Non-uniform Beam Research Articles

Simplified site response analysis for regional seismic risk assessments

Seismic risk assessment on a regional scale requires an estimation of ground motion intensity and its spatial variation over large geographical areas. This task is particularly challenging in the presence of very soft soil deposits where significant amplification occurs at specific frequencies that are often referred to as local resonances. It is often not feasible to conduct a nonlinear site response analysis at thousands of sites of interest across a region such as an urban area due to the computational effort involved and the required detailed information on soil profiles at each site. Thus, in regional seismic risk analyses the hazard is typically represented by a field of response spectral ordinates estimated by ground motion models. This study proposes a simplified site response analysis procedure to improve the characterization of spectral ordinates to be used in regional seismic risk assessments. In the proposed procedure, reduced-order models are used to conduct site response analysis using very few parameters to transform response spectra at rock outcrop into response spectra at the surface of each of the sites. A particular emphasis is given to soft-soil sites characterized by low shear-wave velocities overlaid on rock or firm soils. A non-uniform continuous shear beam model with a parabolic variation of shear modulus along the depth and modal damping is used in combination with Random Vibration Theory. It is shown that the proposed approach provides better estimates than those of ground motion models or physics-based ground motion simulation models which often cannot capture the local resonances. Site-specific validation of the shear beam model and overall simplified site response analysis is performed at four soft-soil sites in San Francisco, California. Despite its simplicity, it is shown that the proposed procedure adequately captures the amplitudes and spectral shapes of the motions. It is also shown that in conjunction with ground motion models or physics-based models, the proposed simplified site response analysis procedure can be used to efficiently simulate the seismic hazard on a regional scale through an example of a regional seismic hazard analysis of 160 softer soil sites in Downtown San Francisco during the Loma Prieta earthquake.

Analysis of free vibration characteristics of non-uniform graphene-reinforced axially functionally graded nanocomposite beam

This paper presents the changing effect on the free vibration behaviour of graphene-reinforced axially functionally graded nanocomposite (Gr-AFG) beam with non-uniform geometry. A generalized finite element method (GFEM) and Euler Bernoulli’s theory have been used for parametric analysis and the subsequent formulation of governing differential equations. The proposed approach (for uniform geometry) has been compared with the existing gradation schemes, e.g. ‘X’, ‘O’, ‘UD’, ‘A’, and ‘V’, and found to be well in agreement. Furthermore, a new harmonic ‘M’ function has been introduced in the current analysis, and the non-uniformity in the geometry has been implemented by varying the cross-section along the axial direction of the beam. The ‘M’ gradation in the axial direction results in an increase in the natural frequency with an increase in the slenderness ratio for the first four modes. Compared to uniform geometry, the non-uniform AFG-M beam with exponential geometry variation shows a lower fundamental frequency for all values of the slenderness ratio of the beam with an increase in the value of the exponent under C–S and C–C end support conditions.

Shape optimization of a non-uniform piezoelectric bending beam for human knee energy harvester

Abstract Scavenging energy from the human body to provide a sustainable source for electronic devices has gained significant attention. Recently, scientists have focused on harnessing biomechanical energy from human motion. This study was dedicated to developing and optimizing a non-uniform piezoelectric bending beam-based human knee energy harvester. The bimorph non-uniform piezoelectric bending beam consisted of a non-uniform carbon fiber substrate and piezoelectric macro fiber composites. Compared to the uniform piezoelectric bending beam, the non-uniform piezoelectric beam can optimize the shape to improve the average strain, thus improving the energy harvesting efficiency. In this study, eight shape functions, including ellipse, sin, tanh, exponential function, parabola, trigonometric line, and bell curves, were investigated and optimized. The bell curve bending beam was selected and fabricated due to its good performance. Then, a benchmark platform was developed to test the deflection curve and reaction force when the nonuniform bending beam was compressed. Finally, to validate the design, experimental testing on three subjects was conducted when they were equipped with the harvester and walked on a treadmill. Testing results indicated that the non-uniform bending beam-based energy harvester can improve the energy harvesting efficiency by 28.57% compared to the uniform beam-based energy harvester. The output power can reach 18.94 mW when walking at 7.0 km h−1.

