Natural Frequency Of System Research Articles

Suppressing the nonlinear vibrations of a compressor blade via a nonlinear saturation controller

A nonlinear saturation controller (NSC) is applied in this work to reduce the oscillations of a rotating blade dynamical system running at unsteady rotating speed. The controller is coupled quadratically to the main system by designing its natural frequency to be one half of the main system natural frequency. This is done to setup an energy bridge between them to make use of saturation phenomenon. That phenomenon is advantageous when the excitation force increases; the whole energy in the main system is channeled to the controller. The two system modes of vibrations are found to be linearly coupled powerfully, so the controller is applied only to the first mode and, consequently, the second mode tracks it. The multiple scales perturbation technique (MSPT) is adopted to derive the steady state equations that describe the modulations of amplitudes and phases of the system before and after control. Then, a stability analysis is achieved via Lyapunov’s indirect method to determine the stable and unstable solutions depending on the real parts of the Jacobian matrix eigenvalues. Time history and different response curves of the controlled system are included for showing the controller effect. Eventually, validation curves and comparison with previously published work are included.

A curved edge diffraction-utilized displacement sensor for spindle metrology.

This paper presents a new dimensional metrological sensing principle for a curved surface based on curved edge diffraction. Spindle error measurement technology utilizes a cylindrical or spherical target artifact attached to the spindle with non-contact sensors, typically a capacitive sensor (CS) or an eddy current sensor, pointed at the artifact. However, these sensors are designed for flat surface measurement. Therefore, measuring a target with a curved surface causes error. This is due to electric fields behaving differently between a flat and curved surface than between two flat surfaces. In this study, a laser is positioned incident to the cylindrical surface of the spindle, and a photodetector collects the total field produced by the diffraction around the target surface. The proposed sensor was compared with a CS within a range of 500 μm. The discrepancy between the proposed sensor and CS was 0.017% of the full range. Its sensing performance showed a resolution of 14 nm and a drift of less than 10 nm for 7 min of operation. This sensor was also used to measure dynamic characteristics of the spindle system (natural frequency 181.8 Hz, damping ratio 0.042) and spindle runout (22.0 μm at 2000 rpm). The combined standard uncertainty was estimated as 85.9 nm under current experiment conditions. It is anticipated that this measurement technique allows for in situ health monitoring of a precision spindle system in an accurate, convenient, and low cost manner.

Exact free vibration analysis for mechanical system composed of Timoshenko beams with intermediate eccentric rigid body on elastic supports: An experimental and analytical investigation

The purpose of this article is to investigate the changes in the magnitude of natural frequencies and their associated modal shapes of Timoshenko beam with respect to different system design parameters. This beam includes an intermediate extended eccentric rigid mass mounted on two elastic segments. The equilibrium equations which govern the transverse and rotational motions are derived. The application of the developed system frequency equation is demonstrated by several illustrative examples. Several end and intermediate conditions are considered. The influence of, rotary inertia, shear deformation, axial load, eccentric mass and elastic segments step ratio on the system natural frequencies and mode shapes are conducted. Several sets of new results are presented. Comparison of the present model results with the experimental data for shaft integrated with intermediate rigid mass demonstrates the accuracy of the analysis in practical applications. The present model is valid for several industrial applications, such as mechanical, structural, naval and for wider range of applications.

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.

Mass ratio effect on vortex induced vibration of a flexibly mounted circular cylinder, an experimental study

The effect of mass ratio (m∗=the mass of oscillating body/the mass of displaced fluid) on vortex induced vibration of an elastically mounted rigid circular cylinder over a wide range of Reynolds numbers (1.7×104<Re<7×104) in a high damping system is studied in this paper. The cylinder is limited to a transverse oscillation and is carried inside a 14m long water channel for constant velocities. The amplitude of response depends on various parameters such as mass ratio, damping ratio, natural frequency of system and Reynolds number. Here we considered three different circular cylinders with the same diameter and length, but distinct masses (m∗=1.6, 2.3 and 3.4). The experiments carried out in a towing-tank water channel and the following results achieved: the peak amplitude of oscillation principally depends on the mass ratio and it increases with decrease of m∗. For systems with constant mass-damping parameters, by reducing the mass ratio, the maximum amplitude of oscillation remains constant, while the range of synchronization increases considerably. Higher Reynolds numbers in our experiments led to reach the same maximum amplitude of oscillation as some of the previous studies (A∗=1), even with higher mass-damping parameter (m∗ζ) in our system. The high damping in our experiments, resulted in disappearance of the lower branch in the amplitude response graphs.

