Closely-spaced Natural Frequencies Research Articles

Effect of Modal Interactions on Friction-Damped Self-Excited Vibrations

Abstract It is well-known that nonlinear dry friction damping has the potential to bound the otherwise unboundedly growing vibrations of self-excited structures. An important technical example are the flutter-induced friction-damped limit cycle oscillations of turbomachinery blade rows. Due to symmetries, natural frequencies are inevitably closely spaced, and they can generally be multiples of each other. Not much is known on the nonlinear dynamics of self-excited friction-damped systems in the presence of such internal resonances. In this work, we analyze this situation numerically by regarding a two degrees-of-freedom system. We demonstrate that in the case of closely-spaced natural frequencies, the self-excitation of the lower-frequency mode gives rise to non-periodic oscillations, and the occurrence of unbounded behavior well before reaching the maximum friction damping value. If the system is close to a 1:3 internal resonance, limit cycles associated with much higher frictional damping appear, however, most of these are unstable. If more than one mode is subjected to self-excitation, the maximum resistance against self-excitation is at least given by the damping capacity of the most weakly friction-damped mode. These results are of high technical relevance, as the prevailing practice is to analyze only periodic limit states and argue the stability solely by the slope of the damping-amplitude curve. Our results demonstrate that this practice leads to considerable mis- and overestimation of the resistance against self-excitation, and a more rigorous stability analysis is required.

Condition assessment of structure with tuned mass damper using empirical wavelet transform

Tuned mass damper (TMD) has been one of the most commonly used passive vibration control devices over the past few decades. While an optimally designed TMD can significantly suppress the structural vibration, detuning often occurs due to various reasons such as change in operating conditions or variation in primary structure properties, resulting in degradation of TMD’s performance. In order to restore its performance, it is necessary to estimate the modal properties of the primary structure, and perform the re-tuning process. Such an exercise requires powerful signal processing methods to successfully extract the structural modes in the presence of closely-spaced modes. This study focuses on the identification of modal frequencies and damping of the structure installed with a TMD. In view of the advantages and limitations of existing modal identification methods, this paper provides a new technique that combines the second-order blind identification (SOBI) method with the empirical wavelet transform (EWT) to delineate closely-spaced frequencies. While the SOBI method does not guarantee the separation of closely-spaced modes and suffers from the limitation of generating mode-mixed modal responses, the EWT operates on the modal responses estimated by the SOBI and yields the closely-spaced natural frequencies. The proposed method is illustrated using a six-story simulation model with a wide range of detuning cases. An experiment on a three-story bench-scale model equipped with a TMD is also conducted to validate the applicability of the proposed method.

Performance analysis of differential-frequency microgyroscopes made of nanocrystalline material

The performance of a microgyroscope consisting of a rotating microbeam made of nanocrystalline silicon connected to a proof mass and subjected to electric actuation is investigated. The microgyroscope is assumed to operate in frequency-based mode; that is, the measurement of the rotation rate is extracted from the shift in the frequency response along the drive and sense directions. Closed-form expressions of the natural frequencies and mode shapes of the coupled dynamical system are derived. The onset of the base rotation is observed to split the common natural frequency of the two bending modes into a pair of closely-spaced natural frequencies. This frequency shift used as the output parameter detecting the rotation rate. A sensitivity analysis of this parameter to the rotation rate when varying the material properties of the microbeam and electric actuation is then performed. When applying the same DC voltage for the drive and sense modes, the differential frequency is found to vary linearly with the base rotation. Incorporating the fringing field in the electrostatic forcing and varying the grain size of the nanocrystalline silicon have insignificant impact on the calibration curve for this case. Breaking the symmetry of the microgyroscope in terms of the applied DC voltage along the drive and sense directions is observed to reduce significantly the sensitivity of the differential frequency to the base rotation rate. The larger the DC voltage bias is, the lower the sensor sensitivity is. Furthermore, under these operating conditions, the results show that the variations of the differential frequency with the rotation rate undergo nonlinear trends. Considering higher order modes is found to reduce the impact of the DC voltage bias on the calibration curve of the microgyroscope.

OUTPUT-ONLY MODAL IDENTIFICATION OF TENSEGRITY STRUCTURES

Tensegrity systems are a special class of spatial reticulated structures that are composed of struts in compression and cables in tension. In this paper, the performance of stochastic subspace algorithms for modal identification of complex tensegrity systems is investigated. A sub-class algorithm of the Stochastic Subspace Identification family: the Balanced Realization Algorithm is investigated for modal identification of a tripod simplex structure and a Geiger dome. The presented algorithm is combined with a stabilization diagram with combined criteria (frequency, damping and mode shapes). It is shown that although the studied structures present closely spaced modes, the Balanced Realization Algorithm performs well and guarantees separation between closely-spaced natural frequencies. Modal identification results are validated through comparisons of the correlations (empirical vs. model based) showing effectiveness of the proposed methodology.

