Generalization of the Wittrick-Williams formula for counting modes of flexible structures

The oil and gas, chemical, and process industries employ centrifugal compressors for a wide range of applications. Due to this, the conditions under which centrifugal compressors have to operate, vary significantly from case to case. Gas pipeline compressors, for example, may feature discharge pressures well over 100 bar. In other fields of application, like gas injection, which is used to enhance oil recovery, this quantity can reach considerably higher values. Here, discharge pressures over 600 bar and gas densities over 300 kg/m3 are not uncommon. During the last several decades, comprehensive research was conducted on the impact of high pressure operating conditions on the vibrational behavior of centrifugal compressor wheels. Nowadays, it is well-known that an increase in gas pressure levels leads to a more pronounced interaction between the side cavities and the impeller, which results in a frequency shift of the acoustic and structural modes. For the safe operation of compressors, it is necessary to predict these coupled natural frequencies accurately. The state-of-the-art approach to achieve this objective is the finite element method. While this technique provides high-quality results, the simulation of acousto-mechanical systems is still a time-consuming process that incurs high computational costs. Therefore, finite element models are, in this case, not suitable for probabilistic studies, sensitivity analyses, and comprehensive simulations of the full operating range of the compressor. In 2013, Magara proposed a simplified model based on an annular plate between two cylindrical cavities to solve this problem. While this method reduces the required computational effort significantly, its use is limited to platelike impellers. The authors of the current paper propose a more generalized method to overcome the challenges mentioned above. It uses the uncoupled structural and acoustic modes of the actual impeller and side cavities in a modal superposition to approximate the natural frequencies of the coupled acousto-mechanical system. In this way, the intended design geometries of the impeller and side cavities are considered while maintaining the advantages of Magara’s model regarding the computational effort. In a numerical study, Magara’s method and the generalized model are applied to different systems of increasing complexity. The investigation starts with a simple annular plate in a cylindrical cavity and ends with an actual compressor impeller. At every complexity level, the results of both approaches are compared to a finite element analysis. Moreover, measurement data of a simplified rotor in a cylindrical cavity is used to validate the numerical models. Finally, the paper concludes with a discussion of the limitations and benefits of all employed numerical methods.

Tuned liquid column gas dampers (TLCGD) show excellent energy and vibration absorbing capabilities appropriate for earthquake engineering. The objective of this work is to introduce a new concept of coupled tuned liquid column gas dampers which allow an extended field of applications. In the proposed configuration, several absorbers are connected in a spatial chain generating a multi-degree of freedom damper system which can be tuned to a selected number of structural modes. Because all absorbers vibrate at different natural frequencies simultaneously, the proposed design represents an improvement over conventional TLCGD which can only be tuned to a single structural mode. Another benefit of multi-mode tuning is the significant increase in active liquid mass compared to other damping devices. For a most effective vibration reduction, a numerical tuning process in state space will deliver the free system natural frequency and damping ratio of all absorbers. The capabilities of the new damping device are demonstrated on a simple laboratory structure that is investigated numerically and experimentally. The results illustrate the excellent energy dissipating properties of the proposed setup and emphasize the adequacy of coupled TLGCD for base-isolated structures. A notable benefit of such types of dampers lies in their lack of moving mechanical parts, their cheap and easy implementation into civil engineering structures and low maintenance costs. Possible modifications with respect to natural frequency and even of the damping properties can be performed with little additional expenses, and the additional weight due to the liquid mass may be used as a possible reservoir, e.g., for fire fighting. Apart from that, they exhibit a performance that is comparable to tuned mass dampers.

A new viewpoint on the mechanism of transonic single-degree-of-freedom flutter

In order to study the mechanical characteristics of the linear switched reluctance motors (LSRMs), modal analysis and linear static structure analysis of a LSRM is investigated in this paper. Basic theories of modal analysis and linear static structure analysis are stated. The structure of the LSRM is illustrated. And the modal analysis and linear static structure analysis by finite element method are demonstrated based on ANSYS workbench. The first six natural frequencies with the corresponding mode shapes and the deformation of the LSRM are obtained. The results show that the first six natural frequencies of the LSRM range from 2244.7 Hz to 4033.1 Hz, the deformation of the LSRM is within 0.0020546 mm under the largest normal force, and the easiest deformation part of the LSRM is the mover platform. The results verify that the mechanical design of the LSRM is reasonable.

