Features of the oscillations of linear dynamic systems with dissipative coupling having two degrees of freedom

The natural frequencies of a system of this type lie between its partial frequencies. There are three characteristic regions of the mismatch between partial frequencies. In the central region (near the internal resonance point), the natural frequencies of the system practically coincide, and so do the oscillation amplitudes for these frequencies. In the left and right regions, the natural frequencies approach the partial frequencies. Generally, the motion consists in damped amplitude-phase-modulated oscillations. As time elapses, there occurs frequency tuning of the oscillators to one frequency at which damping goes more slowly, and the oscillations invariably go over to the in-phase mode. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 52–59, January, 2008.     

Theory of coupled mode oscillation in a laser with axial magnetic field

The nonlinear laser theory without noise is applied to calculate amplitudes, frequencies and beat frequencies of two circularly polarized modes in a laser with axial magnetic field. The resonator is allowed to have different damping constants for waves polarized inx- andy-direction. For small beat frequencies frequency locking occurs, and the modes combine to a linearly or elliptically polarized mode. The plane of polarization rotates with increasing magnetic field up to ± π/4.     

Analytical modeling and vibration analysis of internally cracked rectangular plates

This study proposes an analytical model for nonlinear vibrations in a cracked rectangular isotropic plate containing a single and two perpendicular internal cracks located at the center of the plate. The two cracks are in the form of continuous line with each parallel to one of the edges of the plate. The equation of motion for isotropic cracked plate, based on classical plate theory is modified to accommodate the effect of internal cracks using the Line Spring Model. Berger?s formulation for in-plane forces makes the model nonlinear. Galerkin?s method used with three different boundary conditions transforms the equation into time dependent modal functions. The natural frequencies of the cracked plate are calculated for various crack lengths in case of a single crack and for various crack length ratio for the two cracks. The effect of the location of the part through crack(s) along the thickness of the plate on natural frequencies is studied considering appropriate crack compliance coefficients. It is thus deduced that the natural frequencies are maximally affected when the crack(s) are internal crack(s) symmetric about the mid-plane of the plate and are minimally affected when the crack(s) are surface crack(s), for all the three boundary conditions considered. It is also shown that crack parallel to the longer side of the plate affect the vibration characteristics more as compared to crack parallel to the shorter side. Further the application of method of multiple scales gives the nonlinear amplitudes for different aspect ratios of the cracked plate. The analytical results obtained for surface crack(s) are also assessed with FEM results. The FEM formulation is carried out in ANSYS.     

On the separation of internal and boundary damage in slender bars using longitudinal vibration frequencies and equivalent linearization of damaged bolted joint response

The problem of detecting localized large-scale internal damage in structures with imperfect bolted joints is considered. The proposed damage detection strategy utilizes the structural damping and an equivalent linearization of the bolted lap joint response to separate the combined boundary damage from localized large-scale internal damage. The frequencies are found approximately using asymptotic analysis and a perturbation technique. The proposed approach is illustrated on an example of longitudinal vibrations in a slender elastic bar with both ends clamped by bolted lap joints with different levels of damage. It is found that while the proposed method allows for the estimation of internal damage severity once the crack location is known, it gives multiple possible crack locations so that other methods (e.g., mode shapes) are required to obtain a unique crack location.     

The use of roving discs and orthogonal natural frequencies for crack identification and location in rotors

A variety of approaches that have been developed for the identification and localisation of cracks in a rotor system, which exploit natural frequencies, require a finite element model to obtain the natural frequencies of the intact rotor as baseline data. In fact, such approaches can give erroneous results about the location and depth of a crack if an inaccurate finite element model is used to represent an uncracked model. A new approach for the identification and localisation of cracks in rotor systems, which does not require the use of the natural frequencies of an intact rotor as a baseline data, is presented in this paper. The approach, named orthogonal natural frequencies (ONFs), is based only on the natural frequencies of the non-rotating cracked rotor in the two lateral bending vibration x–z and y–z planes. The approach uses the cracked natural frequencies in the horizontal x–z plane as the reference data instead of the intact natural frequencies. Also, a roving disc is traversed along the rotor in order to enhance the dynamics of the rotor at the cracked locations. At each spatial location of the roving disc, the two ONFs of the rotor–disc system are determined from which the corresponding ONF ratio is computed. The ONF ratios are normalised by the maximum ONF ratio to obtain normalised orthogonal natural frequency curves (NONFCs). The non-rotating cracked rotor is simulated by the finite element method using the Bernoulli–Euler beam theory. The unique characteristics of the proposed approach are the sharp, notched peaks at the crack locations but rounded peaks at non-cracked locations. These features facilitate the unambiguous identification and locations of cracks in rotors. The effects of crack depth, crack location, and mass of a roving disc are investigated. The results show that the proposed method has a great potential in the identification and localisation of cracks in a non-rotating cracked rotor.     

