Showing papers on "Natural frequency published in 2001"

Non-linear vibrations and stability of an axially moving beam with time-dependent velocity

Abstract: Non-linear vibrations of an axially moving beam are investigated. The non-linearity is introduced by including stretching effect of the beam. The beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity. Approximate solutions are sought using the method of multiple scales. Depending on the variation of velocity, three distinct cases arise: (i) frequency away from zero or two times the natural frequency, (ii) frequency close to zero, (iii) frequency close to two times the natural frequency. Amplitude-dependent non-linear frequencies are derived. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

Free vibrations of two side-by-side cylinders in a cross-flow

Abstract: Free vibrations of two side-by-side cylinders with fixed support (no rotation and displacement) at both ends placed in a cross-flow were experimentally investigated. Two fibre-optic Bragg grating sensors were used to measure the dynamic strain, while a hot wire and flow visualization were employed to examine the flow field around the cylinders. Three T/d ratios, 3.00, 1.70 and 1.13, were investigated, where T is the centre-to-centre cylinder spacing and d is the diameter; they give rise to three different flow regimes. The investigation throws new light on the shed vortices and their evolution. A new interpretation is proposed for the two different dominant frequencies, which are associated with the narrow and the wide wake when the gap between the cylinders is between 1.5 and 2.0 as reported in the literature. The structural vibration behaviour is closely linked to the flow characteristics. At T/d = 3:00, the cross-flow root-mean-square strain distribution shows a very prominent peak at the reduced velocity Ur ≈ 26 when the vortex shedding frequency fs, coincides with the third-mode natural frequency of the combined fluid–cylinder system. When T/d < 3:00, this peak is not evident and the vibration is suppressed because of the weakening strength of the vortices. The characteristics of the system modal damping ratios, including both structural and fluid damping, and natural frequencies are also investigated. It is found that both parameters depend on T/d. Furthermore, they vary slowly with Ur, except near resonance where a sharp variation occurs. The sharp variation in the natural frequencies of the combined system is dictated by the vortex shedding frequency, in contrast with the lock-in phenomenon, where the forced vibration of a structure modifies the vortex shedding frequency. This behaviour of the system natural frequencies persists even in the case of the single cylinder and does not seem to depend on the interference between cylinders. A linear analysis of an isolated cylinder in a cross-flow has been carried out. The linear model prediction is qualitatively consistent with the experimental observation of the system damping ratios and natural frequencies, thus providing valuable insight into the physics of fluid–structure interactions.

Feedback stability limits for active isolation systems with reactive and inertial actuators

Abstract: Some of the compromises inherent in using a passive system to isolate delicate equipment from base vibration can be avoided using fully active skyhook damping. Ideally, a secondary force, which is made proportional to the absolute equipment velocity by a feedback controller, acts only on the equipment and so the response of the system under control, between the secondary force input and the collocated velocity output, i.e., the plant response, is proportional to the driving point mobility of the mounted equipment. The frequency response of the plant is guaranteed to have a positive real part under these ideal conditions, and so the feedback system is unconditionally stable for any positive real feedback gain. In practice, the actuator generating the secondary force must either react off the base structure or an inertial mass. In both of these cases the plant response is no longer guaranteed to be positive real and so the control system may become unstable at high gains. Expressions for the overall plant responses are derived for both of these arrangements, in terms of the dynamic response of the individual parts of the isolation system. When using a soft mount, the stability of the reactive system is found to be surprisingly tolerant of the additional contributions to the plant response from the reactive force. In order for the inertial system to be stable with a high feedback gain, however, the natural frequency of the actuator must be well below the natural frequency of the equipment on the mounts. Experimentally measured plant responses are compared with those predicted from theory for both types of actuator and the performance of practically implemented feedback controllers is discussed.