Simple models for rope substructure mechanics: application to electro-mechanical lifts

Mechanical systems modelled as rigid mass elements connected by tensioned slender structural members such as ropes and cables represent quite common substructures used in lift engineering and hoisting applications. Special interest is devoted by engineers and researchers to the vibratory response of such systems for optimum performance and durability. This paper presents simplified models that can be employed to determine the natural frequencies of systems having substructures of two rigid masses constrained by tensioned rope/cable elements. The exact solution for free un-damped longitudinal displacement response is discussed in the context of simple two-degree-of-freedom models. The results are compared and the influence of characteristics parameters such as the ratio of the average mass of the two rigid masses with respect to the rope mass and the deviation ratio of the two rigid masses with respect to the average mass is analyzed. This analysis gives criteria for the application of such simplified models in complex elevator and hoisting system configurations.

Optimized re-entry input shapers

The paper deals with the design of optimized input shapers for non-vibrational control of flexible mechatronic systems. The described method is based on a combination of advantages from two approaches – precomputed control curves and on-line shapers. The strategy has two steps. Primarily, an optimized precomputed curve is found as a solution to the point-to-point control problem with respect to any requested optimization goals. Then it is transformed into an on-line shaper with the re-entry property. The resulting shaper transforms any arbitrary input signal to a non-vibrational one. In contrast to other techniques, the shaper length is not determined from the system natural frequency. The shaper can be easily modified with respect to position, velocity, acceleration or jerk control. The theoretical results are verified by experiments using a laboratory crane.

The effect of external vertical acceleration on the dynamic behaviors of a single nuclear-coupled boiling channel

This study integrates a single nuclear-coupled boiling channel model previously developed by the authors with the external force method to illustrate the qualitative dynamic behaviors of a two-phase flow system subject to external vertical accelerations. With the normal operating condition of an ABWR as a reference case, the effect of external vertical acceleration on the system transients may depend on their imposed amplitudes and frequencies. Periodic and quasi-periodic oscillations can appear in the system as the frequency ratios between two vertical waves. The parametric effects on the seismic-induced oscillations are performed in the present system without nuclear coupling. The resonance oscillations could be triggered if the external vibration frequency is equal to the system natural frequency. The strength of resonance effect may depend on the inherent stability characteristics of the initial states for both the thermal-hydraulic and nuclear-coupled boiling systems. Complex seismic-induced oscillations can be predicted in the nuclear-coupled boiling system with strong void-reactivity feedback. The seismic-induced oscillations can experience various types of oscillations, from periodic ones to chaos, as the inlet subcooling is increased with the reference void-reactivity coefficient being doubled.

Phase coherence induced by correlated disorder.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the coupling strengths and natural frequencies. When these two kinds of disorder are uncorrelated (and when the positive and negative couplings are equal in number and strength), it is known that phase coherence cannot occur and synchronization is absent. Here we explore the effects of correlating the disorder. Specifically, we assume that any given oscillator either attracts or repels all the others, and that the sign of the interaction is deterministically correlated with the given oscillator's natural frequency. For symmetrically correlated disorder with zero mean, we find that the system spontaneously synchronizes, once the width of the frequency distribution falls below a critical value. For asymmetrically correlated disorder, the model displays coherent traveling waves: the complex order parameter becomes nonzero and rotates with constant frequency different from the system's mean natural frequency. Thus, in both cases, correlated disorder can trigger phase coherence.