Beam-like structures are the most common components in real engineering, while single side damage is often encountered. In this study, a numerical analysis of single side damage in a free-free beam is analysed with three different finite element models; namely solid, shell and beam models for demonstrating their performance in simulating real structures. Similar to experiment, damage is introduced into one side of the beam, and natural frequencies are extracted from the simulations and compared with experimental and analytical results. Mode shapes are also analysed with modal assurance criterion. The results from simulations reveal a good performance of the three models in extracting natural frequencies, and solid model performs better than shell while shell model performs better than beam model under intact state. For damaged states, the natural frequencies captured from solid model show more sensitivity to damage severity than shell model and shell model performs similar to the beam model in distinguishing damage. The main contribution of this paper is to perform a comparison between three finite element models and experimental data as well as analytical solutions. The finite element results show a relatively well performance.

In this paper, a novel PARAllel FACtor (PARAFAC) decomposition based Blind Source Separation (BSS) algorithm is proposed for modal identification of structures equipped with tuned mass dampers. Tuned mass dampers (TMDs) are extremely effective vibration absorbers in tall flexible structures, but prone to get de-tuned due to accidental changes in structural properties, alteration in operating conditions, and incorrect design forecasts. Presence of closely spaced modes in structures coupled with TMDs renders output-only modal identification difficult. Over the last decade, second-order BSS algorithms have shown significant promise in the area of ambient modal identification. These methods employ joint diagonalization of covariance matrices of measurements to estimate the mixing matrix (mode shape coefficients) and sources (modal responses). Recently, PARAFAC BSS model has evolved as a powerful multi-linear algebra tool for decomposing an nth order tensor into a number of rank-1 tensors. This method is utilized in the context of modal identification in the present study. Covariance matrices of measurements at several lags are used to form a 3rd order tensor and then PARAFAC decomposition is employed to obtain the desired number of components, comprising of modal responses and the mixing matrix. The strong uniqueness properties of PARAFAC models enable direct source separation with fine spectral resolution even in cases where the number of sensor observations is less compared to the number of target modes, i.e., the underdetermined case. This capability is exploited to separate closely spaced modes of the TMDs using partial measurements, and subsequently to estimate modal parameters. The proposed method is validated using extensive numerical studies comprising of multi-degree-of-freedom simulation models equipped with TMDs, as well as with an experimental set-up.

At medium frequency range (MFR), numerous vibroacoustic problems deal with systems of large number of mixes where classical modal methods must cope with resolution of huge linear system. In the idea of reducing theses models a mode hybridization method proposed in [1] gives one equivalent mode when modes of the complex structure are known. Another method [2],[4] based on reducing size of dynamic model in the MFR, gives a prediction of the vibration response of a structure by using a limited number of modes of the homogeneous master structure in order to interpolate the response of modes of whole structure with attached heterogeneities. The method presented here gives a prediction of plate vibration using a wave decomposition instead of modal calculation. It allows one to consider homogeneous structure of complex geometry without making mode calculations before. The corresponding wave function of the Fourier transform of the displacement is approximated the best by achieving the extremalization of the Hamiltonian of the plate. Quadratic velocity of the structure is derived and the convergence study of the method shows that global quantities involving frequency average can be obtained with reduced models of small size

