Beam Theory Research Articles

Dynamic analysis and stability of variable length vertical pipe containing fluid using Timoshenko’s beam theory

In this article, the dynamics of pipes containing fluid is investigated according to Timoshenko’s beam theory. The problem’s equations were extracted with the help of Hamilton’s principle, and then their responses were obtained using Galerkin’s numerical approach. The modeling of the vertical pipe containing fluid was done using Timoshenko’s beam theory. It is prominent to mention that the considered vertical pipe containing fluid has variable length. The results of this analysis will be compared with previous researches, which were according to Euler–Bernoulli theory. Also, the gravity effect on the pipe stability will be analyzed. Finally, by examining the impact of rotational inertia and shear deformation separately, it was determined that the effect of shear deformation on the pipes’ dynamic behavior containing fluid will be more than the effect of rotational inertia.

Study on coupled mode flutter parameters of large wind turbine blades

As the size of wind turbine blades increases, the flexibility of the blades increases. In actual operation, airflow flow can cause aerodynamic elastic instability of the blade structure. Long blades may experience coupled mode flutter due to the bending torsion coupling effect, leading to blade failure. Based on Euler Bernoulli beam theory combined with Theodorsen non directional aerodynamic loads, a blade flutter characteristic equation is established through finite element method. Taking NREL 5 MW wind turbine blades as an example, analyze the influence of parameter changes in different regions of the blades on flutter characteristics. Research has found that paramter changes in the tip region of blade have the greatest impact on flutter characteristics. The vibration frequency shows an overall upward trend with the increase of waving stiffness and torsional stiffness. The flutter velocity of the three regions tends to stabilize as the bending stiffness decreases. The blade flutter speed increases with the increase of torsional stiffness. The radius of gyration is inversely proportional to the flutter frequency and flutter velocity. The impact of centroid offset on blade structure flutter frequency is minimal, but the centroid offset in the tip region has a greater impact on flutter velocity. Increasing the torsional frequency can prevent coupled mode flutter and provide a theoretical basis for blade flutter prevention design.

The Effect of Hyperelasticity and Nonlinearity on the Dynamic Behaviors of Hyperelastic Functionally Graded Beams on Nonlinear Elastic Foundation

Hyperelastic functionally graded materials have a wide range of application prospects in soft robotics and biomedical fields. This paper investigates the nonlinear free and forced vibrations of a hyperelastic functionally graded beam (HFGB) based on higher-order shear deformation beam theory. The geometrical nonlinearity is considered by using the von-Kármán’s nonlinear theory. The three-material-parameter free energy function named as Ishihara model is employed to characterize the hyperelastic material. The power-law gradient form along the thickness direction is adopted. The HFGB is resting on the elastic foundation. The Winkler, Pasternak and nonlinear stiffness coefficients are considered. The time-harmonic external force is applied to the HFGB. The nonlinear governing equations for the vibration of the HFGB are derived by using Hamilton’s principle, and are subsequently transformed into ordinary differential equations via Galerkin’s method. The nonlinear free vibration and primary resonance of the HFGB are investigated analytically by employing the extended Hamiltonian method and multiple scales method, respectively. The results indicate that the power-law index, slenderness ratio, material properties, and elastic foundation parameters have significant influences on the nonlinear frequency of free vibration as well as the frequency–response and force–response curves of forced vibration. The phase plane method is employed to analyze the system’s stability states under various excitation amplitudes. The relative error between the results of the current computational model and the published literature is less than 0.1 percent.

Implications of vibration characteristics towards mass detection using graphene nanoribbon-based bridge configured mass sensor

Owing to its astounding mechanical properties, graphene nanoribbon (GNR) based nano-resonator is aimed as mass sensor with doubly clamped nano beam subjected to pair of point masses adsorbed at different positions over its surface. Though pristine GNR is revealed with non-piezoelectric behaviour inherently, it can be turned into piezoelectric by suitable adatoms. The present analysis investigates the performance of monolayer based (F-, Li- and K-) doped GNRs in the form of vibration attributes such as eigen-frequencies, Q-factor and Force sensitivities. Euler-Bernoulli thin beam theory is applied to model the doped GNR. The obtained results highlight the shift in the eigen-frequency due to adsorbed masses and their positions over the surface of GNRs. Moreover, the Q-factor and Force sensitivities are also determined for the proposed mass sensors. The results indicate that the eigen-frequencies at higher modes are insignificant to the Q-factor. For the fundamental mode, the magnitude of Q-factor varies in the range of 4.39–4.52 for the proposed mass sensor with respect to the positions and magnitudes of pairs of masses. The force sensitivity for the doped GNR is attained at a level of aN/√Hz. Though, force sensitivity rises with the eigen-modes, the sensitivity with the minimum magnitude to be preferred for a sensor as it reflects to early detection of mass. This suggests the fundamental mode to operate the mass sensor. For the fundamental mode, the sensitivity of Li doped GNR is ranging from 103.60–117.58 aN/√Hz, while it is ranging from 109.67 to 124.45 aN/√Hz and 125.31–142.20 aN/√Hz for K- and F- doped GNRs, respectively. The results of the present work are compared with the available literatures. From the results, significant effect of adatom is observed on the vibration characteristics of the GNRs. The findings of the present work can significantly be used to design the GNR based bridge configured mass sensor.