Design, analysis, and experimental studies of a novel PVDF-based piezoelectric energy harvester with beating mechanisms

Wireless sensor networks become increasingly important for structural health monitoring and other related applications. Consequently, energy harvesters to convert environment wasted vibration energy into electricity for powering those sensor nodes become essential device have been realized by using MEMS technology. To achieve optimal performance, energy harvester must be operated at resonance state. However, the vibration frequencies of environmental vibrations are usually much less than that of those harvesters and this fact could be a major barrier for optimizing harvesting performance. In this paper, a new piezoelectric energy scavenging concept is proposed and demonstrated to convert environmental vibrations into electricity. Unlike previous MEMS-based piezoelectric harvesters, which suffer from frequency matching between environmental vibration and the system natural frequency, this work proposes a novel beating design using polymer piezoelectric materials in collaborating with a beating mechanism. I.e., by creating impact force via the low frequency vibration provided by the mechanism, it is possible to excite system natural frequency by the environmental vibrations and to operate the system at the resonance state. By experimental characterization, the overall harvesting efficiency of the proposed design is much greater than that from the conventional design. It shows that the proposed design could potentially improve the future design for piezoelectric energy harvesters significantly. In short, this primary study shows a promising scheme for guiding future development of high efficient piezoelectric harvesters.

Research of Machining Vibration Restraint Method for Compressor Blade

Machining vibration is one of the most important reasons that influences the tool life, machining efficiency and the final surface quality, the performance and the reliability of aero-engine is affected directly by compressor blades. To restrain machining vibration of compressor blade in multi-axis machining, the mechanism of vibration and milling force during machining thin-walled blades is analyzed. A model of machining vibration and a vibration differential equation are established. Response function of blade reflects that the amplitude is influenced by clamping stiffness of blade, the natural frequency and damping of blade-fixture system. The paper presents a method of tension clamping to increase stiffness and natural frequency of system. Experiments show that the method can reduce clamping deformation, increase blade stiffness and natural frequency of blade-fixture system. It can reduce machining vibration eventually.

Micro-vibration isolation measurement platform using permanent magnet negative-stiffness devices and air springs

Environmental micro-vibration affects the accuracy of precision measurement devices. Passive isolation system can isolate most external vibration, decreasing its stiffness will improve isolation effect. Permanent magnet negative-stiffness device (PMNSD), which is mainly constructed from three permanent magnets, can decrease the system dynamic stiffness of isolation system without any effect to the static stiffness, when it is used parallel with positive-stiffness isolator. The mechanism of its negative stiffness is analyzed, and its stiffness is calculated by theory method and finite element method, it proves that the stiffness will decrease with the decrease of the initial air gap of magnets. A micro-vibration isolation platform using PMNSDs and air springs is constructed, it can carry many kinds of measurement instruments which weight 0-2000 Kg, keeping the system natural frequency under 1 Hz, it has good isolation capacity for micro-vibration over 2 Hz. The PMNSD will have greatly practical applications for its tunable negative stiffness, zero static stiffness, contact free, long life and so on.

Analysis of the dynamic characteristics of a slant-cracked cantilever beam

In this study, the dynamic characteristics of a slant-cracked cantilever beam are studied based on a new finite element (FE) model where both plane and beam elements are used to reduce the computational costs. Simulation studies show that the proposed model has the same system natural frequencies and vibration responses as those in the pure plane element model but is computationally more efficient. Based on the new model, the effects of loads such as gravity Fg, excitation force amplitude F0 and direction angles of excitation force φ, and crack parameters including slant crack angle θ, dimensionless crack depth s and dimensionless crack location p, on system dynamics have been analyzed. The results indicate that (1) the gravity has a more significant effect on the sub-harmonic resonance responses than on the super-harmonic resonance and resonance responses; (2) The amplitudes of the system responses at both excitation force frequencies fe and its harmonics such as 2fe and 3fe increase almost linearly with the increase of the excitation force amplitude F0; (3) Under the constant excitation force in the flexural direction, the tensile and compressive forces along the longitudinal direction can lead to opposite breathing behaviors of the crack within the super-harmonic and sub-harmonic resonance frequency regions; (4) Vibration is most severe under the straight crack angle (θ=90°) and near the straight crack angle such as θ=100° and 110°, and the vibration responses under smaller or larger crack angles such as θ=30° and θ=150° become weaker; (5) The resonance at 2fe is sensitive to the faint crack signals when s is small and p is large. In addition, the significant vibration responses at the multiple frequency of 3fe and the fractional frequency of 0.5fe can be regarded as a distinguishable feature of the serious crack with large s and small p.