Linear and nonlinear frequency shifts in acoustical resonators with varying cross sections

Abstract: The frequency response of a nonlinear acoustical resonator is investigated analytically and numerically. The cross-sectional area is assumed to vary slowly but otherwise arbitrarily along the axis of the resonator, such that the Webster horn equation provides a reasonable one-dimensional model in the linear approximation. First, perturbation theory is used to derive an asymptotic formula for the natural frequencies as a function of resonator shape. The solution shows that each natural frequency can be shifted independently via appropriate spatial modulation of the resonator wall. Numerical calculations for resonators of different shapes establish the limits of the asymptotic formula. Second, the nonlinear interactions of modes in the resonator are investigated with Lagrangian mechanics. An analytical result is obtained for the amplitude-frequency response curve and nonlinear resonance frequency shift for the fundamental mode. For a resonator driven at its lowest natural frequency, it is found that whether hardening or softening behavior occurs depends primarily on whether the nonlinearly generated second-harmonic frequency is greater or less than the second natural frequency of the resonator. A fully nonlinear one-dimensional numerical code is used to verify the analytical result.

In‐plane vibrations of shear deformable curved beams

Abstract: This paper presents the exact dynamic stiffness matrix for a circular beam with a uniform cross-section. The stiffness matrix is frequency dependent, and the natural frequencies are those that cause the matrix to become singular. Using this matrix the exact natural frequencies of circular beams with various boundary conditions are calculated and compared with available results in the literature. Copyright © 2001 John Wiley & Sons, Ltd.

Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity

Abstract: In this study, the non-linear transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with variable velocity are investigated. The pipe is on fixed supports and the immovable end conditions result in the extension of the pipe during vibration and hence introduce further non-linear terms to the equation of motion. The velocity is assumed to be a harmonic function about a mean velocity. These systems experience a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved analytically by direct application of the method of multiple scales (a perturbation technique). Principal parametric resonance cases are investigated in detail. Non-linear frequencies are derived depending on amplitude. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.

A Piezoelectric Valve-Less Pump-Dynamic Model

Abstract: A complete dynamic model for the simulation of the valve-less piezoelectric pump performance is presented, In this model the piezoelectric action is considered as a periodic force acting on a pumping membrane. The natural frequency of the pump is calculated as well as its performance as a function of the driving frequency, The effect of the deviation of the driving frequency from the natural frequency on the pump performances is clearly shown. Also, it is demonstrated that the effect of the liquid mass in the pump nozzles on the natural frequency of the system is very high owing to the high acceleration of the fluid in the nozzles. Comparison with experiments shows a very good agreement with a minimal number of adjusting parameters.

Adaptive tuning for vibrational gyroscopes

Abstract: We propose an adaptive control scheme to tune the natural frequency of a microelectromechanical (MEMS) resonator for a vibrational gyroscope to a desired frequency. This is an attractive alternative to the phase locked loop, since it introduces feedback, which can reduce the effects of imprecise fabrication. In particular, we use adaptive control to set the natural frequency of the drive axis.