Model-based wavefront measurement and compensation in atmospheric turbulence.

The model-based method can measure phase aberration without special wavefront detectors. However, the influence of non-uniform beam intensity distribution was not considered, leading to non-negligible system errors. Moreover, no experiments were employed to verify its capability and practicability in atmospheric turbulence. This paper proposes the aberrated pupil method, which can enhance the accuracy of phase recovery by introducing an aberrated pupil function into the traditional model-based method. The effectiveness of the model-based phase measurement and compensation method was experimentally investigated and verified in the application of free-space optical communication. Compared with the traditional model-based method, the aberrated pupil method can reduce the coupling power loss caused by turbulence from 2.59 to 0.87 dB, resulting in a reduction of 1.72 dB. With coherent communication experiments, the model-based phase recovery method significantly improved the communication power budget by about 13.5 dB. The proposed method can measure and compensate for the phase aberration to significantly improve the communication quality of free-space optical communication.

Exponential stability of a non-uniform Euler-Bernoulli beam with axial force

The aim of this paper is to investigate the boundary stabilization of a non-uniform Euler-Bernoulli beam with axial force generated by the equation ρ(x)utt+(σ(x)uxx)xx−(q(x)ux)x=0 defined in (0,1)×(0,∞), where the coefficients ρ(x)>0, σ(x)>0, and q(x)≥0. By adopting a new technique based on a qualitative theory of fourth-order linear differential equations, we prove that the spectrum of the system operator is confined to the open left-half complex plane and that all the associated eigenvalues are geometrically simple. Using this result and the Riesz basis approach, we obtain the exponential stability and the optimal decay rate of the system. This study extends the earlier paper by B.-Z. Guo [SIAM J. Control Optim., 40 (2002), pp. 1905–1923], where no axial force acting on the system (i.e., q≡0).

Free vibration analysis of an O-pattern graphene reinforced axial functionally graded polymer matrix nano-composite non-uniform beam

Recently, the demand for nanocomposites in the form of functionally graded materials (FGM) has increased because of their improved weight-to-stiffness ratios, less delamination effects, and ability to have desired qualities at the right location. Additionally, compared to typical composites, static qualities like strength and elasticity are superior. In this research work, a model for an axially graphene-reinforced functionally graded polymer matrix nanocomposite non-uniform beam is prepared, to obtain the dynamic behavior of the beam in form of its Natural Frequencies. Along the length of the beam, the graphene Nano reinforcement is dispersed in an epoxy polymer matrix as “O” pattern using a function. Material modulus at each location of the beam is modelled using Halpin-Tsai micromechanics theory, the mass density and Poisson’s Ratio of the beam are determined using rule of mixture. The geometry non-uniformity of the beam is modelled using an exponential function. Using MATLAB software code, simulation and parametric analysis of the beam are performed for various slenderness ratios and varied boundary conditions. The non-uniform beam result is obtained after the result for a uniform beam is used to validate it. In result it is analyzed that for the particular geometric and reinforcement configuration of the beam, as the non-uniformity in the beam geometry is increases, the fundamental frequency decreases, and the slenderness ratio has also the same effect on its fundamental frequency.

Analysis of the transverse vibration of a multistepped FGM beam resting on a Winkler foundation in a thermal environment and carrying concentrated masses

This paper examines a novel case study involving the free vibration of a stepped FGM beam-mass system resting on elastic foundations and subjected to a nonlinear or uniform thermal load. The present study examines the complex interaction of thermal effects on dynamic behavior by varying the cross-section and length ratios, material parameters and mass magnitudes. After adopting Euler-Bernoulli's theory, the dynamic behavior of non-uniform FGM beams was investigated using the generalized finite element method (GFEM) and the Rayleigh quotient finite element method (RQFEM). In line with the objective set, relevant results derived from parametric studies, showcasing accentuated variation in dimensional frequency curves and bending moments, have been highlighted and discussed.

