Time-averaged Power Flow Research Articles

Power flow analysis of the three-beam system on double parameter elastic foundations based on the wave propagation approach

This article investigates the power flow of the three-beam system on Winkler–Pasternak elastic foundations under elastic boundary conditions. The time-averaged power flow of the beam on elastic foundations with elastic boundary conditions is calculated based on the internal shear force and bending moment and the wave propagation approach. The reliability and accuracy of the calculation method are verified by comparing the vibration acceleration calculated by calculation examples and the finite element method. In addition, the effects of structural parameters and boundary conditions on the power flow of the three-beam system are investigated. This contributes to the analysis of the dynamic characteristics of the beam on double parameter elastic foundations under complex boundary conditions.

Research of energy characteristics and fault position detection for cracked rotor system

Early crack faults will introduce subtle local changes in the whole rotor system, which may bring difficulties to the fault detection process due to external disturbing factors. Meanwhile, judging the damaged position of rotor systems might be tough only through displacement or velocity. To overcome these issues, the harmonic balance-energy power flow analysis (HB-EPFA) is presented to investigate the rotor system with a transverse crack. Besides, a novel indicator (Time-averaged power flow) is proposed to evaluate the crack faults. For the energy characteristic analysis, the cracked rotor appears a special amplitude-jumping phenomenon, even in the case of shallow crack depth. The influence of crack depth on power flow values under different rotating speeds is obtained. Furthermore, the existence of cracks will lead to the rotation shaft presenting an abrupt energy-changing zone, which can be used to detect crack position. Relevant experiments are applied to validate the proposed detection method, and the experimental results agree with the theoretical analysis. The conclusions achieved in this paper can provide a reference for fault diagnosis of rotor systems in engineering practice.

Vibration suppression and energy flow tailoring of coupled metal and composite plates with curvilinear fibers

This study investigates the vibration transmission in coupled metal and composite plates with curvilinear fibers from the power flow viewpoint. The time-averaged power flow variables are obtained using a substructure method to assess the levels of vibration transmission of coupled structures with various configurations and fiber angles. It is demonstrated that by adopting tailored curvilinear fiber designs, the vibration transmission through the coupling edge of the plates can be effectively suppressed. Variable fiber angle designs are shown to exert significant effects on the dominant-excited mode and can be exploited to achieve desirable energy transmission paths for low-vibration transmission and effective suppression.

A novel diagnosis indicator for rub-impact of rotor system via energy method

Rub-impact between rotor and stator is a common mechanical fault for rotating machinery, which will bring risks into the normal operation. Detecting the severity of the rub-impact fault accurately is helpful to prevent serious mechanical accidents. However, traditional methods based on the displacements or spectrum analysis may occur some differences toughly distinguished with the naked eyes, and eventually cause inaccurate judgment in some cases. To overcome such shortcomings, this paper proposes a novel diagnosis indicator for rotor systems with rub-impact fault. The dynamic model of the rotor system is established by the finite element method. Based on the vibration power flow theory, the novel indicator is developed by considering the force, velocity, and phase information in a time interval (Time-averaged input power flow). Through theoretical analysis of the rub-impact faults, the energy responses and the law between input power and rub-impact of the rotor system are obtained. Meanwhile, the severity detection of faults by using the proposed indicator has been achieved. The corresponding tests are carried out to validate the accuracy of the theoretical results. The experimental results reveal the proposed indicator can help judge the different severity of rub-impact faults, which means the lower time-averaged input power flow indicates a serious fault. To evaluate the application effect of the proposed indicator in engineering, the rub-impact fault of the GE-T700 power turbine rotor is investigated through theoretical analysis. And the diagnosis results are consistent with the above consequences. The conclusions achieved in this paper can provide a reference for fault diagnosis of rotor systems in engineering practice.

