Structural Power Flow Research Articles

Energy dissipation and power flow analysis based on acoustic black hole laminated beams

To enhance the understanding of the dynamic performance of composite beams, assessing their energy accumulation and dissipation capabilities is crucial. The purpose of this paper is to establish a dynamic analysis model of acoustic black hole (ABH) laminated beams based on the Timoshenko beam theory and isogeometric method and to analyze structural intensity and power flow. The vibration governing equation for the ABH laminated beam is derived by calculating its potential and kinetic energy. The convergence of the results is confirmed through multiple mesh refinements, while the model's accuracy and applicability are substantiated through comparisons with the literature, traditional finite element simulations, and experimental data. This investigation into the power flow and structural intensity under varying parameter conditions elucidates the energy transfer mechanisms within ABH laminated beams, offering valuable insights for the deployment of ABH technologies in practical engineering applications.

Vibration behavior and power transmission of coupled plate structures with embedded acoustic black holes joined at an arbitrary angle

The Acoustic Black Hole (ABH) technology is widely recognized for its passive vibration and noise attenuation capabilities, characterized by lightweight and high efficiency. Exploring the dynamic behavior and energy transmission characteristics of the ABH coupled-plate system is a novel endeavor that provides a theoretical foundation for applying ABH structures in complex coupled systems. In this work, a semi-analytical dynamic model for the ABH coupled-plate system (comprising an ABH plate and a uniform plate) with an arbitrary connection angle is proposed by using the Rayleigh-Ritz framework. To ensure the continuity of high-order spatial derivatives at the edges, the modified version of the Fourier series is employed to describe the vibration displacement fields for transverse and in-plane vibration components. This enables the proposed model to determine the necessary derivatives of relevant quantities, namely the structural intensity and power flow, which facilitates the description of energy transmission behaviors in the ABH coupled-plate structure. Firstly, the convergence characteristics of the model are studied and the accuracy of modal information and dynamic response is verified through the comparison of those calculated from the finite element method (FEM), additionally, the experiment is carried out to verify the proposed model. Subsequently, several numerical examples are presented to study the vibrational characteristics of the ABH coupled-plate system and the good vibration attenuation of the ABH coupled-plate is observed as well. Finally, the vibrational energy transmission results are obtained and the influence of some important system parameters such as coupling angles on energy transmission characteristics in such systems are investigated. This work can shed some new light on ABH structures with potential application to such complex coupled-plate structures.

Vibro-Acoustic Analysis of an Underwater Cylindrical Shell with Internal Structures and a Comparison with Experiment

Abstract: In this work, the low-frequency vibration response and full-band acoustic radiation characteristics of an underwater reinforced cylindrical shell with internal structures are studied by combining the FEM with SEA. The stiffened cylindrical shell contains internal structures such as the F-shape plates and the support valve frames. The exciting sources have two different exciting forces corresponding to two experimental conditions. In the low-frequency band, the FEM was employed, and in the medium and high-frequency bands, the SEA was used. A comparison of the numerical results and the experiment shows that they agree well. The FEM and SEA give better results at [1,1k] Hz and [1k,10k] Hz, respectively. Due to mesh quality limitations, the FEM is not favorable for medium and high-frequency calculations. The SEA focuses on the structural mean power flow but cannot obtain position-specific vibrational responses. The results show that the internal excitation source mainly causes the structural vibration and sound radiation and are closely related to the free vibration characteristics of the structure. In addition, with the increase in frequency, the circumferential sound pressure level of the underwater structure has more substantial directivity.