The smoothness of the universe

We study the evolution of primordial fluctuations in the Big Bang in the context of grand unified theories of elementary particle interactions. We show that dissipation due to grand unified viscosity strongly damps rotational modes of all frequencies, compressional modes of all except very low frequencies, and radiative modes of high frequencies. The damping mainly occurs at temperatures > 10¹⁵GeV, and does not significantly increase the entropy. Compressional modes which originate at high temperatures and have very low frequencies persist down to much lower temperatures and could eventually evolve into galaxies.     

Combination resonance of parametrically excited coupled second order systems with non-linear damping

The parametrically excited oscillation of two coupled second order systems with nonlinear damping is investigated. The coupling is through the small parametric excitation terms in which the off-diagonal terms dominate, indicating combination resonance type motion with primary instability regions near the forcing frequencies $\text{ω}{}_{\mathrm{F}}\text{= Ω}{}_{\mathrm{i}}\text{+ Ω}{}_{\mathrm{j}}$, with each mode oscillating near its own natural frequency, $\text{Ω}{}_{\mathrm{i}}$. Through an asymptotic analysis simple formulae are obtained for evaluating the steady state amplitudes in the first instability region and the non-dimensional solution surfaces are plotted. These are found to be analogous to those for a single second order system and represent master plots. The type of damping considered here includes the important case of velocity squared, fluid dynamic type damping. The theoretical results are compared with those from an experiment in which this type of damping was observed and measured. The measured lateral bending-torsion amplitudes are well predicted by the theory, the results of which are also in agreement with the numerical integration results.     

MODELLING OF TRANSVERSE VIBRATION OF SHORT BEAMS FOR CRACK DETECTION AND MEASUREMENT OF CRACK EXTENSION

The method of detection of location of crack in beams based on frequency measurements is extended here to short beams, taking into account the effects of shear deformation and rotational inertia through the Timoshenko beam theory and representing the crack by a rotational spring. Methods for solving both forward (determination of frequencies of beams knowing the crack parameters) and inverse (determination of crack location knowing the natural frequencies) problems are included. A method to estimate crack extension from a change in the first natural frequency is presented. Both numerical and experimental studies are given to demonstrate the accuracy of the methods. The accuracy of the results is quite encouraging.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.     

Amplification/damping processes of atmospheric acoustic-gravity waves in horizontal winds with linear shear

The time-varying frequencies and amplification/damping rate (ADR) of atmospheric acoustic-gravity waves (AGW) in unbounded horizontal winds with linear shear (HWLS) in the linear approach is analytically obtained for the three-dimensional case. The numerical solutions for the temporal evolution of the AGW amplitudes point at the possibility of the energy transformation between the obtained acoustic, gravity and shear wave modes.     

A recursive approach to prony parameter estimation

The Prony series method of fitting a series of complex exponentials to a time series can be applied to many acoustic and vibration signals. For example, in the analysis of a structure's response to transient excitation the Prony series method can be used to find the natural frequencies, damping ratios, amplitudes and relative phases of the modes of the structure. One of the main problems with this method is the need to calculate and invert matrices. In this paper an algorithm that combines the Prony series method with the recursive least squares algorithm is described. This eliminates the need to invert any matrices and also requires only part of the data to be available at one particular time. The method is applied to analyze a simulated structure's response and also to analyze the response of a beam to transient excitation.     

CRACK DETECTION IN BEAM-TYPE STRUCTURES USING FREQUENCY DATA

A practical method to non-destructively locate and estimate size of a crack by using changes in natural frequencies of a structure is presented. First, a crack detection algorithm to locate and size cracks in beam-type structures using a few natural frequencies is outlined. A crack location model and a crack size model are formulated by relating fractional changes in modal energy to changes in natural frequencies due to damage such as cracks or other geometrical changes. Next, the feasibility and practicality of the crack detection scheme are evaluated for several damage scenarios by locating and sizing cracks in test beams for which a few natural frequencies are available. By applying the approach to the test beams, it is observed that crack can be confidently located with a relatively small localization error. It is also observed that crack size can be estimated with a relatively small size error.     

A note on the interaction between a Helmholtz resonator and an acoustic mode of an enclosure

A single Helmholtz resonator is coupled to an enclosure and tuned to the natural frequency of one of its low order acoustic modes. The effect on the free, and forced, vibrations of the fluid in the enclosure is analyzed. The conditions necessary for the resonator to increase the damping of the two resultant modes, and to control the room response to excitation at frequencies within the range embracing both natural frequencies, are investigated. A simple design graph is presented.     

IDENTIFICATION OF DAMPING: PART 4, ERROR ANALYSIS

In previous papers (S. ADHIKARI and J. WOODHOUSE 2001 Journal of Sound and Vibration243, 43-61; 63-88; S. ADHIKARI and J. WOODHOUSE 2002 Journal of Sound and Vibration251, 477-490) methods were proposed to obtain the coefficient matrix for a viscous damping model or a non-viscous damping model with an exponential relaxation function, from measured complex natural frequencies and modes. In all these works, it has been assumed that exact complex natural frequencies and complex modes are known. In reality, this will not be the case. The purpose of this paper is to analyze the sensitivity of the identified damping matrices to measurement errors. By using numerical and analytical studies it is shown that the proposed methods can indeed be expected to give useful results from moderately noisy data provided a correct damping model is selected for fitting. Indications are also given of what level of noise in the measured modal properties is needed to mask the true physical behaviour.     

