線形系の1/2次分数調波・和差調波振動(機械力学・計測制御IV)

HIGHER-ORDER MAC FOR THE CORRELATION OF CLOSE AND MULTIPLE MODES

In this paper, an extension of Modal Assurance Criterion (MAC) is proposed. It considers the correlation between a modal vector and a subspace spanned by several modal vectors, instead of the usual correlation between two modal vectors. Such higher-order MAC provides additional insight whenever multiple or close eigenvalues exist. In principle, any number of modal vectors can be considered to span the subspace. A closed-form expression is derived for the correlation of a modal vector with a two-dimensional subspace, referred to as S2MAC (subspace of order 2 Modal Assurance Criterion). This allows a better understanding in situations involving double eigenvalues or pairs of close natural frequencies, as highlighted by the discussed test cases.

1002 近接した固有振動数をもつ二自由度非線形系の 1/2 次分数調波振動

1002 近接した固有振動数をもつ二自由度非線形系の 1/2 次分数調波振動

418 近接した固有振動数をもつ二自由度非線形系の1/2次分数調波振動の分岐現象

418 近接した固有振動数をもつ二自由度非線形系の1/2次分数調波振動の分岐現象

Forced Vibrations in a Nonlinear Two-Degree-of-Freedom System with Comparatively Close Natural Frequencies (Comparison of Six Kinds of Approximation)

This paper deals with harmonic oscillations in a nonlinear two-degree-of-freedom system in which natural frequencies are comparatively close. Six kinds of approximated differential equations are derived and the bifurcation analysis are performed using these equations. By comparing the results of theoretical analyses with the results of the numerical simulation, it is clarified that the 2nd-order approximation in the method of multiple scales using the modified detuning parameters is the best approach to analyze the vibration characteristics of this system with a high accuracy. As a result, it is found that undergoing Hopf bifurcation the periodic solution experiences period doubling bifurcations and results in chaos, and that the bifurcation points obtained by this method are in good agreement with the results of numerical simulations.

Use of speckle interferometry and modal assurance criterion for identification of component modes

One of the problems encountered in performing modal analysis is the identification of different modes occurring at close natural frequencies. This paper proposes to face such a problem by an original application of the modal assurance criterion (MAC) that is typically used to verify the reliability of numerical structural models by evaluating the correspondence between numerical and experimental mode shapes. To properly apply the MAC, experimental mode shapes must be retrieved acquiring data at a high number of experimental points suitably distributed on the surface of the structure. Using usual modal analysis techniques based on pointwise transducers, difficulties can be encountered to retrieve mode shapes with the required spatial resolution. In order to overcome such difficulties, Speckle interferometry techniques can be used. This paperproposes a procedure based on the application of a highly reliable MAC evaluated by time average Speckle interferometry for recognising the modes contributing to the stationary vibration patterns. The results obtained from the modal analysis of a thin square plate have shown that the proposed procedure is capable of pointing out the component modes.

Forced Oscillations in a Nonlinear Two-Degree-of-Freedom System with Comparatively Close Natural Frequencies.

This paper deals with nonlinear forced oscillations in a two-degree-of freedom system in which natural frequencies are comparatively close. Two kinds of approximated differential equations for amplitudes and phase angles of a harmonic oscillation and of a combination resonance are derived. The resonance curves and their stability analyses, and the local bifurcation analysis are made by using these equations. The numerical simulation analyses are performed to obtain time histories, their spectra, Poincare maps and Lyapunov exponents by using the original equations of motion. As a result, it is found that Hopf bifurcations can occur on the branch of the resonance curves for the harmonic oscillation, and that chaotic vibrations may occur after the appearance of a period doubling bifurcation, which are generated from the Hopf bifurcation point. The differences of the accuracy between these two kinds of approximated differential equations are also discussed.

Application of the mode isolation algorithm for experimental modal analysis to a system with close natural frequencies

Experimental modal analysis techniques in current use have difficulty with modes whose natural frequencies are very close or whose drive point mobility is very low. How well these issues are addressed by the Mode Isolation Algorithm (MIA), see M. V. Drexel and J. H. Ginsberg, ‘‘Modal parameter isolation using state space mode isolation,’’ in Proceedings of the 19th International Modal Analysis Conference, Orlando, Florida, February 5–8, 2001, is the topic of this presentation. MIA is an iterative procedure that employs frequency domain response data. The application of MIA to a system consisting of a beam with three suspended spring-mass-dashpot subsystems is described. Judicious choice of the system parameters leads to two modes having close natural frequencies, and two other modes in which impulse excitation at the free end of the beam results essentially in vibration of the subsystems away from the end. Synthetic impulse response data for the analysis is contaminating with white noise. It is shown that MIA accurately identifies all natural frequencies and modal damping ratios if the frequency resolution of the response data is sufficiently fine. Furthermore, the two normal mode shapes associated with the proximate natural frequencies are correctly identified, but the two with low drive point mobility are not. [Work supported by the National Science Foundation, Grant No. CMS-0098539.]