Dynamic modeling of vibration behavior and power flow of a plate structure embedded with an ABH indentation

A two-dimensional (2D) semi-analytical model is established for the analyses of vibration behavior and energy transmission of the plate structure with a power-law-profiled thickness variation, referred to as an Acoustic Black Hole (ABH) plate. Under the general Rayleigh-Ritz framework, a modified version of 2D Fourier series is employed as the admissible function of the transverse displacement of the ABH plate featuring a thickness variation. Thanks to such modified Fourier series, a high-order spatial derivative continuity over the entire solution domain is available to express the internal forces and bending moments, further calculating the corresponding structural intensity and power flow within the ABH plate. The proposed model allows an accurate prediction of the modal characteristics and dynamic response, which is verified by the FEM and experiment, respectively. Several numerical analyses are proceeded and the typical dynamic phenomena, including the wave compression and efficient vibrational suppression of the 2D ABH plate are observed. The structural intensity (SI) vector arrows of the ABH plate are given, which represent the magnitude of the vibrational energy and the flowing direction in a visual pattern. The energy transmission behaviors and the manipulation of power flow by the ABH are revealed in an energy perspective. The typical energy focalization and dissipation phenomena of the ABH plate are analyzed in the frequency domain. Finally, the enhancement of the energy transmission by the ABH feature on the plate is explored by the normalized power flow. It is noted that the proposed model offers a visual energy transfer analysis tool for the 2D ABH structures, which may provide a novel perspective in various studies of ABH structures.

Vibrational Power Flow Analysis for the Sandwich Cylindrical Shell Structure with a Metal–Rubber Core in the Thermal Environment

This research is designed to investigate the vibrational characteristics of a sandwich cylindrical shell structure with an elastic-porous metal–rubber core in the thermal environment by the power flow method. The finite element models of the homogeneous and sandwich cylindrical shell structures are established for harmonic response analysis completed by the mode superposition method. Further, the structural power flow is calculated and visualized based on the results of the finite element analysis. A comparison is made with the results for a plate available in the literature validating the effectiveness of the power flow visualization method. The effects of some key factors, such as the frequency, the metal–rubber core, the length-to-radius ratio, and the temperature on the power flow of the sandwich cylindrical shell structure, are analyzed using power flow cloud pictures and vector diagrams. The results reveal that the frequency affects the distribution of power flow, and the metal–rubber core can dissipate energy, whereas the length-to-radius ratio and temperature do not have a significant influence on the power flow of the sandwich cylindrical shell structure with a metal–rubber core (SCSMR). This work should be valuable for the noise and vibration control application of the SCS-MR.

Vibration Power Flow and Transfer Path Analysis of Two-Dimensional Truss Structure by Impedance Synthesis Method

The violent vibration of truss structures may cause fatigue, faults, or even an accident. Aiming to analyze the vibration power flow and transfer path of two-dimensional truss structures in the mid and high-frequency domain, this paper proposed a fast dynamic calculation method—the impedance synthesis method (ISM)—which is based on an analytical equation with litter elements. Firstly, the global coordination vibration impedance of a Timoshenko beam truss is derived; Secondly, a dynamic model of a two-dimensional truss structure is built up with a single truss beam by force balance and geometric continuity; then, real and imaginary parts of dynamic responses and force in simple and periodically truss structures are verified by compared with FEM results, respectively; finally, the transfer path analysis (TPA) method is applied to separate the contribution of different transfer paths of power flow in periodical truss structures. The results show that the TPA method can easily find the line spectrum frequency of power flow, which should be considered in vibration control. This method can also be expanded to three-dimensional, honeycomb, and other truss beam structures.

Random vibration analysis and modal energy characteristics of fiber-reinforced composite beams

In aerospace and automotive industries, structural designs are required to meet specific weight limits whilst maintaining strength and stiffness. The material of choice for such applications is fiber-reinforced composites because of their high strength-to-weight and stiffness-to-weight ratios. Composite materials used in these applications are often subjected to random dynamic loads and if the excitation sources are large enough, excessive vibration can lead to unwanted noise levels and fatigue failure. A commonly used design tool for modeling the vibration transmission and power flow in structures excited by random forces is statistical energy analysis. In order to incorporate fiber-reinforced composites within a statistical energy analysis methodology, a detailed understanding of how the modal energy levels vary when excited by a broadband random excitation source is needed. In this paper, analytical expressions are derived for the modal energy levels of a fiber-reinforced composite beam coupled in bending and torsion subjected to spatial white noise or the so-called rain-on-the-roof loading. It is shown that the modal energy levels of the fiber-reinforced composite beam are markedly nonuniform across the modal spectrum. This finding is in contrast with the well-known classical result that an isotropic beam subjected to the same type of excitation is characterized by a uniform modal energy spectrum, a result sometimes referred to as the equipartition of energy. The equipartition of energy forms the basis of statistical energy analysis and the results of this work indicate that fiber-reinforced composite beams cannot be reliably included into a conventional statistical energy analysis methodology. It is also shown that the mechanism responsible for the nonuniform modal energy distribution is related to the physical process by which power is transmitted across the boundaries. Simulations are also carried out to illustrate that the distribution characteristics of the modal energy are identical for a range of classical boundary conditions.

