Single Mode Shape Research Articles

Multiwave Vibrations in Mistuned Cascades With Tip-Shroud Friction

Abstract Measurements taken during aero engine tests and in the field showed that flutter vibrations of shrouded blades can feature rich wave content (multiwave flutter vibrations). In a previous work, we demonstrated that this behavior can be explained by the nonlinear interaction of aeroelastically unstable traveling wave modes. The resulting vibrations are quasi-periodic. In this work, we show that the nonlinear modal interaction is not strictly needed, but actually mistuning alone can explain the multiwave form of flutter vibrations. The resulting vibrations are periodic and dominated by only a single mode shape of the mistuned system. However, unrealistically high mistuning intensities are needed to obtain significant contributions of multiple wave forms under the considered strong inter-blade coupling. Thus, we conclude that mistuning cannot explain the rich wave content observed in the measurements. Moreover, mistuning tends to hamper the nonlinear modal interactions and, thus, the occurrence of quasi-periodic multiwave flutter vibrations. This implies that intentional mistuning is not only useful to stabilize flutter but might also play an important role in developing flutter-tolerant blade designs.

Vortex-induced vibration of elastically connected multiple circular cylinders in a side-by-side arrangement in steady flow

Vortex-induced vibration (VIV) of multiple circular cylinders elastically connected together in a side-by-side arrangement subject to steady flow is investigated numerically at a low Reynolds number of 150 and a mass ratio of 2. Simulations are conducted for two-, five- and ten-cylinder systems over a wide range of reduced velocities. The aim of the study is to identify the high-amplitude response range of the reduced velocity for the multiple degree of freedom vibration system and identify the difference between the responses of the single- and multiple-degree-of-freedom vibrations. Unlike the single cylinder case, distinct lock-in between the response frequency and any of the structural natural frequencies in a wide range of reduced velocity is not observed in the multiple-cylinder cases. Instead, the response frequency increases continuously with increasing reduced velocity. High response amplitudes are found when the response frequency is between the first and the highest modal frequencies. In a multiple-cylinder system, the single-mode response, where the vibration is dominated by one mode, can be only found in low reduced velocity range. In the single-mode branch, the dominance of a single mode shape in the response can be clearly identified except at the boundary reduced velocity between two modes. The maximum response amplitude occurs in the multiple-mode response and interaction between the vortices in the wake of the cylinders is strong when the response amplitudes are high.

Wall proximity effects on the flow past cylinder with flexible filament

The response of thin, elastic filament attached to a circular cylinder is numerically investigated under uniform flow at different wall proximities, GD (0.5, 0.6, 0.8, 1, 1.5, 2 and 2.5). The Reynolds number based on the diameter of the cylinder is 200. Numerical simulations were conducted for a range of reduced velocities Ur (3,4,5 and 6), where Ur = U∞fnD which appropriately captures the synchronization range of vortex-induced vibration(VIV) of the structure. Ur is varied by changing the density of the material. The force coefficients, the frequency of shedding/flapping are observed to be the function of GD and Ur. The positive vortices diffuses faster as GD decreases. The skewness in the amplitude attained by the flapping filament on either side of the cylinder causes unsteady shedding of vortices. Filament flaps in frequency that is closer to its natural frequency at higher GD and at lower GD, the frequency of flaps for all Ur is similar. For lower gap ratios, 0.5 and 0.6, the modulations in flapping amplitude is observed. The modulation is predominant as the Ur increases. During the growth and damping phase of the amplitudes in each modulation cycle, the vortex shedding is observed to be oblique towards the wall. The filament flaps in single mode shape. The frequency of the flapping depends on the Ur and GD. The frequency of the flapping is in resonance with shedding frequency.

A comprehensive analysis on the discretization method of the equation of motion in piezoelectrically actuated microbeams

In many of microdevices a part of a microbeam is covered by a piezoelectric layer. Depend on the application a DC or AC voltage is applied between upper and lower side of the piezoelectric layer. A common method in many of previous works for evaluating the response of these structures is discretizing by Galerkin method. In these works often single mode shape of a uniform microbeam i.e. the microbeam without piezoelectric layer has been used as comparison function, and so the convergence of the solution has not been verified. In this paper the Galerkin method is used for discretization, and a comprehensive analysis on the convergence of solution of equation that is discretized using this comparison function is studied for both clamped-clamped and clamped-free microbeams. The static and dynamic solution resulted from Galerkin method is compared to the modal expansion solution. In addition the static solution is compared to an exact solution. It is denoted that the required numbers of uniform microbeam mode shapes for convergence of static solution due to DC voltage depends on the position and thickness of deposited piezoelectric layer. It is shown that when the clamped-clamped microbeam is coated symmetrically by piezoelectric layer, then the convergence for static solution may be obtained using only first mode. This result is valid for clamped –free case when it is covered by piezoelectric layer from left clamped side to the right. It is shown that when voltage is AC then the number of required uniform microbeam shape mode for convergence is much more than the number of required mode in modal expansion due to the dynamic effect of piezoelectric layer. This difference increases by increasing the piezoelectric thickness, the closeness of the excitation frequency to natural frequency and decreasing the damping coefficient. This condition is often indefeasible in microresonator system. It is concluded that discreitizing the equation of motion using one mode shape of uniform microbeam as comparison function in many of previous works causes considerable errors.

Output-only analysis of structures with closely spaced poles

In the framework of the operational modal analysis, several approaches have been developed for estimating the modal parameters, i.e., natural frequencies, damping ratios, and mode shapes. Specifically, a technique capable to evaluate the biased (i.e., unscaled by a constant or an almost constant function) frequency response functions, FRFs, has been proposed. Assuming that only the responses of the structure are disposable, the technique allows one to estimate biased FRFs starting from the power spectral densities, PSDs, and applying the Hilbert transform. This paper deals with the estimates of the modal analysis parameters mentioned above. It is possible to obtain each single mode shape, from the singular vectors achieved by applying the singular value decomposition to the FRF matrix evaluated at the spectral line corresponding to the selected natural frequency. A special attention will be devoted to structures with coupled modes, i.e., closely spaced modes. Once the FRFs have been obtained, the natural frequencies and damping ratios could be achieved either in the frequency domain or in the time domain. Experimental tests, carried out on beams, plates and on the AB-204 helicopter blade, will be presented.

Optimal seismic analysis of degrading planar frames using a weighted energy method to associate inelastic mode shapes: Part I optimal parameters

The objective of this paper is to compute three optimal parameters that are subsequently used to formulate the pre-yielded and post-yielded portions of an equivalent single degree of freedom system (E-SDOF) that is used to predict the seismic target demands in planar frames. The procedure uses an optimal number of inelastic mode shapes from a structure’s capacity (pushover) curve to account for any significant higher-mode effects (HME) and predict the inelastic demands. Using a variant inertial load pattern, weighted energy gradients under the capacity curve are used to define an optimal ductility parameter, which is in turn used to combine the inelastic (and elastic) mode shapes into a single mode shape. This is used to determine the pre-yielded portion of the E-SDOF system, where the post-yielded portion is determined using an inelastic modes parameter. The procedure also utilizes a reduction factor parameter to adjust the one-second spectral acceleration demand. The three optimal parameters are established using several buildings, whose responses are generally influenced by specific material strain hardening and plastic flow rules, and by the dissipated energy due to the yielding of the individual members. Using this methodology, the predicted target displacement demands are very reasonably predicted when compared to a nonlinear time-history analysis, which enables the parameters to later be used in the formulation of other buildings’ E-SDOF systems.

Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.