Vibration transmission and energy flow analysis of variable stiffness laminated composite plates

This study investigates the vibration transmission and energy flow tailoring of variable stiffness laminated composite (VSLC) plates with curvilinear fibres subjected to harmonic excitation. The free vibration characteristics of VSLC plates are firstly studied. The steady-state dynamic responses are then obtained based on the first-order shear deformation theory. The effects of variable angle fibres on the time-averaged input power and kinetic energy of the plate are examined using the power flow analysis approach. The time-averaged power flow density vector is used to determine the primary energy transmission paths, which are shown to depend on the dominant mode excited by the applied force. Curvilinear fibre designs are found to have a significant influence on the main energy flow transmission paths and vibration level. For enhanced vibration suppression, the fibre angles can be tailored to modify the vibration transmission paths according to a specific excitation.

Energy flow and performance evaluation of inerter-based vibration isolators mounted on finite and infinite flexible foundation structures

This study investigates the vibration power flow behavior and performance of inerter-based vibration isolators mounted on finite and infinite flexible beam structures. Two configurations of vibration isolators with spring, damper, and inerter as well as different rigidities of finite and infinite foundation structures are considered. Both the time-averaged power flow transmission and the force transmissibility are studied and used as indices to evaluate the isolation performance. Comparisons are made between the two proposed configurations of inerter-based isolators and the conventional spring-damper isolators to show potential performance benefits of including inerter for effective vibration isolation. It is shown that by configuring the inerter, spring, and damper in parallel in the isolator, anti-peaks are introduced in the time-averaged transmitted power and force transmissibility at specific frequencies such that the vibration transmission to the foundation can be greatly suppressed. When the inerter is connected in series with a spring-damper unit and then in-parallel with a spring, considerable improvement in vibration isolation can be achieved near the original peak frequency while maintaining good high-frequency isolation performance. The study provides better understanding of the effects of adding inerters to vibration isolators mounted on a flexible foundation, and benefits enhanced designs of inerter-based vibration suppression systems.

Acousto-elastic radiation force on a fluid cylindrical inclusion embedded in a linear elastic medium

In the framework of linear elasticity, the longitudinal acousto-elastic time-averaged radiation force exerted on a cylindrical fluid-filled inclusion of arbitrary size embedded in an unbounded elastic matrix is derived based upon the integration of the time-averaged power flow density over a surface enclosing the particle. The cases of compressional and shear plane progressive elastic waves are considered. Mathematical expressions for the longitudinal radiation force efficiencies are derived and cast in exact series expansions in cylindrical coordinates. The elastic scattering solution, which is utilized to derive the acousto-elastic radiation force expressions, is determined. The analytical expressions for the acousto-elastic radiation force clearly show the contributions related to mode conversion P → S for the incident compressional waves, and S → P for shear wave incidence (where P and S denote, respectively, the compressional and shear waves). Another method, based on the extinction energy efficiency and the scattering asymmetry parameter, is shown to provide the same exact expression of the force obtained based upon the power flow density formalism. Computations for the dimensionless radiation force efficiencies and their cross-polarized (P → S and S → P) components demonstrate that mode conversion contributes negatively, positively or neutrally depending on the mass density of the particle, its size, and the properties of the surrounding elastic medium. In addition, elastic resonances are manifested in the acousto-elastic radiation force plots. This analysis provides a rigorous exact formalism using exact partial-waves series to compute the acousto-elastic radiation forces of compressional or shear waves on an elongated cylindrical particle encased in a linear elastic medium. The analytical series expressions and numerical predictions can be used as a priori information in optimizing and designing acousto-elastic radiation force experiments in materials science, elastography, vibro-acoustography, nondestructive evaluation, and the activation/excitation of a fluid cylinder embedded in an elastic medium, to name a few examples. The electromagnetic/optical analogue for materials exhibiting circular dichroism is also noted.

Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism

This study investigates the use of a spring-bar mechanism (SBM) in a vibration suppression system to improve its performance. The SBM, comprising bars and springs, is configured with a conventional linear spring-damper isolator unit. The dynamic response, force transmissibility, and vibration energy flow behaviour are studied to evaluate the vibration suppression performance of the integrated system. It is found that the SBM can introduce hardening, softening stiffness, or double-well potential characteristics to the system. By tuning the SBM parameters, constant negative stiffness is achieved so that the natural frequency of the overall system is reduced for enhanced low-frequency vibration isolation. It is also found that the proposed design yields a wider effective isolation range compared to the conventional spring-damper isolator and a previously proposed isolator with a negative stiffness mechanism. The frequency response relation of the force-excited system is derived using the averaging method and elliptical functions. It is also found that the system can exhibit chaotic motions, for which the associated time-averaged power is found to tend to an asymptotic value as the averaging time increases. It is shown that the time-averaged power flow variables can be used as uniform performance indices of nonlinear vibration isolators exhibiting periodic or chaotic motions. It is shown that the SBM can assist in reducing force transmission and input power, thereby expanding the frequency range of vibration attenuations.

Radiation force of stationary elastic compressional and shear plane waves on a cylinder encased in a linear elastic solid

The assembly of acoustically-engineered composite (meta)materials rely on the acousto-elastic radiation force of stationary waves to trap solid particles and form a stable structure. Accurate modeling of the time-averaged force on a particle is needed for experimental design purposes and basic physical understanding of the trapping effect and particle clustering. This work presents a rigorous formalism based on the integration of the time-averaged radial power flow density to derive analytically and calculate numerically the acousto-elastic radiation force induced by compressional (P) and shear (S) plane standing waves incident upon a circular elastic cylinder embedded in a linear lossless solid. Initially, the elastic scattering by the cylinder is determined and then used to derive the acousto-elastic radiation force. Particular emphasis is made on the contributions of elastic mode preservation (P → P, S → S) as well as mode conversion (P → S, S → P) to the radiation force of standing waves. Numerical computations for the dimensionless radiation force efficiency and its components demonstrate the importance of compressional-to-shear and shear-to-compressional mode conversions in scattering that occur in linear elastic media. The analytical formalism and exact solutions of the radiation force can be used to validate numerical methods and the results of the simulations can be utilized a priori in optimizing and designing acousto-elastic radiation force experiments involving standing waves and particle assembly in materials science, elasticity imaging methods, activation of implantable devices, non-destructive evaluation and other related applications.

Vibration transmission and power flow of laminated composite plates with inerter-based suppression configurations

This paper investigates the vibration transmission and power flow behaviour of harmonically excited laminated composite plates attached with different inerter-based suppression configurations. The substructure approach based on analytical and numerical methods is employed to obtain the steady-state dynamic response. Power flow analysis is carried out to determine the time-averaged power flow input and transmission as well as the kinetic energy of the plate. The power flow density vector is used to show explicitly the vibration transmission paths within the composite plate. It is shown that the fibre orientation and stacking sequences can have significant effects on the time-averaged power flow characteristics as well as the dominant vibration transmission paths. It is also shown that passive spring, damper and inerter elements may be attached to the plate to modify its vibration response and transmission according to specific design requirements. The proposed inerter-based suppression device with two different configurations can reduce the vibration level over a wide frequency range for vibration suppression. The findings may provide insights for the enhanced dynamic designs of laminated composite plates, and the suppression of their vibrations using inerter-based devices.

Vibration transmission and power flow in impact oscillators with linear and nonlinear constraints

This paper investigates the vibration transmission and power flow behaviour of impact oscillators comprising linear or quasi-zero-stiffness (QZS) nonlinear constraints. The steady-state responses of the systems subjected to a harmonic excitation are obtained by the harmonic balance (HB) approximations and numerical integrations. The level of vibration transmission within the systems is quantified by the force transmissibility and time-averaged power flow variables. The effects of the stiffness and damping properties of the constraints on the dynamic response and the vibration transmission are investigated and compared. For a single-degree-of-freedom (SDOF) impact oscillator with a linear constraint, it is found that high constraint stiffness may reduce the response amplitude but lead to high force transmission to the constraint. For a two-degree-of-freedom (2DOF) impact oscillator with a linear constraint at the coupling interface, there may be three peaks in the curves of the relative displacement between the masses and also in those of the force transmissibility to the secondary mass. The addition of the constraint may also result in significantly larger amount of the time-averaged dissipated power at the interface. A higher spring stiffness of the linear constraint can increase the peak values of the relative displacement and force transmissibility to the secondary mass, but may reduce the time-averaged transmitted power to the secondary oscillator. The nonlinear constraint can be designed to tailor the magnitude of force transmissibility and the amount of power flow transmission within impact oscillators, comparing with those of the corresponding linear constraint case. The study provides some enhanced understanding of the influence of stiffness and damping properties of linear and nonlinear constraints on vibration transmission, and benefits dynamic designs of impact oscillators for better performance.

