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AbstractThis study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length. [ABSTRACT FROM AUTHOR]

In this study, a distributed rail pad (RP) model of rail supported by double elastomer layers for rigid superstructures like light metro is presented, and a semi-analytical solution is derived via the Galerkin's method. Apart from existing distributed RP models that contain multiple individual spring-dampers, the proposed method offers locally continuous rail bedding. The new model assumes that the rail bedding is not pointwise and connects the continuous bedding and lumped baseplate mass with a derived function, which is the first novelty of the paper. Results of the single-spring model (SSM) and extended locally continuous support models (ELCSM) are obtained and compared to indoor and outdoor measurements in the time and frequency domains. The proposed model shows similar results to the conventional model for dynamic response; however, the clear difference between the models is seen at high resonant frequencies because of the stiffer bedding of the rail. Another point is that rail stresses and displacements can be calculated more accurately compared to SSM, due to including the length of RP. The findings of the study highlight the importance of considering RP length and stiffness when designing and maintaining railway tracks and provide insights into how to optimize these parameters for better performance and durability. [ABSTRACT FROM AUTHOR]

AbstractThe shear rotary inertia caused by the shear deformation affects the natural frequencies and the vibration characteristics of bridges. To guarantee the safety of bridges, it is necessary to investigate the shear rotary inertia as a crucial content. Unlike previous studies, the motivation of this paper is to improve the Timoshenko beam model and further explore the influence of the shear rotary inertia on flexural–torsional natural frequencies and flexural–torsional vibration properties of bridges. Based on the Timoshenko beam theory, a modified dynamic equation of the Timoshenko beam is established. Its natural frequencies are obtained, and its analytical solutions under moving harmonic loads are derived using the integral transform method. In order to further understand the shear rotary inertia, the other four beams (namely, the Euler–Bernoulli beam, the shear beam, the Rayleigh beam, and the Timoshenko beam) are introduced to make comparisons with the modified Timoshenko beam. Subsequently, in the parameter analysis, the natural frequencies and vibration responses of five different beams with various lengths are analyzed numerically. It is discovered that the shear deformation and the shear rotary inertia have great effect on the high-mode vertical and bridge flexural–torsional frequencies, and it can improve the accuracy of calculating the natural frequencies of beams by about 6.00%. Furthermore, the shear deformation effect on vertical dynamic responses under harmonic moving loads intensifies with the beam spans increasing. The shear beam theory and the Timoshenko beam theory are appropriate for the vibration analyses of short beams, and the Rayleigh beam theory and the Timoshenko beam theory are suitable for the vibration analyses of long beams, while the modified Timoshenko beam theory, being more generally, is suitable for both beams. [ABSTRACT FROM AUTHOR]

The Acoustic Black Hole (ABH) is an effective structure for harvesting and dissipating energy from flexural waves. It works by reducing the thickness of the ABH wedge in a power law relationship with m ≥ 2, which rapidly decreases the local phase velocity of flexural waves and leads to an energy convergence effect. However, the vibration performance may differ if the tapered wedge does not meet the requirements of a typical ABH. This article introduces and discusses two types of "pseudo-ABH" wedges: linear-degenerate ABH and step-degenerate ABH. These wedges do not follow the power-law and continuity conditions of typical ABH, respectively. To establish the dynamic models of pseudo-ABH wedges, the lumped-mass transfer matrix method (LTMM) and traveling wave analysis are introduced and validated using analytic solutions of the non-uniform one-dimensional Euler-Bernoulli beam and finite element method. Meanwhile, a good match between the results of the Timoshenko and Euler-Bernoulli beam theory also indicates the correctness of our model. The energy ratio and reflection coefficient are defined to characterize the energy accumulation effect and vibration absorption performance of the pseudo-ABH wedges. The vibration performance of the linear-degenerate ABH is closer to the typical ABH than the step-degenerate ABH. The step-degenerate ABH has more effective energy accumulation and reduction performance for individual frequencies. However, as the number of segments in the step-degenerate ABH increases, its dynamic performance becomes closer to that of the typical ABH. The novelty of this article is to study vibration energy accumulation and absorption characteristics of pseudo acoustic black hole wedges using LTMM model, disclose the differences of linear and step degenerate ABH for their performance, and recommend the step degenerate ABH in place of the typical ABH of a power law thickness relationship. [ABSTRACT FROM AUTHOR]