Feedback control design for intelligent structures with closely-spaced eigenvalues

Large space structures may have resonant low eigenvalues and often these appear with closely-spaced natural frequencies. Owing to the coupling among modes with closely-spaced natural frequencies, each eigenvector corresponding to closely-spaced eigenvalues is ill-conditioned that may cause structural instability. The subspace to an invariant subspace corresponding to closely-spaced eigenvalues is well-conditioned, so a method is presented to design the feedback control law of intelligent structures with closely-spaced eigenvalues in this paper. The main steps are as follows: firstly, the system with closely-spaced eigenvalues is transformed into that with repeated eigenvalues by the spectral decomposition method; secondly, the computation for the linear combination of eigenvectors corresponding to repeated eigenvalues is obtained; thirdly, the feedback control law is designed on the basis of the system with repeated eigenvalues; fourthly, the system with closely-spaced eigenvalues is regarded as perturbed system on the basis of the system with repeated eigenvalues; finally, the feedback control law is applied to the original system, the first order perturbations of eigenvalues are discussed when the parameter modifications of the system are introduced. Numerical examples are given to demonstrate the application of the present method.

Optimal placement and tuning of multiple tuned mass dampers for suppressing multi-mode structural response

The optimal design of multiple tuned mass dampers (multiple TMD\'s) to suppress multi-mode structural response of beams and floor structures was investigated. A new method using a numerical optimizer, which can effectively handle a large number of design variables, was employed to search for both optimal placement and tuning of TMD\'s for these structures under wide-band loading. The first design problem considered was vibration control of a simple beam using 10 TMD\'s. The results confirmed that for structures with widely-spaced natural frequencies, multiple TMD\'s can be adequately designed by treating each structural vibration mode as an equivalent SDOF system. Next, the control of a beam structure with two closely-spaced natural frequencies was investigated. The results showed that the most effective multiple TMD\'s have their natural frequencies distributed over a range covering the two controlled structural frequencies and have low damping ratios. Moreover, a single TMD can also be made effective in controlling two modes with closely spaced frequencies by a newly identified control mechanism, but the effectiveness can be greatly impaired when the loading position changes. Finally, a realistic problem of a large floor structure with 5 closely spaced frequencies was presented. The acceleration responses at 5 positions on the floor excited by 3 wide-band forces were simultaneously suppressed using 10 TMD\'s. The obtained multiple TMD\'s were shown to be very effective and robust.

Galloping of tower-like structure with two closely-spaced natural frequencies

Certain cable-stayed bridge towers have in-plane vibration modes of closely-spaced natural frequencies. A study on wind-induced galloping of such a structure is presented. The structure is modeled as a linear two-degree-of-freedom system and nonlinear quasi-steady wind force is assumed. The present analysis indicates that stable galloping mode is either one of the two modes depending upon the initial disturbances given to the structure. The coexistence of two modes in galloping is unstable; however, for towers of certain properties, e.g. unsymmetrically distributed mass, the multi-mode galloping is stable. The results agree well with the observations in wind tunnel experiments using three dimensional models of bridge tower.

EFFECT OF TURBULENCE ON THE MODAL INTERACTION IN GALLOPING OF STRUCTURE WITH TWO CLOSELY-SPACED NATURAL FREQUENCIES

Effect of turbulent wind on modal interaction in galloping of structure with two closely-spaced natural frequencies is studied. Numerical simulation indicates that the presence of certain intensity of turbulence causes the modal interaction, i. e. the dominant vibration mode alternately changes from one mode to the other in time domain. The modal responses seemingly are the combination of galloping response with time-varying amplitude and the random buffeting responses. This is also confirmed by wind tunnel observations.

GALLOPING OF STRUCTURE WITH TWO CLOSELY-SPACED NATURAL FREQUENCIES

Certain structures have the vibration modes of closely-spaced natural frequencies. The galloping behaviour of such a structure is studied. A cable-stayed bridge tower whose the natural frequency of the in-plane second mode is close to that of the in-plane first mode is employed as the case study. The structure is modeled as a linear two-degree-of-freedom system with proportional damping, and nonlinear quasi-steady wind force is assumed. An asymptotic modal analysis on galloping is conducted. The results indicate that the galloping of this tower is the steady-state motion either in one of the two modes, depending upon the initial disturbances, and that the coexistence of two modes in galloping, i. e. multi-mode galloping, is unstable. However, for structures with certain properties, e. g. unsymmetrically distributed mass, only the multi-mode galloping is stable. These results agree well with the observations in wind tunnel experiments.