Mode structures and excitation conditions for the low-frequency modes (LFMs) have been investigated in experimental advanced superconducting tokamak (EAST) plasmas with q min ⩽ 2. Two different stages/categories of the LFM instabilities are observed during the oscillation of annular/central collapse events: (I) the upward sweeping frequency of LFMs; (II) the upward frequency jumpsof LFMs. The annular/central events are triggered by the m/n = 2/1 double tearing modes with different q-profiles, while the LFMs are characterized by higher mode numbers m/n = 4/2, 6/3, …, where m and n are the poloidal and toroidal mode numbers, respectively. The maximum radial coverage of the LFMs is located in the annular region of 1.97 2.07 m with the normalized minor radius 0.2 0.4, while the higher-frequency (or upward sweeping frequency) branch is more localized to the radial position of 2 2.02 m (q min). The frequency characteristics of upward sweeps or upward jumps of the LFMs are mainly attributed to the change in the q-profile, e.g. the upward sweeping frequency in stage I is caused by q min decreasing. Accordingly, the linear wave properties of LFMs in EAST with weak/reversed magnetic shear are studied numerically and analytically based on a general fishbone-like dispersion relation. Without considering the contribution of energetic ions, it is shown that the LFM with Alfvénic polarization is an MHD-unstable kinetic ballooning mode with frequency of the order of the ion diamagnetic drift frequency. Several important factors for the excitation of LFM instability are analyzed: (1) the role of energetic ions is unimportant, and the LFMs can be excited under the two conditions of with/without energetic ions; (2) the higher τ = T e/T i with larger η i = L ni /L Ti are required, namely the normalized pressure gradient α ∝ (1 + τ)(1 + η i) should be large enough to overcome the stability effect of finite field line bending; (3) the weak/reversed shear q-profile with q min ⩽ 2 and suitable S ≡ (r/q)(q″)1/2 are required.

Configurations with nearby multiple resonant surfaces have broad spectra of linearly unstable coupled tearing modes with dominant high poloidal mode numbers m. This was recently shown for the case of multiple q=1 resonances [Bierwage et al., Phys. Rev. Lett. 94 65001 (2005)]. In the present work, similar behavior is found for double tearing modes (DTM) on resonant surfaces with q⩾1. A detailed analysis of linear instability characteristics of DTMs with various mode numbers m is performed using numerical simulations. The mode structures and dispersion relations for linearly unstable modes are calculated. Comparisons between low- and higher-m modes are carried out, and the roles of the inter-resonance distance and of the magnetic Reynolds number SHp are investigated. High-m modes are found to be destabilized when the distance between the resonant surfaces is small. They dominate over low-m modes in a wide range of SHp, including regimes relevant for tokamak operation. These results may be readily applied to configurations with more than two resonant surfaces.

The non-linear evolution of the interchange instability with a poloidal mode number m = 1 and a toroidal mode number n = 1 is investigated in a straight helical system with the ιh= 1 surface in a vacuum magnetic configuration like that of Heliotron E, where ιh, denotes the rotational transform. In the non-linear reduced MHD equations derived by using the stellarator expansion, a thermal-diffusion term and an external-heating term are added to include rapid energy loss and continued heating during the non-linear evolution of the instability. They are solved as an initial and boundary value problem for resistive cylindrical plasmas, in order to explain the sawtooth-like oscillation of electron temperature and density observed in the high-beta current-free Heliotron E experiment. – Without the heating term, the m = 1 pressure-driven mode becomes unstable and changes the pressure profile significantly for central beta, β(0) ≳ 2%, with an initial profile of p(r) ∝ (1 −(r/a)2)2. When a surface-like current increases around the ιh, = 1 resonant surface, because of the diamagnetic effect of the deformed pressure profile, reconnections of the magnetic field lines occur, and the resulting flux surfaces show an m = 2 mode structure. In the case of both thermal diffusion and external heating, the pressure behaviour oscillates between peaked profiles unstable against the m = 1 pressure-driven mode and broad profiles that are stable against it. – These results are consistent with the experimental data.

Elastic support is a commonly used method of vibration isolation and vibration reduction. The research on the mode of underground arch structure with elastic support can improve the battlefield survival ability of personnel and equipment in underground protection engineering. In this paper, the ANSYS/LS-DYNA nonlinear explicit dynamic finite element analysis program is used to numerically simulate the mode of the elastically supported underground arch structure, and the natural vibration circle frequency variation curve and mode diagram of the structure are obtained. The results show that the elastic support reduces the natural frequency of the structure, prolongs the natural vibration period of the structure, and has a good vibration isolation and vibration reduction effect; the critical stiffness coefficient causes the first and second modes of the structure to be easily excited at the same time, which should be avoided in structural design.