The interest in phononic crystals and acoustic metamaterials has been an intensive subject of research in recent years. Finding a robust way to significantly expand or actively control the bandgap has received extensive attention. In this study, we propose a prestressed metamaterial beam attached with multiply local resonators connected by actively tunable piezoelectric springs. The Euler–Bernoulli beam theory and Timoshenko beam theory are applied in the theoretical analysis of the system. Further, the spectral element method is utilized to analytically compute the dispersion relation and transmission ratio and excellent agreement with reference to the benchmark is reported. The influences of an external axial force on the bandgap range and attenuation behavior are further studied. Subsequently, the effect of resonator number and mass on the local resonance bandgap structure is investigated in two parametric studies. The active control of bandgap range and frequency is then verified. By analyzing frequency response function, the tunable transmission ratio of a supercell can be observed. To conclude, this paper not only provides a guideline for designs of wave attenuation with multiple frequency regimes in a one-dimensional system, but it can also be extended to sub-wavelength wave manipulation designs. [ABSTRACT FROM AUTHOR]

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes. [ABSTRACT FROM AUTHOR]

Analytical methods for calculating the buckling loads in laminated panels often assume uniform edge loads without any openings, overlooking real-world conditions with diverse non-uniform edge loads and varying cutout sizes. This study aims to explore the buckling behavior of composite panels with and without stiffeners under different non-uniform edge load distributions by developing a robust and computationally efficient finite element (FE) formulation. Buckling loads are determined for different plate aspect ratios, number of stiffeners under various non-uniform edge-loading cases. Furthermore, it addresses the effects of different layup schemes and stiffener eccentricities.The analysis employs a 9-noded heterosis plate element and a compatible 3-noded beam element, incorporating shear deformation and rotary inertia for both the plate and stiffeners. Additionally, due to the non-uniform stress distribution in the stiffened panels, a unique dynamic technique has been implemented to account for the stability performance by employing two sets of boundary conditions. The study demonstrates that maintaining stiffener eccentricity (est) within the range of 2.0 to 3.0 notably improves buckling strength depending on the type of edge loads. [ABSTRACT FROM AUTHOR]

The free vibration dynamic modellings of sandwich beams with periodic lattice truss cores are established by combining the Bernoulli–Euler beam theory and Timoshenko beam theory, and the fundamental frequency and unit cell mass of lattice sandwich beams are optimized based on the non-dominated sorting genetic algorithm-II (NSGA-II). Firstly, the vibration control equation of lattice sandwich beams is derived using Hamilton's principle. Then, it is compared and verified by modal experiment and finite element numerical simulation. On this basis, the influence of different truss cores on the structure's natural frequency is further analyzed. Finally, taking the sheet thickness, core thickness, and radius of the truss as the optimization design variables and taking the structural fundamental frequency maximization and mass minimization as the optimization objectives, the Pareto front is obtained by NSGA-II, and the optimization results are verified by theoretical and numerically. The research results show that the theoretical method in this article can well predict the natural frequencies of sandwich beams with different truss cores. The optimized sandwich structure has higher fundamental frequency with lighter weight, which effectively improves the anti-vibration ability. [ABSTRACT FROM AUTHOR]

• A new modeling approach for compressor blades is proposed by the Timoshenko beam theory. • The dynamic model of cracked blades is proposed considering the stiffness nonlinearity induced by breathing cracks. • The modeling method is verified by experiments and the proposed dynamic model is validated using ANSYS software. 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 f n1 , f n2 , and f n4 increase as the pre-twisted angle and stagger angle increase. Due to the breathing effect, some integer frequencies appear in the spectrum including 0 f e and 2 f e , and the variations in the amplitude of frequencies 0 f r and 2 f r 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. [ABSTRACT FROM AUTHOR]

In the current study, the large deflection of nanobeam according to the modified couple stress theory (MCST) with surface effects (SE) was investigated by applying finite element method (FEM). For modeling of the nanobeam, we applying the Timoshenko beam theory (TBT), and for the solution of the problem, we used a total lagrangian finite element method. Also, shear deformation and axial displacements of nanobeams have been considered in this model. The FEM has been used for the solution of the problem. With the present model, small to large deflection of nanobeam with arbitrary loadings and boundary conditions can be studied. The numerical results for the small deflection of nanobeam achieved by the present model are well matched with results reported in the previous researches. Effects of the ratio of material length scale parameter to thickness on the large deflection of nanobeams have been investigated. Also, the influence of considering different beam constitutive theories in the large deflection analysis of nanobeams has been examined. The results achieved in this study show the relative importance of considering surfaces and size effects in the analysis of nanobeam. [ABSTRACT FROM AUTHOR]