Free vibrations of two side-by-side cylinders in a cross-flow

Abstract: Free vibrations of two side-by-side cylinders with xed support (no rotation and displacement) at both ends placed in a cross-flow were experimentally investigated. Two bre-optic Bragg grating sensors were used to measure the dynamic strain, while a hot wire and flow visualization were employed to examine the flow eld around the cylinders. Three T=d ratios, 3.00, 1.70 and 1.13, were investigated, where T is the centre-to-centre cylinder spacing and d is the diameter; they give rise to three dierent flow regimes. The investigation throws new light on the shed vortices and their evolution. A new interpretation is proposed for the two dierent dominant frequencies, which are associated with the narrow and the wide wake when the gap between the cylinders is between 1.5 and 2.0 as reported in the literature. The structural vibration behaviour is closely linked to the flow characteristics. At T=d =3 :00, the cross-flow root-mean-square strain distribution shows a very prominent peak at the reduced velocity Ur 26 when the vortex shedding frequency fs, coincides with the third-mode natural frequency of the combined fluid{cylinder system. When T=d <3:00, this peak is not evident and the vibration is suppressed because of the weakening strength of the vortices. The characteristics of the system modal damping ratios, including both structural and fluid damping, and natural frequencies are also investigated. It is found that both parameters depend on T=d. Furthermore, they vary slowly with Ur, except near resonance where a sharp variation occurs. The sharp variation in the natural frequencies of the combined system is dictated by the vortex shedding frequency, in contrast with the lock-in phenomenon, where the forced vibration of a structure modies the vortex shedding frequency. This behaviour of the system natural frequencies persists even in the case of the single cylinder and does not seem to depend on the interference between cylinders. A linear analysis of an isolated cylinder in a cross-flow has been carried out. The linear model prediction is qualitatively consistent with the experimental observation of the system damping ratios and natural frequencies, thus providing valuable insight into the physics of fluid{structure interactions.

Investigation of Resonant Frequency and Amplitude of Vibrating Footing Resting on a Layered Soil System

Abstract: Influence of layering and presence of rigid boundary in the soil mass on natural frequency and resonant amplitude are studied experimentally by conducting model block vibration tests in vertical mode. Tests are conducted on different layered beds prepared in a tank using a model footing. A Lazan type mechanical oscillator is used for inducing vibration and two different materials (sand and sawdust) are used to form a layered system. In total, 180 tests are conducted in different layering combinations and different static and dynamic loading combinations, and several important observations are reported. Damping factors are found to be within 6.5% for the entire test series, which indicates that radiation damping is insignificant in the test system. It is also found that layering including layer position and thickness has significant effect on natural frequency. Observed natural frequencies are also compared with the predicted one based on static equivalent stiffness; encouraging agreement between observed and predicted values are found.

Broadband piezoelectric shunts for passive structural vibration control

Abstract: This paper reports an advanced passive piezoelectric shunt damping technique called broadband piezoelectric shunting. The shunt circuit for the broadband shunting is built with a reactance neutralizing circuit, which is designed with operational amplifier circuits to maximize the parallel reactance of the shunt circuit over a wide frequency band. It is different from the earlier shunt circuits, either used for the single- or multiple-mode shunting, which were built with inductors or simulated inductor circuits. In these earlier shunt circuits the anti-resonant frequency of the shunt circuit, with the inclusion of the piezoelectric transducer, was tuned with the inductor to a specific frequency to match with the natural frequency of the mode to be controlled. We first describe how the concept of the broadband shunt circuit was conceived and developed. We next report the shunt damping experiments performed on three difference structures: a thin aluminum cantilever beam, a two-wing aluminum cantilever beam, and a small-scale thermoplastic composite fairing. We have successfully demonstrated broadband amplitude reduction of structural vibration modes over a wide frequency band using the broadband shunt-damping technique. More detailed experimental results will be presented.

Natural Frequencies of Rotating Uniform Beams with Coriolis Effects

Abstract: A Dynamic Finite Element (DFE) for vibrational analysis of rotating assemblages composed of beams is presented in which the complexity of the acceleration, due to the presence of gyroscopic, or Coriolis forces, is taken into consideration. The dynamic trigonometric shape functions of uncoupled bending and axial vibrations of an axially loaded uniform beam element are derived in an exact sense. Then, exploiting the Principle of Virtual Work together with the nodal approximations of variables, based on these dynamic shape functions, leads to a single frequency dependent stiffness matrix which is Hermitian and represents both mass and stiffness properties. A Wittrick-Williams algorithm, based on a Sturm sequence root counting technique, is then used as the solution method. The application of the theory is demonstrated by two illustrative examples of vertical and radial beams where the influence of Coriolis forces on natural frequencies of the clamped-free rotating beams is demonstrated by numerical results.