This paper investigates the flow past a flexible splitter plate attached to the rear of a fixed circular cylinder at low Reynolds number 150. A systematic exploration of the plate length ( $L/D$ ), flexibility coefficient ( $S^{*}$ ) and mass ratio ( $m^{*}$ ) reveals new laws and phenomena. The large-amplitude vibration of the structure is attributed to a resonance phenomenon induced by fluid–structure interaction. The modal decomposition indicates that resonance arises from the coupling between the first and second structural modes, where the excitation of the second structural mode plays a critical role. Due to the combined effects of added mass and periodic stiffness variations, the two modes become synchronised, oscillating at the same frequency while maintaining fixed phase difference $\pi /2$ . This further results in the resonant frequency being locked at half of the second natural frequency, which is approximately three times the first natural frequency. A reduction in plate length and an increase in mass ratio are both associated with a narrower resonant locking range, while a higher mass ratio also shifts this range towards lower frequencies. A symmetry-breaking bifurcation is observed for cases with $L/D\leqslant 3.5$ , whereas for $L/D=4.0$ , the flow remains in a steady state with a stationary splitter plate prior to the onset of resonance. For cases with a short flexible plate and a high mass ratio, the shortened resonance interval causes the plate to return to the symmetry-breaking stage after resonance, gradually approaching an equilibrium position determined by the flow field characteristics at high flexibility coefficients.

During the 9 March 2018 event with two consecutive interplanetary shocks compressing the dayside magnetosphere, the azimuthal mode structure and frequency spectrum of ultra low frequency magnetic pulsations are resolved using a cross‐spectral analysis based on high‐fidelity multi‐probe Magnetospheric Multiscale Mission (MMS) magnetometer data. The results based on the MMS 4 and MMS 3 pair of measurements show that shock arrival leads to low mode () magnetic fluctuations in the Pc4‐5 regimes, and smaller spatial scale fluctuations implied by the dominant high mode numbers are observed after both shock signatures hit and passed the magnetosphere. Detailed evolution of the mode structure is also shown for the first shock to reveal the development of high mode structure from a bump‐on‐tail distribution at to a dominant peak at in about 10 min. In addition, an interesting change of sign in from negative to positive is observed as MMS crosses ∼11 MLT pre‐noon, which is consistent with the picture of wave generation by dayside magnetopause compression and then anti‐sunward propagation. For both shocks, the contribution of higher frequency waves (Pc‐4 range compared with Pc‐5) to the total wave power is found to be negligible before and after the shock impact, but it becomes more significant during the shock impact.

Mode structures during MHD events in the TST-2 spherical tokamak are revealed by means of singular value decomposition. Soft X-ray and magnetic coil signals show growths of modes with toroidal mode numbers n = 1 and 2. Slow evolution of the soft X-ray profile indicates beginning of localized crash. Occurrence of these events is correlated with the spatial gradient of the soft X-ray profile, suggesting that these events are driven by the pressure gradient. Increases in impurity ion temperatures (CIII and OV) are observed. These are positively correlated with the increment of the plasma current during these events.

This research presents the numerical analysis of the triply coupled flap-wise, cord-wise and torsional vibrations of flexible rotating blades. Euler-Bernoulli bending and St. Venant torsion beam theories are considered to derive the governing differential equations of motion. Based on Finite Element Methodology (FEM), the cubic "Hermite" shape functions are implemented where the solution of the equations results in a linear engine problem. Then, the Dynamic (frequency dependent) Trigonometric Shape Functions (DISF's) for beam's uncoupled displacements are derived. The application of the Dynamic Finite Element (DFE) approach to the solution of the governing equations is then presented. The DFE formulation, based on the weighted residual method and the DTSF's results in a nonlinear engine problem representing eigenvalues and engine modes of the system. The applicability of the DFE method is then demonstrated by illustrative examples, where a Wittrick-Williams root counting technique is used to find the system's natural frequencies. The DFE approach, an intermediate method between FEM and "Exact" formulation, is characterized by higher convergence rates, and can be advantageously used when multiple natural frequencies and/or higher modes of beam-like structures are to be evaluated.