• An effective method is proposed to analyse the longitudinal mechanical response of shield tunnel during its construction. • The influence of the shearing dislocation of circumferential joints on the uplift of shield tunnel is fully considered. • A coefficient is introduced to accurately calculate the dynamic buoyancy generated by the synchronous grouting. • The effect of the equivalent shear stiffness of shield tunnel on the lining uplift is quantitatively analysed. During shield tunnelling, longitudinal deformation occurs in the shield tunnel structure owing to the buoyancy generated by the synchronous slurry, which has adverse effects on the assembly quality and even the structural integrity of the segmental ring structure. However, the current longitudinal beam-like model for analysing the uplift response during shield tunnel construction adopts the Euler–Bernoulli beam to describe the shield tunnel. This model does not consider the shearing dislocation effect, causing errors in the structural longitudinal deformation analysis. In this context, this study proposes a shield tunnelling uplift model based on the Timoshenko beam theory, which can effectively consider the shearing dislocation effect, aiming to evaluate the longitudinal response of shield tunnels more accurately. A Pasternak two-parameter foundation with springs reflecting the resistance of the overburden with different thicknesses was adopted to simulate the supporting effects of the synchronous grouting layer and surrounding soil layer composite on the tunnel, considering the influences of both the shearing effect of the foundation and tunnel buried depth on the shield tunnel uplift. The dynamic buoyancy coefficient, determined through an inverse analysis of the field measurement data, was introduced to accurately calculate the magnitude of the dynamic buoyancy generated by the flow of synchronous slurry. The proposed model was validated by comparing its results with actual measurements. The uplift characteristics of the shield tunnel during construction were obtained by using the proposed model in an actual project. It was found that the longitudinal analysis model for shield tunnel uplift based on the Euler–Bernoulli beam theory underestimates the uplift and overestimates the internal forces generated during shield tunnelling. [ABSTRACT FROM AUTHOR]

The study focuses on the free vibration analysis of beams made of axially functionally graded materials (AFGM) with curvilinear variable cross-sections along their length. The beams encompass various shapes, including concave and convex conic sections, with axial material properties varying according to polynomial and exponential laws. The equations of motion are derived using Hamilton's principle within the framework of Timoshenko beam theory. These governing equations, subjected to various boundary conditions, are solved using the differential transform method (DTM). The proposed solution technique is validated by comparing computed natural frequencies with the existing literature and results obtained using three-dimensional finite element analysis in ABAQUS. The incorporation of material gradients into the beam finite element models was achieved using the user-defined material subroutine (UMAT). Additionally, a comprehensive study is conducted to examine the influence of various factors on the natural frequencies of functionally graded beams. These factors include parameters of material laws, types of variable beam shapes, slenderness ratio, and specific boundary conditions. This study provides a thorough understanding of the modal dynamics of the considered beams, offering valuable insights into the behavior of FGM structures. [ABSTRACT FROM AUTHOR]

In the current paper, size‐dependent mechanical analysis of functionally graded porous (FGP) nanobeams resting on Pasternak foundation in thermal environment is presented based on nonlocal strain gradient theory and Gurtin‐Murdoch surface elasticity theory. The nanobeam is modeled based on Euler‐Bernoulli beam theory, Timoshenko beam theory and Reddy's third‐order shear deformation theory and the set of the governing equations are solved for a clamped‐clamped FGP nanobeam using generalized differential quadrature method. Numerical results show that as porosity parameter increases, static deflection increases and critical buckling load decreases, but its effect on the fundamental frequency strongly depends on the porosity distribution pattern. It was revealed that among all studied porosity distribution patterns, the minimum value of the static deflection and maximum values of critical buckling load and fundamental frequency can be achieved using symmetric distribution pattern of pores. It is concluded that tensional residual stress reduces static deflection and increases both critical buckling load and fundamental frequency and temperature rise has reverse effects. [ABSTRACT FROM AUTHOR]

Integral forms of Eringen's nonlocal theory are employed to model nanoscale constitutive relations without possible potential paradoxes. The Timoshenko beam theory as well as the element-free Galerkin method are accepted to model rotating nanobeams and develop numerical solutions of critical rotating angular velocities for nanobeams. Optional setting angles as natural defects are taken into account in buckling analyses. Important design parameters, including length-to-height ratios, ratios of hub radii to lengths, setting angles, and additional supports are carefully assessed between local and nonlocal beams. [ABSTRACT FROM AUTHOR]