Mechanical properties and regulatory strategy of twinned tetrahedral lattice structures

Given the outstanding mechanical performance of lattice metamaterials, novel structural forms and performance modulation strategies have garnered considerable attention. Herein, we develop a twined tetrahedral lattice structure (TTL) with a bending defect inside the unit cell. Mechanical properties before and after the introduction of defects in TTLs were studied using experiments and numerical simulations. The developed structure exhibits strictly stretching-dominated properties in the absence of bending defects. Whereas, the bending-dominant component of TTLs is elevated with increasing bending angle that can increase energy absorption efficiency. Moreover, the relative elastic modulus and initial peak stress can be reduced by even more than 70 % and 50 %, respectively, as the bending angle increases. Besides, a theoretical model for the relative elastic modulus of TTL at θ = 0° was established based on the displacement method on the Representative Volume Element (RVE), including both Euler-Bernoulli beam theory and Timoshenko beam theory. However, the two solutions show consistent results for strictly stretching-dominated structures. The accuracy of solutions is verified by experiments and simulations. The plateau stress theoretical model for TTLs was derived from the deformation process of the RVE based on plastic hinge theory. The theoretical solutions under different θ align well with numerical and experimental results. Overall, TTLs have superior relative elastic modulus, relative yield strength, and specific energy absorption compared to common lattice structures. The developed TTLs provide a new perspective on the property regulation of mechanical metamaterials.

Lightweight damping layer with acoustic black hole profile bonded to a beam for broadband vibration reduction

Inspired by the thickness reduction and wave focusing characteristics of the acoustic black hole (ABH) effect, a so-called ABH-damping layer (ABHD) is proposed for the broadband vibration reduction of a host structure, different from the common-used uniform thickness damping layer (UNID). The ABHD has a lighter weight and shows better vibration suppression ability. It can overcome the defect that ABHs cannot serve as the main load-bearing component due to the stiffness reduction. The primary beam structure and damping layer both follow the Timoshenko beam theory. The null space method handles the continuity between the two-layer structures. Analyses reveal the difference in vibration suppression characteristics between UNID and ABHD, further exhibiting the wave-gathering properties of ABH. The mass reduction ratio of ABHs is presented under different damping material loss factors. Finally, an experiment is conducted to verify the correctness of the proposed method and the feasibility of designing the ABHD. The damping layer’s weight of ABHD is reduced by 30.8% less than that of UNID, when the material loss factor is equal to 1.0. This study provides a new method for lightweight, strong vibration suppression performance and wide-band vibration reduction in engineering with the application of ABH.

Bandgap accuracy and characteristics of fluid-filled periodic pipelines utilizing precise parameters transfer matrix method

The phononic crystal theory provides a novel approach for effectively controlling the bending vibrations in fluid-filled pipelines. This paper innovatively proposes the precise parameters transfer matrix method to investigate the band calculation accuracy and bandgap characteristics of fluid-filled periodic pipelines with various beam types. Firstly, the differential equations for bending vibration of fluid-filled pipelines are established based on deformation and force analysis. The parameters of the system state are precisely represented by the structural form of multiplying the constitutive matrix with the derivative matrix. Combined with Bloch's theorem, the novel precise parameters transfer matrix method for calculating the band structure is proposed. Secondly, the validity of this method is verified through a comparison with finite element simulation results. A detailed analysis is provided regarding the mechanism of bandgap formation and the effect of fluid filling on the band structure. Then, the influence of shear deformation, moment of inertia, and their coupling on the band calculation accuracy for fluid-filled periodic pipelines is studied based on various beam theories. Finally, it delves into the bandgap characteristics of fluid-filled periodic pipelines under different material parameters, structural parameters, and excitation conditions. This research offers valuable insights for the structural design and vibration damping application in fluid-filled periodic pipelines, providing theoretical support for accurately determining their bandgaps.