Vibration-based identification of rotating blades using Rodrigues' rotation formula from a 3-D measurement

In this study, the geometrical setup of a turbine blade is tracked. A research-scale rotating turbine blade system is setup with a single 3-axes accelerometer mounted on one of the blades. The turbine system is rotated by a controlled motor. The tilt and rolling angles of the rotating blade under operating conditions are determined from the response measurement of the single accelerometer. Data acquisition is achieved using a prototype wireless sensing system. First, the Rodrigues\' rotation formula and an optimization algorithm are used to track the blade rolling angle and pitching angles of the turbine blade system. In addition, the blade flapwise natural frequency is identified by removing the rotation-related response induced by gravity and centrifuge force. To verify the result of calculations, a covariance-driven stochastic subspace identification method (SSI-COV) is applied to the vibration measurements of the blades to determine the system natural frequencies. It is thus proven that by using a single sensor and through a series of coordinate transformations and the Rodrigues\' rotation formula, the geometrical setup of the blade can be tracked and the blade flapwise vibration frequency can be determined successfully.

Vibration‐based damage assessment of steel structure using global and local response measurements

SummaryDamage assessment of a structure involves acquiring and identifying dynamic characteristics of the structure and using these characteristics to evaluate behavior and performance. In this study, an unsymmetrical three‐story steel structure (fabricated with one weak column in the first floor) was tested on shaking table and subjected to a series of earthquake excitations with increasing level of excitation back to back. Besides, white noise excitation was also applied in between the earthquake excitation to serve as the reference state. Both the traditional sensing system (accelerometer and linear variable differential transformer) and the local optical tracker system were implemented in the structure to collect the vibration‐based responses. For operational modal analysis, structural response from white noise excitation will be used in this study. First, the traditional system identification using global response data is used (multivariate autoregressive (AR)‐model) to extract system natural frequencies and mode shapes from all different set of white noise responses after earthquake excitation. The migration of AR‐coefficient ellipse error from each sensor response was used to identify the damage location. Second, blind source separation technique was used to identify the modal contribution of the structure from each test, which provide information to detect the damage severity. Finally, from the local optical tracker array data, the principal component analysis was applied to quantify the earthquake‐induce local stress of the structural member. Combine the result from damage detection using global measurement and the identified local element stress, one can locate and quantify the damage. Copyright © 2015 John Wiley &amp; Sons, Ltd.

On the statistical properties of random causal frequency response functions

In nuclear physics it is known that under broad restrictions a random scattering matrix element H satisfies a condition known as the analyticity-ergodicity (AE) requirement, which states that E[f(H)]=f(E[H]), where f is some function of H, and E[] represents the ensemble average. A scattering matrix element is directly analogous to a vibrational frequency response function, and it is of significant interest to consider whether the AE requirement is also applicable to random engineering systems. The proof of the AE condition in nuclear physics rests on the assumptions that H is causal and ergodic: causality implies that a Lorentzian frequency average satisfies the AE equation, and ergodicity implies that Lorentzian frequency averages are equal to ensemble averages. In vibrational systems it is readily shown that a typical frequency response function is non-stationary and non-ergodic, so that the Lorentzian and ensemble averages can differ significantly, and this means that the standard proof of the AE requirement breaks down. The question then arises as to whether the AE requirement might nonetheless apply to vibrational systems. It is shown in the present paper that the requirement does apply providing that the random point process representing the system natural frequencies is at least locally stationary (which is a much weaker condition than local stationarity of the frequency response function), and a number of the implications of this result are explored.