Wave Scattering and Power Flow in Straight-Helical-Straight Waveguide Structure

The knowledge of wave scattering and power flow in waveguide structures is important for many engineering applications. In this paper, power flow and scattering in a straight-helical-straight waveguide structure are investigated using the wave and finite element (WFE) method. For simple (straight or helical) waveguides, wave scattering (and subsequently the power flow and scattering) can be resolved analytically. This is not the case for complex waveguides such as laminated or sandwiched waveguides or waveguides with noncanonical cross-sections. In such cases, the WFE method is used to model the wave behavior in each waveguide in the structure. The power flow is then studied by considering how waves reflect and transmit at the boundaries that join the straight waveguides with the helical waveguide. We present three numerical examples but analytical solutions can be obtained for the first example only; for the second and third examples, the WFE is used in earnest since the wave behavior and subsequently the power flow would at best be extremely difficult to formulate analytically.

Optimization of autonomous underwater vehicle structure shape based on the characteristics of power flow distribution

Power flow method (PFM) is firstly applied to the structural analysis of autonomous underwater vehicle (AUV). By finite element analysis, the numerical model of AUV is established. The propulsive force of the propeller is calculated and the dynamic responses of AUV are discussed. The characteristics of power flow distribution in AUV are studied, and the optimization of AUV shape by PFM is developed. The results show that the performances of AUV's anti-vibration are improved. The present work may lay the foundation for the propagation characteristics of structural vibration power flow under the multi-source load excitations, and also provide a novel optimization approach for AUV shape.

An Investigation of Vibrational Power Flow in One-Dimensional Dissipative Phononic Structures

Owing to their ability to block propagating waves at certain frequencies, phononic materials of self-repeating cells are widely appealing for acoustic mitigation and vibration suppression applications. The stop band behavior achieved via Bragg scattering in phononic media is most commonly evaluated using wave propagation models which predict gaps in the dispersion relations of the individual unit cells for a given frequency range. These models are in many ways limited when analyzing phononic structures with dissipative constituents and need further adjustments to account for viscous damping given by complex elastic moduli and frequency-dependent loss factors. A new approach is presented which relies on evaluating structural intensity parameters, such as the active vibrational power flow in finite phononic structures. It is shown that the steady-state spatial propagation of vibrational power flow initiated by an external disturbance reflects the wave propagation pattern in the phononic medium and can thus be reverse engineered to numerically predict the stop band frequencies for different degrees of damping via a stop band index (SBI). The treatment is shown to be very effective for phononic structures with viscoelastic components and provides a clear distinction between Bragg scattering effects and wave attenuation due to material damping. Since the approach is integrated with finite element methods, the presented analysis can be extended to two-dimensional lattices with complex geometries and multiple material constituents.

Mechanics of longitudinal and flexural locally resonant elastic metamaterials using a structural power flow approach

Elastic metamaterials are sub-wavelength structures with locally resonant components that contribute to the rise of tunable stop bands, i.e. frequency ranges within which waves do not propagate. A new approach is presented here to model and quantify this stop band behavior by evaluating structural vibrating power flowing in the different constituents of locally resonant metamaterials. It is shown that the patterns of power propagation resemble, to a great extent, steady-state wave profiles derived from displacement fields, and can thus be used to develop an algorithm that numerically predicts stop band frequencies for any given realization with a finite length and a known number of repeating cells. The approach is presented here in the context of one-dimensional metamaterials with single and multiple internal resonators and is applied to two traditional examples constituting both longitudinal and flexural type structures. The presence of dissipative elements is taken into consideration since the active component of vibrational power is shown to depend on the damping matrix of the finite element description. The presented approach can be further extended to complex metamaterials with multi-dimensional locally resonant configurations to locate critical energy transmission paths within the media of such structures.