Novel method for determination the dynamic elastic modulus of composite wind turbine blades

A novel method, a non-destructive test, is developed to obtain the dynamic elasticity modulus of composite wind turbine blades. The proposed method is based on the free vibration theory of cantilever beams, the sectional analysis of laminated composites, the laminated composite theory, and the area contribution method. In response, the static elasticity modulus and the area properties of a hollow cross-section are obtained. On the other hand, by employing an equivalent beam that integrates all the cross-sections of a wind turbine blade. The method is tested with an equivalent blade, and all incidences of nonlinearity are avoided to obtain linear vibration modes; it is demonstrated that the free vibration test is applicable. Finally, it is observed that the first three natural frequencies were related to the dynamic elasticity modulus, where the third frequency shows an error of less than 1.2 %. This study revealed that the method for dynamics elasticity modulus has good potential to be used in the design of wind turbine blades as a quality control and non-destructive test.

Energy harvesting from vibrations of a beam under moving mass using auxetic cantilever beams: Theoretical and experimental investigations

Recent research aims to enhance piezoelectric energy harvesting for low-power electronic circuits like sensors and wireless systems using cantilever energy harvesters with piezoelectric substrates featuring auxetic structures such as reentrant, missing rib, and double arrowhead patterns. Employing the Wentzel–Kramers–Brillouin (WKB) method is a novel method to analyze these structures based on Euler–Bernoulli beam theory. The study comprises theoretical analysis, finite element simulations, and experimental tests. The highest power output is achieved when consecutive masses excite the bridge using the double arrowhead auxetic unit cell substrate, yielding power outputs of 1.152 μ W / N . The effect of increasing unit cells has also been investigated. Highlights Creating unique analytical methods for auxetic substrate use in energy harvesters. Analyzing three auxetic harvester substrates, emphasizing their unique performance. Auxetic harvesters were studied under various bridge simulator base excitations. The study examines three auxetic structures' effect on harvested power. Study of moving masses' influence on bridge simulator, weights, and velocities.

A data-driven procedure for one-dimensional dynamic analysis of thin-walled beams with arbitrary cross-sections

This paper develops a one-dimensional dynamic model for thin-walled beams with arbitrary complex cross-sections in a data-driven way. In order to consider complicated deformations in the framework of a beam theory, a universal node system is first created to fit the deformed shape of a thin-walled cross-section, whether prismatic or curved, with or without symmetry. This helps to reduce the three-dimensional displacement field of the thin-walled beam to one dimension, and results in a preliminary higher-order beam model with limited precision loss. For the purpose of largely condense DOFs of the new model, a data-driven approach is proposed to identify cross-section deformation modes through the principal component analysis of free vibration deformation data of an unconstrained thin-walled beam. As a result, a compact set of core deformation modes are obtained and selected to update the preliminary model, leading to the refined higher-order beam model of high accuracy and efficiency. Examples are presented to illustrate the concrete implementation and check the effects of data-related controlling parameters of the proposed procedure. We also verify through numerical examples that the proposed model agrees well with plate/shell and other beam theories, and has remarkable advantages in physical interpretation, modeling simplicity and computation efficiency.

Exploring structures of small anionic nickel-ethanol clusters with infrared spectroscopy.

Small anionic nickel clusters with ethanol are investigated with a combination of mass-selective infrared photodissociation spectroscopy in a molecular beam and density functional theory simulations at the BLYP/6-311g(d,p) and TPSSh/def2-TZVPP level. In this context, the O-H stretching vibration of the ethanol is analyzed to obtain information about the structural motif, the geometry of the metal core, and the spin state of the clusters. For the [Ni2(EtOH)]- and [Ni3(EtOH)]- clusters, we assign quartet states of motifs with a hydrogen bond from the ethanol to the linear nickel core. The aggregation of a further ethanol molecule, yielding the [Ni3(EtOH)2]- cluster, results in the formation of a cooperative hydrogen bond network between the nickel core and the two ethanol molecules.

Newton vs. Euler–Lagrange approach, or how and when beam equations are variational

AbstractThere is a clear and compelling need to correctly write the equations of motion of structures in order to adequately describe their dynamics. Two routes, indeed very different from a philosophical standpoint, can be used in classical mechanics to derive such equations, namely the Newton vectorial approach (i.e., roughly, sum of forces equal to mass times acceleration) or the Euler–Lagrange variational formulation (i.e., roughly, stationarity of a certain functional). However, it is desirable that whichever derivation strategy is chosen, the equations are the same. Since many structures of interest often consist of slender and highly flexible beams operating in regimes of large displacement and large rotation, we restrict our attention to the Euler-Bernoulli assumptions with a generic initial configuration. In this setting, the question that arises is: What conditions must the constitutive assumptions satisfy in order for the equations of motion obtained by Newton’s approach to be identical to the Euler–Lagrange equations derived from an appropriate Lagrangian, natural or virtual, for any arbitrary initial configuration? The aim of this paper is to try to answer this basic question, which indeed does not have an immediate and simple answer, in particular as a consequence of the fact that bending moment could be related to two different notions of flexural curvature.