Energy generation from friction-induced vibration of a piezoelectric beam

The primary challenge in harnessing vibration energy with piezoelectric materials is the discrepancy in frequency between the energy source and the energy generator, which lowers the efficiency of energy harvesting. To address the challenge, a piezoelectric beam under friction-induced vibration (FIV) is designed, modeled, and studied for the first time to realize the pronounced FIV contributing to energy generation by adapting the vibrations of the continuum structure to align close to its resonant frequencies. The Stribeck friction model is applied to characterize the variation of friction based on the relative sliding velocity vr between contacting objects. The analytical solution is derived to solve the dynamic responses, and transient charging simulation validated by the experiment is utilized to assess energy output. Furthermore, parameter studies are conducted based on the validated model with regard to the material properties of the beam and piezoelectric material, electrode connections of piezoelectric patches, and friction model parameters to investigate their influences on energy output. Considering the same dimensional properties, materials with low Young's modulus E and density ρ are desired for the host structure to facilitate large dynamic strain in piezoelectric materials. With an exponential decay factor C = 8, representing optimal material contact interface, pronounced higher FIV mode can be induced leading to higher output power. A root mean square charging power PeRMS of 42.4 mW and a peak instant charging power Pe − peak of 263 mW can be achieved. In the current study, the model is implemented on a beam coupled with one piezoelectric patch, which has potential applicability to non-uniform beams with various layouts of piezoelectric patches. The presented model enables efficient optimization of continuum structural design for higher piezoelectric energy generation under friction.

Exact solution for mode shapes of cracked nonuniform axially functionally graded beams

This paper established the exact formulas for mode shapes of cracked nonuniform axially functionally graded (AFG) beams. The formula of the mode shape of cracked nonuniform AFG beam is derived in the form of a power presented as a recurrent relation. The power series solution of cracked beam is the same with the intact beam where the effect of cracks contributes to only the first four coefficients of the power series. The derivation of the formula of mode shapes is presented in details. Numerical simulations show that these effects on the natural frequency and mode shape are quite significant. The natural frequency and mode shape are more sensitive to cracks located at positions with low elastic modulus, small cross section and high mass density. For the case of tapered AFG cantilever beam, it is important to consider not only cracks located close to the fixed end but also cracks located close to the free end, especially when the free end has lower elastic modulus and higher mass density.

A stochastic approach to delays optimization for narrowband transmit beam pattern in medical ultrasound

In ultrasound imaging, state-of-the-art methods for transmit beam pattern (TBP) optimization have well-known drawbacks like non-uniform beam width over depth, significant side lobes, and quick energy drop-out beyond the focal region. To overcome these limitations, we develop a TBP optimization approach by focusing on its narrowband approximation and considering transmit delays as free variables instead of linked to specific focal depths. We formulate a non-linear Least Squares problem to minimize the difference between the TBP corresponding to a set of delays and the desired one, modeled as a 2D rectangle elongated along the beam axis. Three metrics are defined to quantitatively evaluate the results. Results obtained by synthetic simulations show that the main lobe width is considerably more intense (+31.5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+31.5\\%$$\\end{document} on average) and uniform (+28.7%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$+28.7\\%$$\\end{document} on average) over the whole depth range compared to classical focalized Beam Patterns. The application of the method to elastography shows improvements in the ultrasound energy concentration along a desired axis.

Mechanical metastructure with embedded phononic crystal for flexural wave attenuation

Destructive interference-based metamaterials have shown excellent characteristics in elastic wave manipulation and vibration attenuation. Nevertheless, challenges persist in the application due to limited space and lightweight design, as current metastructures require additional beam structure. To simplify the design of metamaterials for flexural wave manipulation, this paper presents a new class of embedded phononic crystal for manipulating flexural wave propagation in both one and two-dimensional space by taking advantage of destructive interference, which can effectively suppress the mechanical vibration of a beam structure with a broad band gap. The flexural wave dispersion characteristic in a non-uniform beam structure is derived based on the Euler–Bernoulli beam theory, and an embedded phononic structure with the mechanism of destructive interference is presented to demonstrate its effectiveness in mitigating mechanical vibration. Subsequently, four typical units of embedded phononic structures are designed for attenuating flexural wave propagation in a beam structure. Finally, both numerical simulations, including one and two-dimensional phononic crystals, and physical experiments are implemented to evaluate the performance of the presented metastructure for flexural wave manipulation, which indicates that the proposed embedded phononic structures can effectively mitigate structural vibration in the low-frequency domain. To the best of our knowledge, it is the first attempt to design the metabeam with embedded phononic structures by taking advantage of destructive interference.