A wavelet based finite element method (WBFEM) using B-spline wavelet on the interval is developed in this paper for modeling multi-span pipes conveying fluid with different support conditions. The pipe element is considered as a beam element which is formulated based on Euler–Bernoulli and Timoshenko beam theories. The generalized eigenvalue problem is formulated and solved to obtain the modal characteristics of the multi-span pipe. In comparison to the conventional finite element method, the developed WBFEM discretization scheme uses appreciably less elements for modeling the multi-span pipe. Numerical solutions are obtained and compared with some available results in the literature and those obtained by ANSYS to verify the accuracy and efficiency of the developed method. [ABSTRACT FROM AUTHOR]

Abstract Visual examination of the surface topography, in conjunction with the other sensors, may confirm the existence of chatter. Online chatter detection during real machining operations is possible with the use of sensors, and the presence of noise in their output and restricted bandwidth are the major drawbacks of these sensors. Productivity drops and manufacturing costs go up when there is a lot of chatter in the machining process. In the present paper, an integrated spindle tool system is modeled using finite element method with Timoshenko beam theory including rotational and shear deformation effects. To maximize the average stable depth of cut in an end milling process while simultaneously minimizing the chatter vibration levels, real time and offline strategies have been investigated. Machining experiments performed on Al6061-alloy specimens provide an empirical confirmation of the stability boundaries. The surface topography methods such as scanning electron microscope (SEM) and optical microscope images along with vibration levels are considered, to identify chatter marks under various machining conditions, which helps to assure cutting process stability. Stability lobe diagrams are plotted with these derived conditions and observed at an incremental level in the axial depths of the cut. The methodology shown in this paper improves the machining stability of the end milling with the reduction in the tool tip vibrations.

AbstractA theoretical model of a quartz crystal microbalance carrying a layer of substrate and an array of micropillars based on the classical and modified Timoshenko beam theory was established. By choosing specific substrate thickness and micropillar height, this composite sensor approaches the coupled resonance state, and thus significantly improving the mass sensitivity. The intentionally designed sensor is extremely sensitive to tiny mass. However, with increasing mass, the mass sensitivity decreases sharply and then tends to be constant. The frequency shifts calculated by two models have some differences, but their mass sensitivity is almost the same. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic equation of a rigid body coupled with two flexible beams was established, considering the rotational motion of the rigid body and the elastic deflection of the flexible beams. The coupling effect between the rigid body rotation and flexible beam vibration was studied in detail. The dynamic response of the flexible beam with piezoelectric effect was established based on Hamilton’s principle. Under the premise of ignoring the nonlinear vibration coupling term of flexible beams, only the first two modes of transverse displacement and one directional rotation motion were studied in case studies. It was found that the coupling term caused by rigid body rotation can slightly alter the natural frequency and amplitude of flexible beams, mainly reflected in the first-order mode. With increasing the external excitation on the rigid body, the induced transverse displacement at the tip of the flexible beam increased significantly. Meanwhile, when the flexible beam was under external excitation, the vibration could also induce rigid body rotation. With the inverse piezoelectric effect, one may actively actuate the flexible beam vibration and rigid body rotation or control the undesired motion of the system. The control effect can be adjusted by varying the applied voltage excitation on the piezoelectric patches as well as patch positions. The conclusions drawn from this study can be applied to active precise control of space facilities such as satellites. Based on the dynamic modeling of rigid–flexible multi-body systems, this paper completed the study of the coupling effects of rigid body rotation and flexible body vibration, and proposed an application idea for completing small-angle maneuvers of rigid–flexible multi-body systems through piezoelectric control. [ABSTRACT FROM AUTHOR]

Visual examination of the surface topography, in conjunction with the other sensors, may confirm the existence of chatter. Online chatter detection during real machining operations is possible with the use of sensors, and the presence of noise in their output and restricted bandwidth are the major drawbacks of these sensors. Productivity drops and manufacturing costs go up when there is a lot of chatter in the machining process. In the present paper, an integrated spindle tool system is modeled using finite element method with Timoshenko beam theory including rotational and shear deformation effects. To maximize the average stable depth of cut in an end milling process while simultaneously minimizing the chatter vibration levels, real time and offline strategies have been investigated. Machining experiments performed on Al6061-alloy specimens provide an empirical confirmation of the stability boundaries. The surface topography methods such as scanning electron microscope (SEM) and optical microscope images along with vibration levels are considered, to identify chatter marks under various machining conditions, which helps to assure cutting process stability. Stability lobe diagrams are plotted with these derived conditions and observed at an incremental level in the axial depths of the cut. The methodology shown in this paper improves the machining stability of the end milling with the reduction in the tool tip vibrations. [ABSTRACT FROM AUTHOR]