Dynamic modeling and verification of rotating compressor blade with crack based on beam element

Aero-engine blades may crack under harsh operating environments. This reduces the reliability of the aero-engine and causes catastrophic accidents. Although dynamic modeling and vibration characteristics of simplified blades with cracks have been analyzed extensively, there are few studies on the actual compressor blade with varying sections and twisted shapes. Based on Timoshenko beam theory, a new modeling approach for the compressor blade is proposed. Next, according to the strain energy release rate and Castingliano theory, the dynamic model of cracked compressor blades is established considering the stiffness nonlinearity induced by the breathing crack. The modeling approach is validated by the measured natural frequencies of an intact blade. Then, the dynamic model of the cracked blade is verified by comparing the modal characteristics and vibration characteristics using ANSYS software. Finally, some results show that the natural frequencies fn1, fn2, and fn4 increase as the pre-twisted angle and stagger angle increase. Due to the breathing effect, some integer frequencies appear in the spectrum including 0fe and 2fe, and the variations in the amplitude of frequencies 0fr and 2fr are monotonic under different aerodynamic amplitudes. The research can provide some help for the modeling of the actual blade and fault diagnosis of cracked blades.

Verification of Euler–Bernoulli beam theory model for wind blade structure analysis

This paper presents a computational tool for analyzing the structural response of wind turbine blades using the advanced Euler–Bernoulli Beam theory. The objective is to achieve efficient computation of blade structural response under aerodynamic loading without sacrificing computational accuracy. This study involves the development of a Python-based computer program as a numerical tool, which utilizes inputs such as blade geometry and aerodynamic loading from Blade Element Momentum theory. The program outputs axially varying moments of inertia, spatial deflections of the blade, and stress distributions within the blade. The numerical tool is verified by comparing the results with both analytical solutions and far more time-consuming commercial finite element solvers. One of the key contribution of this research is the application of the polygon algorithm, which efficiently calculates the moments of inertia of web-stiffened turbine blades along the blade length. It is shown herein that this algorithm significantly reduces computational time while maintaining numerical accuracy as compared to the results obtained from commercially available finite element codes. The study confirms that the advanced Euler–Bernoulli beam theory, when combined with the polygon algorithm, is capable of accurately predicting the structural response of cambered and twisted wind turbine blades during the initial design phase.

Review of Vibration Analysis and Structural Optimization Research for Rotating Blades

AbstractBlades are important parts of rotating machinery such as marine gas turbines and wind turbines, which are exposed to harsh environments during mechanical operations, including centrifugal loads, aerodynamic forces, or high temperatures. These demanding working conditions considerably influence the dynamic performance of blades. Therefore, because of the challenges posed by blades in complex working environments, in-depth research and optimization are necessary to ensure that blades can operate safely and efficiently, thus guaranteeing the reliability and performance of mechanical systems. Focusing on the vibration analysis of blades in rotating machinery, this paper conducts a comprehensive literature review on the research advancements in vibration modeling and structural optimization of blades under complex operational conditions. First, the paper outlines the development of several modeling theories for rotating blades, including one-dimensional beam theory, two-dimensional plate–shell theory, and three-dimensional solid theory. Second, the research progress in the vibrational analysis of blades under aerodynamic loads, thermal environments, and crack factors is separately discussed. Finally, the developments in rotating blade structural optimization are presented from material optimization and shape optimization perspectives. The methodology and theory of analyzing and optimizing blade vibration characteristics under multifactorial operating conditions are comprehensively outlined, aiming to assist future researchers in proposing more effective and practical approaches for the vibration analysis and optimization of blades.