General guide concepts for compact, high-brilliance neutron moderators

The trend in neutron sciences is toward integrating compact, high-brightness moderators into new or upgraded facilities. Transporting neutrons from the source to the sample position with a phase-space distribution tailored to specific requirements is crucial to leverage high source brilliance. We have investigated four guide concepts using Monte Carlo ray tracing simulations: Montel beamline with nested Kirkpatrick–Baez mirrors, curved-tapered beamline with a bender and straight sections, straight-elliptical beamline, and curved-elliptical beamline. The straight-elliptical (curved-elliptical) beamline features two half-ellipse guides connected by a straight (non-straight) guide section. The neutron transport efficiency and phase space homogeneity have been quantitatively compared. Our results show that the straight-elliptical beamline performs best because of few neutron bounces on the guide surface with small reflection angles, minimizing flux loss. The Montel beamline provides the best spatial confinement of neutrons within the desired region; however, there is a high thermal-neutron loss due to large reflection angles. The curved-tapered beamline suffers from significant flux loss due to high bounces, and it shows a non-uniform angular distribution related to broad ranges of bounces and reflection angles. The non-straight guide section of the curved-elliptical beamline increases the phase space inhomogeneity, leading to a spatially non-uniform beam profile. The results apply to general neutron instruments that require transporting thermal and cold neutrons from a compact, high-brilliance moderator to the sample location with a moderate phase-space volume.

A mode localization inspired vibration control method based on the axially functionally graded design for beam structures

Vibration control plays a vital role in engineering structures, especially for low-frequency range, because designing structures with small dimensions makes it difficult to control large-wavelength vibrations. The present study seeks another road to realize low-frequency vibration control from the view of the real vibration responses. Similar to mode shapes, deflection modes are the main distributions of vibration amplitudes on structures in low frequency. Therefore, they are also essential vibration characteristics of structures. This paper proposes an effective passive vibration control method under both non-resonant and resonant excitations based on deflection mode theory and optimal algorithm. In fact, it is just like an axially functionally graded design for structures. The equations of motion for non-uniform beam structures are formulated by Hamilton’s principle. Firstly, a desired deflection mode is artificially designed, and the beam structure is discretely divided into subunits. Then, by adjusting the thickness or elastic modulus of each subunit to make the deflection mode of the beam coincide with that of the desired one. This process is completed by the genetic algorithm (GA). After that, by experimental and simulation analyses, the deflection mode of the optimally designed beam structures coincides with that of the desired one. Therefore, the present vibration control method is verified to be correct and effective.

Loaded tooth contact analysis and meshing stiffness calculation for cracked spiral bevel gears

Tooth cracks may occur in spiral bevel gear transmission system of the aerospace equipment. In this study, an accurate and efficient loaded tooth contact analysis (LTCA) model is developed to predict the contact behavior and time-varying meshing stiffness (TVMS) of spiral bevel gear pair with cracked tooth. The tooth is sliced, and the contact points on slices are computed using roll angle surfaces. Considering the geometric complexity of crack surface, a set of procedures is formulated to generate spatial crack and determine crack parameters for contact points. According to the positional relationship between contact point and crack path, each sliced tooth is modeled as a non-uniform cantilever beam with varying reduced effective load-bearing tooth thickness. Then the compliance model of the cracked tooth is established to perform contact analysis, along with TVMS calculations utilizing three different models. By employing spiral bevel gear pairs with distinct types of cracks as examples, the accuracy and efficiency of the developed approach are validated via comparative analyses with finite element analysis (FEA) outcomes. Furthermore, the investigation on effects of cracks shows that tooth cracks can induce alterations in meshing performance of both entire gear pair and individual tooth pairs, including not only cracked tooth pair but also adjacent non-cracked tooth pairs. Hence, the proposed model can serve as a useful tool for analyzing the variations in contact behavior and meshing stiffness of spiral bevel gears due to different cracks.