Wave Energy Focalization in a Plate With Imperfect Two-Dimensional Acoustic Black Hole Indentation

The acoustic black hole (ABH) phenomenon in thin-walled structures with a tailored power-law-profiled thickness allows for a gradual change of the phase velocity of flexural waves and energy focalization. However, ideal ABH structures are difficult to realize and suffer from potential structural problems for practical applications. It is therefore important to explore alternative configurations that can eventually alleviate the structural deficiency of the ideal ABH structures, while maintaining similar ability for wave manipulation. In this study, the so-called imperfect two-dimensional ABH indentation with different tailored power-law-profiled is proposed and investigated. It is shown that the new indentation profile also enables a drastic increase in the energy density around the tapered area. However, the energy focalization phenomena and the process are shown to be different from those of conventional ABH structure. With the new indentation profile, the stringent power-law thickness variation in ideal ABH structures can be relaxed, resulting in energy focalization similar to a lens. Different from an ideal ABH structure, the energy focalization point is offset from, and downstream of indentation center, depending on the structural geometry. Additional insight on energy focalization in the indentation is quantitatively analyzed by numerical simulations using structural power flow. Finally, the phenomenon of flexural wave focalization is verified by experiments using laser ultrasonic scanning technique.

Quantitative modal analysis of optical power flow and energy loss in photonic structures with a dipole emission source.

Fourier modal method based quantitative analysis method of optical power flow and energy loss in general multi-block photonic structures with an internal dipole emitter is described. The analytic expressions of modal power flow and loss are derived for accurate and efficient quantitative analysis. It is revealed that a few dominating excited photonic modes substantially govern the internal energy flow and energy loss. The optical characteristics of the dominant modes are investigated.

Determining the Power Flow in a Rectangular Plate Using a Generalized Two-Step Regressive Discrete Fourier Series

The evaluation of structural power flow (or structural intensity (SI)) in engineering structures is a field of increasing interest in connection with vibration analysis and noise control. In contrast to classical techniques such as modal analysis, the SI indicates the magnitude and direction of the vibratory energy traveling in the structures, which yields information about the positions of the sources/sinks, along with the energy transmission path. In this paper, a new algorithm is proposed to model operational deflection shapes (ODS). The model is a two-dimensional Fourier domain model that is estimated by using a weighted nonlinear least-squares method. From the wave number-frequency domain data thus obtained, the spatial derivatives that are necessary to determine the structural power flow are easily computed. The proposed method is less sensitive to measurement noise than traditional power flow estimation techniques. A numerical model of a simply supported plate excited by two shakers, phased to act as an energy source and sink, is used as a simulation case. Measurements are executed on a clamped plate excited by an electromagnetic shaker in combination with a damper.

Smart paint sensor for monitoring structural vibrations

A class of smart paint sensors is proposed for monitoring the structural vibration of beams. The sensor is manufactured from an epoxy resin which is mixed with carbon black nano-particles to make it electrically conducting and sensitive to mechanical vibrations. A comprehensive theoretical and experimental investigation is presented to understand the underlying phenomena governing the operation of this class of paint sensors and evaluate its performance characteristics. A theoretical model is presented to model the electromechanical behavior of the sensor system using molecular theory. The model is integrated with an amplifier circuit in order to predict the current and voltage developed by the paint sensor when subjected to loading. Furthermore, the sensor/amplifier circuit models are coupled with a finite element model of a base beam to which the sensor is bonded. The resulting multi-field model is utilized to predict the behavior of both the sensor and the beam when subjected to a wide variety of vibration excitations. The predictions of the multi-field finite element model are validated experimentally and the behavior of the sensor is evaluated both in the time and the frequency domains. The performance of the sensor is compared with the performance of conventional strain gages to emphasize its potential and merits. The presented techniques are currently being extended to sensors that can monitor the vibration and structural power flow of two-dimensional structures.