Quantification of vibration force and power flow transmission between coupled nonlinear oscillators

This paper presents a quantitative investigation on the level of vibration force and power flow transmission between linearly coupled nonlinear oscillators. Both analytical and numerical methods are employed for systematic examinations of the vibration transmission associated with hardening, softening and double-well potential stiffness nonlinearities in the oscillators. The method of averaging is employed to obtain analytical formulations of the steady-state frequency-response relationship. Time-averaged power flow variables are formulated and used to quantify vibration input, dissipation and transmission levels associated with both periodic and chaotic responses. It is shown that the stiffness nonlinearities have significant effects on vibration transmission levels near resonance frequencies and in the low-frequency range. It is also shown that when both oscillators are of double-well potential stiffness, the system may exhibit chaotic motions with higher levels of vibration transmission between the oscillators as compared to the corresponding first-order analytical approximations, because of large super-harmonic components in the response. It is found that there may be multiple possible levels of vibration transmission due to co-existing periodic or chaotic responses. It is shown that when comparing the vibration transmission levels of the co-existing stable responses, the use of time-averaged transmitted power and force transmissibility may lead to different evaluation outcomes. These findings provide a better understanding of the effects of stiffness nonlinearities on the vibration transmission which may be maximized or suppressed based on specific design objectives for enhanced dynamic performance.

Theoretical Modeling of the Internal Power Flow and Absorption Loss of the Air Mode Based on the Proposed Poynting Vector Analysis in Top-emitting Organic Light-emitting Diodes

We propose the Poynting vector analysis of the air mode in a top-emitting organic light-emitting diode (OLED) by combining the transfer matrix method and dipole source term. The spatial profiles of the time-averaged optical power flow of the air mode are calculated inside and outside the multilayer structure of the OLED with respect to the thickness of the semi-transparent top cathode and capping layer (CPL). We elucidate how the micro-cavity effect controlled by the thickness variation of the semi-transparent top cathode or CPL affects the internal optical power and absorption loss inside the OLED multilayer and the external optical power coupled into the air. When the calculated absorption loss and external power obtained by the proposed Poynting vector and currently-used point dipole models are compared, two calculation results are identical, which demonstrates the validity of the two models.

On vibration transmission in oscillating systems incorporating bilinear stiffness and damping elements

This paper investigates the dynamic characteristics and vibration transmission behaviour of a single oscillator and also coupled oscillators incorporating both bilinear stiffness and bilinear damping elements. Both harmonic balance analytical approximations and numerical integrations are used to obtain the steady-state dynamic response and vibration transmission behaviour. Vibration transmission within the oscillating systems is evaluated by time-averaged power flow variables and force transmissibility. For the single oscillator with bilinear elements, it is shown that a small bilinear stiffness ratio is beneficial in the suppression of force transmission in the high-frequency range. A large bilinear damping ratio can reduce the peak force transmissibility and also the peak time-averaged input power. For the coupled oscillators with interfacial bilinear elements, it is also found that a small bilinear stiffness ratio can reduce both the force transmissibility and the time-averaged transmitted power at high excitation frequencies. A combination of a small bilinear stiffness ratio and a large bilinear damping ratio can lead to effective mitigation of vibration transmission in a wide range of excitation frequencies. It is also shown that the inclusion of bilinear elements may cause large superharmonic or subharmonic response components, which result in high vibration transmission level. These results provide a better understanding of the effects of bilinear stiffness and bilinear damping on vibration transmission, and benefit future designs of vibration suppression systems with such elements.