Multiple-scattering effects on single-wavelength lidar sounding of multi-layered clouds

Abstract. We performed Monte Carlo simulations of single-wavelength lidar signals from multi-layered clouds with special attention focused on the multiple-scattering (MS) effect in regions of the cloud-free molecular atmosphere (i.e. between layers or outside a cloud system). Despite the fact that the strength of lidar signals from the molecular atmosphere is much lower compared to the in-cloud intervals, studies of MS effects in such regions are of interest from scientific and practical points of view. The MS effect on lidar signals always decreases with the increasing distance from the cloud far edge. The decrease is the direct consequence of the fact that the forward peak of particle phase functions is much larger than the receiver field of view (RFOV). Therefore, the photons scattered within the forward peak escape the sampling volume formed by the RFOV (i.e. the escape effect). We demonstrated that the escape effect is an inherent part of MS properties within the free atmosphere beyond the cloud far edge. In the cases of the ground-based lidar, the MS contribution is lower than 5 % within the regions of the cloud-free molecular atmosphere with a distance from the cloud far edge of about 1 km or higher. In the cases of the space-borne lidar, the rate of decrease of the MS contribution is so slow that the threshold of 5 % can hardly be reached. In addition, the effect of non-uniform beam filling is extremely strong. Therefore, practitioners should employ, with proper precautions, lidar data from regions below the cloud base when treating data of a space-borne lidar. In the case of two-layered cloud, the distance of 1 km is sufficiently large so that the scattered photons emerging from the first layer do not affect signals from the second layer when we are dealing with the ground-based lidar. In contrast, signals from the near edge of the second cloud layer are severely affected by the photons emerging from the first layer in the case of a space-borne lidar. We evaluated the Eloranta model (EM) in extreme conditions and showed its good performance in the cases of ground-based and space-borne lidars. At the same time, we revealed the shortcoming that can affect practical applications of the EM. Namely, values of the key parameters – i.e. the ratios of phase functions in the backscatter direction for the nth-order-scattered photon and a singly scattered photon – depend not only on the particle phase function but also on the distance from a lidar to the cloud and the receiver field of view. Those ratios vary within a quite large range, and the MS contribution to lidar signals can be largely overestimated or underestimated if erroneous values of the ratios are assigned to the EM.

Nonlinear analysis on electro-elastic coupling properties in bended piezoelectric semiconductor beams with variable cross section

In this paper, the effects of nonlinear constitutive relation on the electro-elastic coupling behaviors in non-uniform piezoelectric semiconductor beams are investigated. On the basis of three-dimensional theory for piezoelectric semiconductor and double power series, one-dimensional bending model is developed. In order to deal with the nonlinear problem, a differential quadrature method based iteration procedure is proposed. Comparing with the results from finite element method, the proposed method has good convergence and high accuracy. In numerical analysis, the effects of nonlinearity and non-uniformity in piezoelectric semiconductor are discussed thoroughly. It is found the introduction of nonlinearity induces the loss of symmetric for all electrical field quantities but little changes the mechanical quantities. While the reduction of width along the length direction produces larger mechanical and electrical responses. Importantly, the effects of nonlinearity on the electrical field quantities are more evident in a piezoelectric semiconductor beam with rapidly varied cross section. Besides, we designed the piezoelectric semiconductor beams subject to double shear forces and a beam with locally varied cross section. In the designed structures, the potential barriers and wells are realized, which provide new approaches adjusting the electro-elastic behaviors in piezoelectric semiconductors. The study in this paper could be the guidance designing high performance electronic devices.

Visible camera-based diagnostic to study negative ion beam profiles in ROBIN ion source

Ensuring ion beam quality is of prime importance concerning the design of large-size neutral beam injectors(NBIs). Homogeneity, divergence, and beam energy are three crucial parameters defining beam quality. ROBIN(RF-Operated Beam source in India for Negative ions) source is one such prototype for extracting H− ion beam from hydrogen plasma. The extracted beam interacts with background neutrals and emits photons. These emissions are proportional to current density and the visible range emissions are used to characterize the beam using CCD cameras. The observed vertical emission profiles depict Gaussian-type asymmetric wings with a tilted flat-top that suggest non-uniform beam groups from the two grid segments. Divergence data extracted from the camera decreases with the acceleration voltage sweep. A strong correlation of the camera divergence data with calorimeter measurements was observed. A numerical beam model is in development to provide more insight into this correlation and pave the way to study beam group interaction.