Aeroelastic instability of long-span bridges: contributions to the analysis in frequency and time domains

According to research currently developed by several authors (including the present ones) arnmultimode approach to the aeroelastic instability can be appropriate for suspension bridges with very longrnspan and so with close natural frequencies. Extending that research, this paper deals in particular with: i)rnthe role of along-wind modes, underlined also by means of the flutter mode representation; ii) the effectsrnof a variation of the mean wind speed along the span. A characterisation of the response in the timerndomain by means of an energetic approach is also discussed.

Eigensolution Method for Structures with Multiple or Close Eigenvalues

A solution method that can calculate the eigenproblem for large structures with multiple or close natural frequencies is presented. The method is formulated by the accelerated Newton-Raphson method for the transformed problem. The method can calculate the natural frequencies and mode shapes without any of the numerical instability that may be encountered in the inverse iteration process with a shift of the well-known methods, such as the subspace iteration method or the determinant search method, which have been widely used for solving an eigenproblem. The efficiency of the method is verified by comparing convergence and solution times for numerical examples with those of the subspace iteration method and the determinant search method.

VIBRATION OF CIRCULAR PLATES ON A FREE FLUID SURFACE: EFFECT OF SURFACE WAVES

The effect of free-surface waves on free vibrations of circular plates resting on a free fluid surface is studied. The solution is achieved by using a perturbation technique and the Hankel transformation method which give a couple of dual integral equations of Titchmarsh type. The fluid is considered inviscid and incompressible and the velocity potential describes its motion. The Kirchhoff theory of plates is used to model the elastic thin plate. The theory is suitable for all axisymmetric plate boundary conditions. Numerical results are given in non-dimensional form for modes up to five nodal circles and diameters for clamped, simply supported and free-edge plates, to be ready-to-use in applications. The effect of free-surface waves on the plate's natural frequencies is significant only when bluging and sloshing modes of the system have close natural frequencies.

Free oscillations of a non-linear cubic system with two degrees of freedom and close natural frequencies

The multiscale method [1] is used to investigate free oscillations of a conservative system with two degrees of freedom having cubic non-linearities (of symmetric nature) and close natural frequencies. Possible oscillation modulation regimes are found which depend on the coefficients of the system of differential equations, the energy and initial conditions.

SYSTEM MODAL IDENTIFICATION USING FREE VIBRATION DATA

System modal identification for structures is a fundamental problem in structural dynamics. The identified natural frequencies, damping ratios and mode shapes can be used to better understand the behaviour of structures, modify structures, design control system for structures, and can also be used in direct analysis of structures. A time domain method for identification of modal parameters for a test structure using free vibration responses is presented in this paper. The method in based on the multidimensional autoregressive model of a vibrating structure. The method is efficient with computing time and has the advantage of being able to identify very close natural frequencies and highly coupled modes. A numerical example and a test case are used to illustrate the usefulness and results of the method.

The effects of detuning upon the forced vibrations of bladed disks

In the construction of steam and gas turbines it is not feasible to attain complete identity of all the blades mounted on one turbine disk due to limitations inherent in manufacturing processes. In this paper, a simplified model is used to study the implications of this real situation from a consideration of the vibration levels which these components might suffer in practice. It has been shown elsewhere that detuning—the presence of small differences between the blades on a bladed disk—gives rise to a natural frequency “splitting” phenomenon in which a pair of modes with close natural frequencies and similar characteristics are formed in place of an apparently single mode of vibration in the perfect, or tuned, system. The present work describes the dynamic characteristics of a system having such pairs of close natural frequency modes when it is forced to vibrate under typical operating conditions—with excitation being derived from the presence of obstructions in the flow. A simplified analysis suggests that the best system is one in which all the blades are identical. It is shown that with the random arrangements of typical blades which will occur in practice, some blades may suffer stress levels which are 20% higher than those of the equivalent tuned system. However, by careful arrangement of the same set of non-identical blades, it is possible to eliminate the risk of such an increase in the peak stress levels. An optimum arrangement of typical blades is proposed.

Experimental Determination of the Complete Dynamical Properties of a Two-Degree-Of-Freedom Model Having Nearly Coincident Natural Frequencies

Existing techniques of resonance testing have shown a marked inability to find the principal modes, natural frequencies and levels of damping in a structure which possesses two or more close natural frequencies (1)§. This paper describes an experimental investigation on a two-degree-of-freedom model of a technique which makes use of dynamical influence coefficients (or receptances) measured at a number of stations on the structure (2) (3) (4) (5). The measured coefficients are used to calculate natural frequencies and modes of vibration, and the mass, damping and stiffness properties of the system. Several model configurations having different natural frequency separations were tested and no special difficulty resulted when natural frequencies were close or even coincident.

Theoretical Investigation into the Effects of Close Natural Frequencies in Resonance Testing

It is difficult (if not impossible) to find the principal modes, natural frequencies and levels of damping of a structure by means of a resonance test if the structure happens to possess two (or more) close natural frequencies. Under the assumption that this condition does exist, the ability of each of the various techniques for extracting the modes and frequencies from resonance test results is investigated.