In this study, the instability and bifurcation diagrams of a functionally graded (FG) porous sandwich beam on an elastic, viscous foundation which is influenced by an axial load, are investigated with an analytical attitude. To do so, the Timoshenko beam theory is utilized to take the shear deformations into account, and the nonlinear Von-Karman approach is adopted to acquire the equations of motion. Then, to turn the partial differential equations (PDEs) into ordinary differential equations (ODEs) in the case of equations of motion, the method of Galerkin is employed, followed by the multiple time scale method to solve the resulting equations. The impact of parameters affecting the response of the beam, including the porosity distribution, porosity coefficient, temperature increments, slenderness, thickness, and damping ratios, are explicitly discussed. It is found that the parameters mentioned above affect the bifurcation points and instability of the sandwich porous beams, some of which, including the effect of temperature and porosity distribution, are less noticeable. [ABSTRACT FROM AUTHOR]

• Development of novel inverse differential quadrature tool for curved beam analysis. • Three-order improvement in accuracy with iDQM compared to DQM solutions. • 98% computational savings versus DQM-based Strong Unified Formulation. • Greater accuracy of higher-order derivatives than lower order Finite elements required for transverse shear stress recovery. Recent advancements in fibre placement technologies have expanded the potential applications of variable stiffness curved composite beams in industries such as aerospace, automotive, and naval engineering. Accurate solution techniques for examining these beams, especially those composed of advanced composite materials, are indispensable. In view of this demand, this study proposes a new high-order computational tool that combines the higher-order accuracy of the emerging inverse differential quadrature method (iDQM) and the simple kinematics of the Timoshenko beam theory for efficient and accurate prediction of the static behaviour of both constant stiffness and variable stiffness curved beam structures. This novel application of iDQM to curved beam analysis is leveraged upon its excellent potential to mitigate differentiation-induced errors by using the so-called indirect approximation strategy. Simple procedures for implementing different orders of iDQM models are presented to analyse curved beam problems, and are independently benchmarked against closed-form Navier's solutions, as well as numerical solutions obtained through the differential quadrature method (DQM) and finite element method (FEM), demonstrating excellent spectral accuracy. Furthermore, the iDQM scheme offers outstanding potential in recovering transverse shear stress, achieving superior accuracy over lower-order FEM in approximating higher-order derivatives. Remarkably, iDQM predictions for variable stiffness curved beams exhibit satisfactory agreement with the Strong Unified Formulation, achieving over 98% computational efficiency. Finally, convergence analysis of iDQM solutions reveal up to three orders of improved accuracy and faster convergence rates compared to the DQM, constituting a new benchmark for curved beam analysis. [ABSTRACT FROM AUTHOR]

This work aims at studying the interfacial behavior of functionally graded coatings (FGCs) on different substrates, here modeled as asymmetric double cantilever beams, in line with the experimental tests. An enhanced beam theory (EBT) is proposed to treat the mixed-mode phenomena in such specimens, whose interface is considered as an assembly of two components of the coating/substrate system bonded together partially by an elastic interface. This last one is modeled as a continuous distribution of elastic–brittle springs acting along the tangential and/or normal direction depending on the interfacial mixed-mode condition. Starting with the Timoshenko beam theory, we determine the differential equations of the problem directly expressed in terms of the unknown interfacial stresses, both in the normal and tangential directions. Different distribution laws are implemented to define the functional graduation of the material in the thickness direction of the specimens, whose variation is demonstrated numerically to affect both the local and global response in terms of interfacial stresses, internal actions, energy quantities and load–displacement curves. The good accuracy of the proposed method is verified against predictions by a classical single beam theory (SBT), with interesting results that could serve as reference solutions for more expensive experimental investigations on the topic. [ABSTRACT FROM AUTHOR]