Closed-form solutions for axially non-uniform Timoshenko beams and frames under static loading

This paper presents the Green’s Functions Stiffness Method (GFSM) for solving linear elastic static problems in arbitrary axially non-uniform Timoshenko beams and frames subjected to general external loads and bending moments. The GFSM is a mesh reduction method that seamlessly integrates elements from the Stiffness Method (SM), Finite Element Method (FEM), and Green’s Functions (GFs), resulting in a highly versatile methodology for structural analysis. It incorporates fundamental concepts such as stiffness matrices, shape functions, and fixed-end forces, in line with SM and FEM frameworks. Leveraging the capabilities of GFs, the method facilitates the derivation of closed-form solutions, addressing a gap in existing methods for analyzing non-uniform reticular structures which are typically limited to simple cases like single-span beams with specific axial variations and loading scenarios. The effectiveness of the GFSM is demonstrated through three practical examples, showcasing its applicability in analyzing non-uniform beams and plane frames, thereby broadening the scope of closed-form solutions for axially non-uniform Timoshenko structures.

Elastostatics of nonuniform miniaturized beams: Explicit solutions through a nonlocal transfer matrix formulation

A mathematically well-posed nonlocal model is formulated based on the variational approach and the transfer matrix method to investigate the size-dependent elastostatics of nonuniform miniaturized beams. The beams are composed of an arbitrary number of sub-beams with diverse material and geometrical properties, as well as small-scale size dependency. The model adopts a stress-driven nonlocal approach, a well-established framework in the Engineering Science community. The curvature of a sub-beam is defined through an integral convolution, considering the bending moments across all cross-sections of the sub-beam and a kernel function. The governing equations are solved and the deflections are derived in terms of some constants. The formulation uses local and interfacial transfer matrices, incorporating continuity conditions at cross-sections where sub-beams are joined, to define relations between constants in the solution of a generic sub-beam and those of the first sub-beam at the left end. The boundary conditions are then imposed to derive an explicit, closed-form solution for the deflection. The solution significantly simplifies the study of nonuniform beams with multiple sub-beams. The predictions of the model for two limiting cases, namely local nonuniform and nonlocal uniform beams, are in excellent agreement with the available literature data. The flexural behavior of nonuniform miniaturized beams, composed of two to five different sub-beams and subjected to different boundary conditions, is studied. The results are presented and discussed, emphasizing the effects of the material properties, nonlocalities, and lengths of the sub-beams on the deflection. It is demonstrated that the flexural response of nonlocal nonuniform beams is more complex than local counterparts. Unlike the local beams, dividing a nonlocal uniform beam into multiple sub-beams and then reconnecting them changes the overall stiffness of the beam. The study highlights the potential to design nonuniform miniaturized beams with specific configurations to control their flexural response effectively.

An analytical method for free vibrations of the fuel rod with non-uniform mass of small modular reactor

The intricate internal structure of fuel rods results in a non-uniform mass distribution, making it imperative to employ analytical methods for accurate assessment. The study utilizes Euler beam theory to derive the transverse vibration equation for beams with varying mass distribution. The approach involves transforming the non-uniform mass beam into a multi-segment beam with concentrated mass points. Modal function relationships between adjacent uniform segments are established based on continuous conditions at connection points. This transformation leads to the conversion of the variable coefficient differential equation into a nonlinear matrix equation. The Newton-Raphson method is then applied to calculate the characteristic equation and mode shapes, essential for determining natural frequencies. To validate precision, the results obtained are compared with those derived from the finite element method. Furthermore, the developed method is employed to assess the impact of gas plenum location and length on the natural frequency of fuel rods. The proposed methodology serves as a rapid design tool, particularly beneficial during the design phase of fuel rods with non-uniform mass distribution, aiding in configuring structural aspects effectively.