On the limiting of vibration amplitudes by a sequential friction-spring element

The possibility of using an additional sequentially connected friction spring element in order to reduce vibration amplitudes both for the self-excited oscillations and for the forced vibrations is discussed in the paper. The analysis is based on the averaging technique for systems with “slave variables” and demonstrates two main effects: damping during slipping in the additional element and fast switching between different natural frequencies due to alternating sticking/slipping phases. Analytic predictions for the oscillations’ amplitudes are obtained as steady state solutions of the equations governing slow motions of the system. The obtained analytic results enable optimal choice of friction in order to achieve maximal damping effect in case of the forced vibrations. The reasonable choice of the friction by the self-excited vibrations is a compromise between the acceptable amplitude and the robustness of the corresponding limit cycle. The asymptotic results are confirmed by numeric simulations.     

Experimental analysis of the linear and nonlinear behaviour of composites with delaminations

An analysis of the linear and nonlinear vibration responses of composites with delaminations is presented. The effect of delamination size on the linear and nonlinear vibration response is studied. The composite material used in this paper is a glass fibre reinforced plastic (GFRP) having delaminations at the plies interfaces. The experimental procedure consists in inducing the specimen on its resonance flexural modes with different excitation levels (amplitudes) for six bending modes and for each delamination length. The presence of the nonlinearity introduced by the delamination was clearly identified by the variation of natural frequencies for increasing excitation levels. Then, nonlinear elastic parameters for progressive delamination length were determined and discussed for the first six bending modes. The linear and the nonlinear elastic parameters were compared in their sensitive modes.     

Audible vibration diagnostics of thermo-elastic residual stress in multi-crystalline silicon wafers

This paper presents audible vibratory mode data obtained by mechanically exciting acoustic modes in multi-crystalline silicon wafers with various levels and distributions of residual stress. Natural frequency data from the silicon wafers is found to correlate with residual stress optical polariscopy measurements. The data is fit with both linear and quadratic models with correlation coefficients of 0.8. The results reveal a dependence of wafer audible mode frequencies on residual stress level that may be useful for solar cell mechanical quality control and breakage inspection.     

IDENTIFICATION OF DAMPING: PART 3, SYMMETRY-PRESERVING METHODS

In two recent papers (Adhikari and Woodhouse 2001 Journal of Sound and Vibration243, 43-61; 63-88), methods were proposed to identify viscous and non-viscous damping models for vibrating systems using measured complex frequencies and mode shapes. In many cases, the identified damping matrix becomes asymmetric, a non-physical result. Methods are presented here to identify damping models which preserve symmetry of the system. Both viscous and non-viscous models are considered. The procedure is based on a constrained error minimization approach and uses only experimentally identified complex modes and complex natural frequencies together with, for the non-viscous model, the mass matrix of the system. The methods are illustrated by numerical examples.     

Vibration analysis of a partially treated multi-layer beam with magnetorheological fluid

Semi-active vibration control based on magnetorheological (MR) materials offers excellent potential for high bandwidth control through rapid variations in the rheological properties of the fluid under varying magnetic field. Such fluids may be conveniently applied to partial or more critical components of a large structure to realize more efficient and compact vibration control mechanism with variable damping. This study investigates the properties and vibration responses of a partially treated multi-layer MR fluid beam. The governing equations of a partially treated multi-layered MR beam are formulated using finite element method and Ritz formulation. The validity of the proposed finite element formulations is demonstrated by comparing the results with those obtained from the Ritz formulation and the experimental measurements. The properties of different configurations of a partially treated MR-fluid beam are evaluated to investigate the influences of the location and length of the MR-fluid for different boundary conditions. The properties in terms of natural frequencies and loss factors corresponding to various modes are evaluated under different magnetic field intensities and compared with those of the fully treated beams. The effect of location of the fluid treatment on deflection mode shapes is also investigated. The forced vibration responses of the various configurations of partially treated MR sandwich beam are also evaluated under harmonic force excitations. The results suggest that the natural frequencies and transverse displacement response of the partially treated MR beams are strongly influenced not only by the intensity of the applied magnetic field, but also by the location and the length of the fluid pocket. The application of partial treatment could also alter the deflection pattern of the beam, particularly the location of the peak deflection.     

Spin-transfer-driven ferromagnetic resonance of individual nanomagnets

We demonstrate a technique that enables ferromagnetic resonance measurements of the normal modes for magnetic excitations in individual nanoscale ferromagnets, smaller in volume by more than a factor of 50 compared to individual ferromagnetic samples measured by other resonance techniques. Studies of the resonance frequencies, amplitudes, linewidths, and line shapes as a function of microwave power, dc current, and magnetic field provide detailed new information about the exchange, damping, and spin-transfer torques that govern the dynamics in magnetic nanostructures.