This paper plans to propose a general numerical model for investigating the free and forced vibration behaviors of the multi-stepped functionally graded porous graphene platelet-reinforced composite (FGP-GPLRC) curved beam with various boundary conditions based on the first-order shear deformation Timoshenko beam theory by employing Rayleigh–Ritz method in conjunction with Jacobi polynomials. Different porosity and GPL distribution types are considered. The general boundary conditions of the multi-stepped FGP-GPLRC curved beam and connecting conditions of the FGP-GPLRC curved beam elements are simulated by employing boundary springs and connecting springs, respectively. The convergence and validation of the proposed numerical model are demonstrated by comparing the proposed results with the corresponding results which come from open literature and finite element software ABAQUS. The free and forced vibration behavior analysis of multi-stepped FGP-GPLRC curved beam is carried out by investigating the influences of material property and geometric parameters on the natural frequency and displacement response in the frequency and time domains of multi-stepped FGP-GPLRC curved beam structure. The investigation results can offer the technique guidance for the design and manufacture of the multi-stepped FGP-GPLRC curved beam. [ABSTRACT FROM AUTHOR]

This paper aims to present better physical understanding of flexural wavenumber spectral behaviors of multi-stepped periodic beams by the method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory. According to the Bloch theorem, the formulation of MRRM is presented and the complex flexural wavenumbers of multi-stepped periodic beams are solved by numerical techniques. The validation of MRRM in analyzing flexural wave behaviors of multi-stepped periodic beams is verified by comparison with the analytical wavenumber spectrum of a homogeneous beam. Numerical examples are provided to illustrate the general wavenumber spectral characteristics of an infinite periodic beam with alternate variation of cross-sections. The effects of the material properties and geometrical parameters of the beam segments are evaluated. The most remarkable finding inspired from the wavenumber spectral curves is that the real wavenumber of the multi-stepped periodic beam does not vary with frequency in the attenuation bands, which indicates that the wavelength of the flexural wave is constant in the attenuation bands. Highlights A multi-stepped periodic beam model with arbitrary finite number of beam segments is established. Complex wavenumber spectrum analysis is performed to reveal the propagation and attenuation behaviors of multi-stepped periodic beams. Similarity and comparative studies on the propagation and attenuation characteristics of beams of a uniform cross-section and of variable cross-sections are carried out to provide better physical understanding. An innovative finding that the wavelength of the flexural wave is constant in the attenuation bands is achieved. [ABSTRACT FROM AUTHOR]

Based on Kirchhoff thin plate and Mindlin thick plate theories, the vibration and energy flow characteristics of clamped stiffened plate are studied by using the analytical model constructed by finite integral transform method. The results show that the energy flow characteristics of the stiffened plate at the beam/plate coupling interface depend on the position of the rib in the vibration modes of the plate. The effects of shear deformation and rotatory inertia on the energy flow across the beam/plate coupling interface of the stiffened plate are further investigated. It is found that the inclusion of rotatory inertia of the beam and plate in the model only affects the energy flow component controlled by the moment coupling but not that controlled by the shear force coupling. Whilst the inclusion of the shear deformation of the beam and plate mainly causes a decreased amplitude of the energy flow for the mode group where the beam is located away from both the nodal and antinodal lines of modes, in addition to the shear deformation of the plate which also leads to an increased amplitude of the energy flow component controlled by the shear force coupling for the mode group where the beam locates at the antinodal line of modes. The understanding of energy flow characteristics of the stiffened plate at the beam/plate interface is essential to effectively control the noise and vibration problems of structures such as transformer tanks and machine covers. [ABSTRACT FROM AUTHOR]

This study aims to analyze the load-bearing characteristics of non-pneumatic tires with composite spokes using experimental and finite element simulation methods and to establish a mechanical analysis model based on the Timoshenko beam theory. Subsequently, experiments were conducted on carbon fiber-reinforced plastics and rubbers to establish the corresponding constitutive model. A finite element model of the non-pneumatic tires with composite spokes was also developed. The main structural and material parameters were selected, and their correlation with the vertical stiffness of the non-pneumatic tires with composite spokes was studied using response surface methodology. The stiffness characteristics of the composite spokes were simplified, and a load-bearing characteristic analysis model was established. The results indicated that among the parameters of the reinforcement plate structure and rubber, the constitutive parameter C10 of the rubber in the spokes had the greatest impact, with a comprehensive influence value of 319.83 N/mm. Under a load of 5000 N, the load-bearing characteristic analysis model results were consistent with those of the finite element simulation, with a maximum relative error of 7.49%. The proposed load-bearing characteristic analysis model can assist in the rapid design and performance prediction of non-pneumatic tires with composite spokes. [ABSTRACT FROM AUTHOR]