Electro-mechanical impedance method for crack detection in metallic plates

Abstract: As a nondestructive evaluation technology, the EM impedance method allows us to identify the structural dynamics directly through in-situ active piezoelectric sensors. Previous work performed on 1-D steel beams structures shown through both theoretical analysis and experimental results that E/M impedance (or admittance) spectrum is a direct identifier of structural dynamics. The scope of presented work was to extend the positive results obtained for 1-D structure onto 2-D structures. Experiment analysis of 1-D and2-D structures has shown that E/M impedance (or admittance) spectrum accurately identifies the natural frequency spectrum of the specimens. Theoretical analysis was performed for particular boundary conditions to model the experimental set-up. Experiments were conducted on simple specimens in support of the theoretical investigation, and on thin-gauge aluminum plates to illustrate the method's potential. The number of specimens was sufficient to form a statistical data set. The aging aircraft panel was instrumented with piezoelectric active sensors and the spectrum of natural frequencies was measured at high frequency range. The changing of the spectrum due to presence of local small crack was noticed.

Passive and active damping characteristics of smart electro-rheological composite beams

Abstract: The flexural vibration of laminated composite beams sandwiched by two electro-rheological (ER) fluid layers has been investigated to maximize the possible damping capacity. The equations of motion are derived for flexural vibrations of symmetrical, multilayer laminated beams with double ER fluid layers. The damped natural frequency, damping ratio, and modal damping of the first bending mode are calculated by means of an iterative complex eigensolution method. For the validation of the finite element modeling method using viscoelastic theory, the predicted dynamic properties are compared to the experimental results and they show a good agreement. The impact of varying the stacking sequence of beams on the stiffness and the damping properties is studied. The active and passive damping ratio and modal damping for the first bending of the beam are calculated for various fiber orientations and various electric fields. When an electric field is applied, the active damping is more effective in a flexible laminated beam than in a stiff beam. The laminated beams have the best performance at the electric field of 0.4 kV mm-1 for damping, and 1.0 kV mm-1 for stiffness. A proper choice of fiber orientation angle and the electric field can maximize the structural damping and/or stiffness. The design strategy of a laminated composite under flexural vibrations incorporating an ER fluid is addressed.

Stability and Natural Vibrations of Shells with Variable Geometric and Mechanical Parameters

Abstract: The bifurcation stability and natural vibrations of shells of revolution with variable geometric and mechanical parameters are studied by using refined models and the variational–difference method. The qualitative and quantitative effects of the external geometry, material properties, and design features on the critical load and natural frequency are evaluated

Apparently first closed‐form solutions for inhomogeneous vibrating beams under axial loading

Abstract: Closed‐form solutions are presented for fundamental natural frequencies of inhomogeneous vibrating beams under axially distributed loading. The mode shape is postulated as coinciding with the static deflection of the associated homogeneous beam without distributed axial loading. Then the inverse problem of determining the stiffness and mass density distributions, producing the above mode shape, is solved. To describe these variations, the family of polynomial functions is used. Several sets of boundary conditions are considered. It is shown that the natural frequency vanishes when the intensity of the axially distributed loading equals the critical buckling value. A linear relationship is established between the square of the natural frequency and the load ratio for all reported sets of boundary conditions, in contrast to uniform beams where the exact linear relationship holds for columns with simply supported and/or sliding ends.

On the Dynamics of a Harmonic Oscillator Undergoing Impacts with a Vibrating Platform

Abstract: This paper deals with a model for repeated impacts of a massattached to a spring with a massive, sinusoidally vibrating table. Thismodel has been studied in an attempt to understand the cantilever-sampledynamics in atomic force microscopy. In this work, we have shown thatfor some values of the frequency of the vibrating table, there arecountably many orbits of arbitrarily long periods and the system issensitive to the initial conditions with which the experiments areconducted. We have also shown existence of complex dynamics in the caseswhen the natural frequency of the spring-mass system is very low; andwhen it is the same as the oscillating frequency of the table.