Global performances of a semi-submersible 5MW wind-turbine including second-order wave-diffraction effects

The global performance of the 5 MW OC4 semisubmersible floating wind turbine in random waves was numerically simulated by using the turbine-floater-mooring fully coupled and time-domain dynamic analysis program FAST-CHARM3D. There have been many papers regarding floating offshore wind turbines but the effects of second-order wave-body interactions on their global performance have rarely been studied. The second-order wave forces are actually small compared to the first-order wave forces, but its effect cannot be ignored when the natural frequencies of a floating system are outside the wave-frequency range. In the case of semi-submersible platform, second-order difference-frequency wave-diffraction forces and moments become important since surge/sway and pitch/roll natural frequencies are lower than those of typical incident waves. The computational effort related to the full second-order diffraction calculation is typically very heavy, so in many cases, the simplified approach called Newman\'s approximation or first-order-wave-force-only are used. However, it needs to be justified against more complete solutions with full QTF (quadratic transfer function), which is a main subject of the present study. The numerically simulated results for the 5 MW OC4 semisubmersible floating wind turbine by FAST-CHARM3D are also extensively compared with the DeepCWind model test results by Technip/NREL/UMaine. The predicted motions and mooring tensions for two white-noise input-wave spectra agree well against the measure values. In this paper, the numerical static-offset and free-decay tests are also conducted to verify the system stiffness, damping, and natural frequencies against the experimental results. They also agree well to verify that the dynamic system modeling is correct to the details. The performance of the simplified approaches instead of using the full QTF are also tested.

Parametric excitations and lock-in of flexible hydrofoils in two-phase flows

This work introduces a reduced-order method to study the parametric excitations and lock-in of flexible hydrofoils caused by unsteady two-phase (cavitating) flow. The reduced-order method is based on a 1-DOF structural model coupled with a van der Pol wake oscillator with empirically derived relations for the variation in lift, cavity-length, and cavity-shedding frequency as a function of a non-dimensional cavitation parameter. The results are compared with several available data from both numerical simulations and experimental measurements. The frequency content of both the predicted and measured vibrations suggested that, in addition to the primary cavity-shedding frequency and the hydrofoil natural frequencies, unsteady two-phase flows may excite additional modulated frequencies due to time-varying fluid-added mass effects. The results show that these frequency modulations might cause the flexible hydrofoil to undergo higher-order resonances, as well as parametric resonances. While the maximum deformations for the primary and higher-order resonances were observed to damp out, parametric resonances might persist even with realistic fluid damping coefficients (4–12%). It was observed that with higher effective foil flexibility, the cavity-shedding frequencies may be significantly modified from the rigid foil trends, and may instead lock-in with the system natural frequencies.

Helicopter Blade Stability Analysis Using Aeroelastic Frequency Response Functions

In the present paper, the aeroelastic stability of helicopter rotor blade is determined using Aeroelastic Frequency Response Function. The conventional methods of aeroelastic stability usually use an iterative procedure while the present method does not require such approach. Aeroelastic Frequency Response Functions are obtained by inverting dynamic stiffness matrix of the aeroelastic system. System response could be obtained through exciting each degree of freedom. The resulting response was then plotted and the behavior of this function was investigated to find out the stability criteria and system natural frequencies. The results of this method are compared with stability boundaries obtained from the conventional p-k method and it can be inferred that, compared to other methods, the present algorithm is of less numerical cost.

Low-frequency impact sound transmission of floating floor: Case study of mortar bed on concrete slab with continuous interlayer

The vibroacoustic behavior of a commonly used floating floor installed in an actual multifamily housing unit was investigated to determine the factors that influence impact sound transmission at low frequencies. A finite element vibration model of the floor structure and an experimental sound field against a rubber ball impact were analyzed in combination. The results indicated that, in addition to isolation of the impact energy above the system's natural frequency, the aspect of coupled and decoupled wave fields of the floating floor influences the impact sound transmission. The coupled wave field below the natural frequency is dominated by the bending wave field of the base slab and exerts a strong influence on the sound field, in which the sound field is dependent on the modal space and impact location of the coupled motion. The decoupled wave fields generated in the floating plate or the base slab above the natural frequency may disturb the vibration isolation. The non-rectangular acoustic cavity is considered to mitigate the influence of axial room modes on the impact sound field.