Analysis of bending vibrations of a three-layered pre-twisted sandwich beam with an exact dynamic stiffness matrix

This paper presents the novel development of the dynamic stiffness matrix for a three-layered symmetric pre-twisted sandwich beam (PTSB), aiming to investigate its free vibration characteristics. The outer layers of the beam are modeled using the Euler–Bernoulli theory, while the core is assumed to deform solely in shear. The boundary conditions and governing partial differential equations of motion are derived based on Hamilton's principle. By applying harmonic variations of displacements, the governing equations of motion are expressed as a tenth-order equation, which is solved to obtain the desired dynamic stiffness matrix. To compute the natural frequencies of in-plane and out-of-plane free vibration for both uniform and PTSBs, the Wittrick–Williams algorithm is employed. The computed frequencies are compared with the results obtained by other authors as well as those obtained from ABAQUS simulations. Various vibration modes of uniform and twisted sandwich beams are plotted and thoroughly discussed. Interestingly, contrary to straight symmetric sandwich beams, the results indicate that flexural displacements in pre-twisted symmetric sandwich beams exhibit coupling in two planes. Additionally, although there are minor changes in vibration frequencies, the mode shapes undergo significant transformations as the pre-twist angle is altered.

An enhanced quasi-3D HSDT for free vibration analysis of porous FG-CNT beams on a new concept of orthotropic VE-foundations

The original primary objective of this study is to conduct a comprehensive investigation into the free vibration behavior of beams with porosity, reinforced by carbon nanotubes (CNTs). These beams are supported by an arbitrary orthotropic variable elastic foundation (AOVEF). The focus lies on understanding how the presence of porosity and CNT reinforcement, coupled with the complex support provided by the AOVEF, influences the vibration characteristics of the beams. In addition, we intend to examine the impact of a number of micromechanical models on the vibration properties of these beams. Four carbon nanotube variables are introduced to symbolize several CNT variations. CNTs’ mechanical properties are examined through a variety of micromechanical models. Shear deformation effects are considered into account in the framework of higher-order shear deformation (HSDT) beam theory. The equations of motion are constructed using Hamilton’s concept and the equation system for the FG-CNT beam with supported ends is solved via the Navier methodology. A new approach in elastic foundation modeling is the Arbitrarily Orthotropic Variable Elastic (AOP-VE) foundation. It builds on the Winkler layer concept but introduces the idea of a variable response along the length of the beam foundation. Unlike the Winkler-Pasternak model, AOP-VE allows for control over the directional properties of the Pasternak foundation. The study looks into multiple aspects, involving various distribution kinds of CNTs, the impact of Winkler and Pasternak factors, mode values, side-to-length ratio, porosity, and angle variation. The results indicate that these parameters have a major effect on the inherent vibration features of FG-CNT beams. The study results in various novel findings that improve the understanding of the subject topic and set a standard for future studies.

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

The Shear Effect of Large-Diameter Piles under Different Lateral Loading Levels: The Transfer Matrix Method

In various analytical models, modeling the behavior of large-diameter monopiles and piles can be challenging due to these foundations with huge body sizes carrying mechanisms of lateral loads to the surrounding soils. In this paper, the transfer matrix method with the Timoshenko beam theory was used to modify the shear rotation of pile sections under different loading stages, including serviceability limit stages and the ultimate loading stage. In this transfer matrix method, a large-diameter pile is considered according to the Timoshenko beam theory, and the recurring variables in the matrix equation are replaced with constants to simplify the calculation steps. Two model test cases were used to verify the accuracy of the method. Then, a series of comparisons between the Timoshenko beam and the Euler–Bernoulli beam theories, with the relative pile–soil stiffness being equal to 0.15, 0.45, and 0.75, was conducted to investigate the differences in pile response after considering the shear deformation. The results show that the effect of shear deformation of large-diameter piles changes with different loading levels. The values of the pile deformation based on the Timoshenko beam theory divided by those of that based on the Euler–Bernoulli beam theory were in the range of 1.0 to 1.10, and they increased slightly with increasing loads, reaching their maximum value, and then rapidly decreased to 1.0 when close to the ultimate lateral load; the maximum value was influenced by the relative pile–soil stiffness. Furthermore, the ratio of the shear rotation of the pile section to the slope of the deflection curve was in the range of 1.0 to 1.10; these also showed similar but more moderate trends compared with the values of pile deformation based on the Timoshenko beam theory divided by those of that based on the Euler–Bernoulli beam theory.

General stability of a triple layer beam with time-varying delay and weak internal damping

Analyzing a triple layer Rao–Nakra beam model which has time-dependent weight function and delay, weak internal damping, with thermal dissipation elements, is our focus. Heat conduction from Fourier’s law was adopted to regulate the thermal dissipation. It is known that when the three equations of the Rao–Nakra beam is directly and globally damped, an exponential stability is achieved. Weak internal damping leads to prolonged resonance, extended vibrational amplitudes and periods, as well as delayed energy dissipation. In this work we will rigorously demonstrate that despite weak internal damping and delay term in the bottom layer’s equation, we can obtain a general stability for the Rao–Nakra beam model. As a result, exponential and polynomial decay can be deduced as particular instances.