Propagation of surface plasmon polaritons in monolayer graphene surrounded by nonlinear dielectric media

The propagation of s-polarized surface plasmon polaritons (SPPs) was investigated in a monolayer graphene sheet surrounded by two dielectric media on each side, so that one or both sides of the media were linear or nonlinear with Kerr-type nonlinearity. The plasmonic properties including the wave propagation index neff, the penetration depth, the time-averaged power flow and the spatial profile of electric and magnetic fields were calculated for the following structures: Linear medium/Graphene/Linear medium (L/G/L), Nonlinear medium/G/L (NL/G/L) and NL/G/NL. The analysis of the nonlinear coefficient effect on the SPP properties showed that increasing the nonlinearity in NL/G/L enhanced neff. However, for a smaller difference between the nonlinearity of layers, neff decreased in NL/G/NL. By comparing between the proposed structures, it was found that while large values of neff can be obtained from L/G/L, its frequency confinement is smaller than that of NL/G/L and NL/G/NL. Furthermore, NL/G/L and NL/G/NL were able to support localized nonlinear modes, leading to enhanced frequency confinement of transverse electric (TE) waves in the presence of nonlinearity. Increasing the nonlinearity in NL/G/L confined the spatial profile of the electric field near the graphene interface, indicating the existence of surface plasmon solitons. The influence of the graphene chemical potential μ on the plasmonic properties of the structures was also investigated. In this case, it was found that the plasmonic properties can be controlled by μ. Our calculations may solve the difficulties in TE surface plasmons for application in optics and plasmonics.

On the generation of magnetic field enhanced microwave plasma line

Microwave linear plasmas sustained by surface waves have attracted much attention due to the potential abilities to generate large-scale and uniform non-equilibrium plasmas. An external magnetic field was generally applied to enhance and stabilize plasma sources because the magnetic field decreased the electron losses on the wall. The effects of magnetic field on the generation and propagation mechanisms of the microwave plasma were tentatively investigated based on a 2-D numerical model combining a coupled system of Maxwell's equations and continuity equations. The mobility of electrons and effective electric conductivity of the plasma were considered as a full tensor in the presence of magnetic field. Numerical results indicate that both cases of magnetic field in the axial-direction and radial-direction benefit the generation of a high-density plasma; the former one allows the microwave to propagate longer in the axis direction compared to the latter one. The time-averaged power flow density and the amplitude of the electric field on the inner rod of coaxial waveguide attenuate with the propagation of the microwave for both cases of with and without external magnetic field. The attenuation becomes smaller in the presence of appropriately higher axial-direction magnetic field, which allows more microwave energies to transmit along the axial direction. Meanwhile, the anisotropic properties of the plasma, like electron mobility, in the presence of the magnetic field confine more charged particles in the direction of the magnetic field line.

Vibration power flow and force transmission behaviour of a nonlinear isolator mounted on a nonlinear base

The vibrational power flow and force transmission characteristics of a two-degree-of-freedom (TDOF) nonlinear system are investigated to examine the performance of a nonlinear isolator. The system consists of a harmonically-excited mass, mounted on a single-degree-of-freedom (SDOF) nonlinear flexible base structure through a nonlinear isolator. Both the isolator and the base exhibit damping and stiffness nonlinearities characterized by cubic damping and cubic restoring forces, respectively. The steady-state response of the system is obtained by using the method of averaging and numerical integrations. The bisection method is developed to solve coupled nonlinear frequency response equations. The time-averaged input, dissipated and transmitted powers of the system in the steady-state motion are formulated. Power transmission ratio is proposed as an isolation performance index and compared with force transmissibility. It is found that softening stiffness nonlinearity in the isolator can reduce power transmission ratio, and suppress the peak values of time-averaged power flow and force transmissibility. In comparison, the hardening stiffness in the isolator can lead to a larger power transmission ratio and larger force transmissibility in the high-frequency range. It is shown that power transmission ratio can be reduced by the damping nonlinearity in the isolator, but is enlarged by that in the base. It is also shown that softening nonlinear isolator may provide a better performance than a hardening isolator when the base is of either hardening or softening stiffness nonlinearity. The paper provides a better understanding of the effects of nonlinearities on the performance of nonlinear vibration isolators and demonstrates their potential application to enhance vibration mitigation.