Needle-punched carbon/carbon (NP C/C) composite is widely used in rocket-engine nozzles and re-entry vehicles. Recyclable technology expedited the research on repeated oxidation and residual mechaincal properties of thermal-protection materials. In this study, the critical longitudinal compression strengths before and after oxidation are derived based on the Timoshenko beam theory. Three repetitions of short-term oxidation cycles and compression experiments are investigated. The average oxidation rate of this composite was 5∼6% in 10 min and kept linear increase. In-plane and out-of-plane compressive strengths of NP C/C composite diminish quasi-linearly due to oxidation at 1000°C, with their moduli decreasing in a periodically slow-sharp pattern. After three oxidation cycles, the levels of residual in-plane modulus and strength were 55.20% and 56.89%, respectively, while the resudual out-of-plane modulus and strength were 44.65% and 47.23%, respectively. The results showed that the material exhibited the pesudo-plastical behaviour after oxidation, cracks grew along the punched conical structures formed by the punching technology. In-plane and out-of-plane modulus were more sensitive than their strengths after first oxidation cycle. [ABSTRACT FROM AUTHOR]

Elastic beam structures connected by hinges are a common basic component of many mechanical systems. In this study, the nonlinear transverse vibration of a beam with one end free and the other end simply supported with a torsion spring is investigated. A torsional quasi-zero-stiffness vibration isolator is employed at the fixed end of the beam for isolating the vibration excitation from the base. Using Hamilton's principle, the differential equation of motion for the transverse vibration of the beam is obtained, and the natural characteristics of the beam are determined. The dynamic equation of the system is derived by using Galerkin truncation method and the harmonic balance method is adopted to obtain the nonlinear vibration response. Compared with the vibration transmissibility of the beam supported by the equivalent linear torsion spring, the quasi-zero-stiffness vibration isolation system shows superior vibration isolation performance. It is found that the beam fixed with a torsional quasi-zero-stiffness isolator can reduce the first resonant frequency and provide a wide effective isolation range at low frequency, which is rarely affected by the strong nonlinearity. The results would provide an innovative method of introducing the torsional quasi-zero-stiffness isolator into the passive vibration control of continuum system. [ABSTRACT FROM AUTHOR]

A structural analysis of framed structures using the finite element method considering both the Bernoulli-Euler and the Timoshenko beam theories can be performed adopting cubic interpolation functions that yield analytical solutions for the displacements. However, these theories may not provide stress results with sufficient accuracy. In such cases, it is necessary to employ higher order beam formulations, that may require a high level of discretization. Therefore, this study proposes an enhanced Reddy beam element, obtained by considering interpolation functions calculated directly from the solution of the differential equation system. This solution minimizes the impact of structural discretization on the analysis, and framed structures can be effectively modeled considering the minimum number of elements required to describe the geometry. The results obtained by the proposed formulation were compared against classical beam theories and the Reddy beam model adopting conventional shape functions, showing the efficacy of the proposed element in simulating the elastic behavior of framed structures in a FEM-like procedure. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the damped nonlinear vibration of cracked functionally graded (FG) graphene platelets (GPLs)-reinforced composite (FG-GPLRC) dielectric beam. The effective material properties of the composites are evaluated by effective medium theory (EMT) and rule of mixture. Governing equations incorporating damping and dielectric properties are derived from an energy method with the framework of Timoshenko beam theory and nonlinear von Kármán strain–displacement relationship. Stress intensity factor (SIF) of cracked FG-GPLRC beam at the crack tip is obtained via finite element method (FEM). Differential quadrature (DQ) and direct iterated methods are utilized to discretize and solve the nonlinear system. Accuracy and convergence of the model and the solution are verified. An extensive numerical study is performed to examine the effects of crack location and depth, damping and attributes of GPL and the applied electric field on the nonlinear vibration behavior of the cracked FG-GPLRC beam. It is found that the frequency ratios of cracked FG-GPLRC beams are more sensitive to the applied electric field when the crack with larger depth is located close to the mid-span. The cracked FG-GPLRC beams with FG distribution profiles exhibit better stability. [ABSTRACT FROM AUTHOR]

In this paper, the flexural wave propagation and its control of a novel piezoelectric composite pipe conveying fluid are investigated. Dual piezoelectric layers used as sensor and actuator are periodically arranged on the pipe, and a feedback amplifying circuit is applied from sensor to actuator, forming a self-powered phononic crystal (PC) control structure. The vibration reduction performance can be actively tuned by adjusting the feedback control gain instead of conventional changing the construction of pipe itself. The pipe is composed of functionally graded material (FGM), in which the material properties vary continuously along the radial direction, and a poroelastic medium is introduced. By using the Timoshenko beam theory and Hamilton's principle, a set of electromechanical coupling equations governing flexural vibration of the pipe is deduced. The band structure, band gap (BG) distribution and frequency response are presented by applying the spectral element technology. Comprehensive parametric studies are carried out. The results obtained validate the excellent vibration control effect of the proposed design, and further demonstrate the significant impacts of material, piezoelectric layers, feedback control and flowing fluid on the BG characteristics. This paper is expected to provide a technological reference for the vibration and elastic wave control of engineering composite pipe structures. [ABSTRACT FROM AUTHOR]