Active vibration control of finite L-shaped beam with travelling wave approach

In this paper, the disturbance propagation and active vibration control of a finite L-shaped beam are studied. The dynamic response of the structure is obtained by the travelling wave approach. The active vibration suppression of the finite L-shaped beam is performed based on the structural vibration power flow. In the numerical calculation, the influences of the near field effect of the error sensor and the small error of the control forces on the control results are all considered. The simulation results indicate that the structural vibration response in the medium and high frequency regions can be effectively computed by the travelling wave method. The effect of the active control by controlling the power flow is much better than that by controlling the acceleration in some cases. And the control results by the power flow method are slightly affected by the locations of the error sensor and the small error of the control forces.

Real‐time active control of structural energy density and structural power flow.

Structural energy density and structural power flow have long been used as metrics in the active control of vibrating structures. The greater portion of this previous work has focused on frequency‐domain methods which incorporate assumptions about the relative contributions of near‐field and far‐field energy components. This paper describes the implementation of filtered‐x‐based time‐domain control schemes which utilize 9‐ and 13‐accelerometer arrays to estimate and control structural energy density and structural power flow, respectively. Experiments were performed on a clamped steel plate excited and controlled by various combinations of loudspeakers and electrodynamic shakers in a frequency range from 25 to 100 Hz. Analog circuitry was used to estimate spatial derivatives and reduce channel count. The development of control laws incorporating the effects of the analog circuitry is presented. Control attentuation results are given, sensor placement is discussed, and implementation challenges are addressed. [This work is supported by NSF Grant 0826554.]

Relationships between structural energy density, power flow, and their influence on acoustic intensity.

Power flow in structures has been a topic of research for the past few decades. Many different methods for determining structural power flow magnitude and direction have been developed and proven. Structural energy density methods have likewise been developed. In this paper, a study of both structural energy density and power flow in a plate and their influence on the acoustic field is presented. Structural energy density and power flow were determined using equations typically used with accelerometer arrays but adapted for use with a scanning laser doppler vibrometer. Experiments were performed from 25 to 100 Hz by exciting the plate at single frequencies but using different excitation points and methods at each frequency. In some cases two sources of excitation were used to alter the power flow response, giving a broader basis for concluding relationships. Both speaker and shakers were used as sources. The acoustic intensity in the room was calculated at approximately 3 in. from the vibrating plate. Relationships between structural energy density, power flow, and the acoustic field variables are presented.

Vibrational power flow analysis of thin cylindrical shell with a circumferential surface crack

In the view of structure-borne sound, the structural wave and power flow characteristics of infinite thin cylindrical in vacuo shell with a circumferential surface crack are investigated. In this paper, the equivalent distributed line spring is designed to model the surface crack in the shell. The local compliance matrix due to the presence of the crack is deduced from fracture mechanics and three modes of the crack stress intensity factors and their coupling are considered in the local compliance matrix. The vibration of the thin shell is described by the Flügge shell equations. Under the excitation of radial harmonic line force, the input power flow and transmitted power flow of uncracked and cracked shells are obtained. The results show that the vibrational power flow of cracked shell changes substantially due to the presence of crack, and the change is strongly related to the depth and location of crack. Contours of input power flow at different frequencies are constructed to identify the location and depth of the crack. It is revealed that the power flow in the cylindrical shell structures can be used as an alternative defect information carrier. This research provides theoretical basis for the crack detection by measuring the vibrational power flow in cracked shell structures.

Structural power flow analysis of Timoshenko beam with an open crack

In this paper, the structural power flow characteristics of cracked Timoshenko beams are presented. The case of an infinite Timoshenko beam with an open crack is considered. The crack in the Timoshenko beam structure is modeled as a rotational spring with a constant flexibility which is deduced from the relationship between the strain energy and stress intensity factor in fracture mechanics. The flexural wave motions of perfect Timoshenko beam and cracked Timoshenko beam are investigated, then the input power flow and transmitted power flow of perfect and cracked beams are calculated. The results show that the vibrational power flow of cracked beam is highly related to the location and depth of the crack. Contours of input power flow with different frequencies are constructed to identify the location and depth of the crack. This method is very promising for the detection and location of structural damage in the future work.