Solution to the Klein–Gordon equation for the study of time‐domain waveguide fields and accompanying energetic processes

In electromagnetics, power flow and energy densities are associated with time-varying electromagnetic fields. For time-harmonic waves, such quadratic characteristics are derived using the complex-valued spatial forms of field vectors resulting in time-averaged power flow and energy densities. Hence, the opportunity to study dynamic processes of the quadratic characteristics is lost in the time-harmonic electromagnetics. To overcome this drawback, the authors consider the problem of a waveform propagation in a hollow waveguide via solving the system of Maxwell's equations with time derivative in an energetic space of ‘real-valued’ functions. The transverse electric and transverse magnetic modal field components are presented as the products of modal basis elements and the modal amplitudes. The vector basis elements are obtained with required physical dimensions, volt per metre and ampere per metre. Thereby, the modal amplitudes are dimension-free scalar functions. The system of evolutionary equations for the modal amplitudes is derived and solved explicitly. The Klein–Gordon equation (KGE) plays a central role herein. A novel set of real-valued solutions to the KGE is obtained and applied for analysis of the energetic processes pertinent to the time-domain signal propagation. The velocity of transportation of the modal field energy is obtained and energetic exchange between the transverse and longitudinal field components is discussed.

Longitudinal Modes along Thin Piezoelectric Waveguides for Liquid Sensing Applications

The propagation of longitudinally polarized acoustic modes along thin piezoelectric plates (BN, ZnO, InN, AlN and GaN) is theoretically studied, aiming at the design of high frequency electroacoustic devices suitable for work in liquid environments. The investigation of the acoustic field profile across the plate revealed the presence of longitudinally polarized Lamb modes, travelling at velocities close to that of the longitudinal bulk acoustic wave propagating in the same direction. Such waves are suitable for the implementation of high-frequency, low-loss electroacoustic devices operating in liquid environments. The time-averaged power flow density, the phase velocity and the electroacoustic coupling coefficient K2 dispersion curves were studied, for the first (S0) and four higher order (S1, S2, S3, S4) symmetrical modes for different electrical boundary conditions. Two electroacoustic coupling configurations were investigated, based on interdigitated transducers, with or without a metal floating electrode at the opposite plate surface. Enhanced performances, such as a K2 as high as 8.5% and a phase velocity as high as 16,700 m/s, were demostrated for the ZnO- and BN-based waveguides, as an example. The relative velocity changes, and the inertial and viscous sensitivities of the first symmetric and anti-symmetric mode, S0 and A0, propagating along thin plates bordered by a viscous liquid were derived using the perturbation approach. The present study highlights the feasibility of the piezoelectric waveguides to the development of high-frequency, integrated-circuits compatible electroacoustic devices suitable for working in liquid environment.

Power flow behaviour and dynamic performance of a nonlinear vibration absorber coupled to a nonlinear oscillator

The vibrational power flow characteristics of a two-degree-of-freedom system are investigated to examine the performance of nonlinear absorbers in vibration attenuation of nonlinear primary oscillators. The nonlinearities in the oscillator and those in the absorber are both characterised by cubic restoring and damping forces. Both analytical approximations and numerical integrations are used to obtain time-averaged power flow variables, as well as kinetic energies of the system. Power absorption ratio and the kinetic energy of the nonlinear oscillator are proposed to quantitatively evaluate the effectiveness of nonlinear absorbers with respect to the existing nonlinearities in the oscillator. Comparing with linear absorbers, it is found that softening (hardening) stiffness absorber provides benefits for vibration mitigation of a hardening (softening) stiffness primary oscillator by enhancing power absorption efficiency and reducing the kinetic energy of the oscillator so that the functioning frequency range of the absorber can be enlarged. Nonlinear cubic damping in the absorber is shown beneficial for vibration suppression as the power absorption ratio becomes large at resonance frequencies so that the peak power flow and kinetic energy levels are reduced. The developed model can be conveniently extended to study other types of nonlinearities in the absorber/oscillator. Conclusions and suggestions are provided for applications.