A simplified mechanical model for investigating pile–pile horizontal vibration is developed by combining the Pasternak model and the Timoshenko beam theory, with the consideration of the influence of axial load at the head of the source pile. The expressions for horizontal displacements and internal forces of the source pile are obtained with respect to the differential transform method and pile–soil boundary continuity conditions. Additionally, considering that the vibration of the source pile affects the dynamic displacements of the receiver pile, the dynamic equilibrium equation of the receiver pile is established. Then, a closed-formed solution for the response of the receiver pile is derived from the obtained expressions. Further, the pile–pile horizontal dynamic interaction factor is derived based on its definition and validated by comparison with existing related analytical solutions. Finally, the effects of the shear deformations of soil and pile, pile slenderness ratio, and axial load on the pile–pile horizontal dynamic interaction factors are examined. It is indicated that a smaller pile-to-soil modulus ratio of the near-surface layer results in a more significant effect of shear deformation of the soil. In particular, when L/d is small (i.e., L/d < 5) or the pile cross section is annular, the effect of shear deformation of the pile is not ignorable. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Vibration of aircraft wings and the dynamic stress concentration at the clamped edge are important research topics due to concerns on the safety of aircrafts. To have a better understanding of the problem, the free and forced vibration response of a ribbed rectangular cantilever plate representing a section of an aircraft wing is investigated in this study. A new analytical solution is developed for the vibration analysis of rib stiffened cantilever plates using Mindlin plate and Timoshenko beam theories alongside the finite integral transform technique. The one- and two-dimensional integral transforms are applied to the governing equations of beams and plates, respectively, where the coupling force components at the interface between the base plate and the beam(s) can be automatically defined during the integral transform. Eventually, the partial differential equations are transformed into a system of linear algebraic equations in which its derivation is rigorous and easily implemented. Good agreements are found between the results of analytical solution, finite element analysis (FEA) and related literature. The solution is then employed to study the vibration suppression of cantilever plates and the shear force at the clamped edge. It is found that the insertion of a pair of orthogonal ribs in the plate can effectively reduce its vibration. An optimum orthogonal ribbing pattern is obtained using multi-objective particle swarm optimization (MOPSO) algorithm, taking into consideration both the vibration suppression of the plate and the maximum induced shear force at the corners of the clamped edge. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a two‐stage theory‐based analytical method for the dynamic performance of pile groups in layered poroelastic saturated cross‐anisotropic soils induced by moving loadings. Among them, the free‐field vibrational analysis of saturated soils is performed by the analytical element‐layer approach (ALEA) and Fourier transformation. Based on the free‐field response, the boundary element (BE) solution for the soil resistance at the soil‐pile interface is derived utilizing the two‐stage theory. Simultaneously, the finite element (FE) solutions for the pile shaft resistance and deformation of pile groups are derived based on the Timoshenko beam theory. Finally, the FE‐BE coupled dynamic equation for deformations and internal loadings of the soil‐pile system is obtained. Thereafter, the reliability of the proposed method is validated by comparing with existing solutions and FE data from ABAQUS. Based on the derived solutions, a comprehensive parametric study is performed to examine the effects of loading amplitude, force speed, soft soil‐layer stiffness, soil anisotropy, and pile length on the dynamic responses of pile groups. [ABSTRACT FROM AUTHOR]

The natural frequency curve veering phenomena have been significant and used in several structural dynamic problems. By formulating a pre-twisted and double-tapered rotating Timoshenko beam as a structural problem, variations in mode interactions among coupled axial, chordwise, and flapwise motions were presented. Rotordynamic effects such as gyroscopic couple, centrifugal stiffening, and spin softening effects were taken into account while establishing the equations of motion. Hamilton's principle was used in deriving the equations of motion and is discretized based on Galerkin finite element method. Using dimensionless parameters, the model was verified with the existing literature for rotation, pre-twist, and double-taper configurations. A comparison of the frequency veering regions among the Euler–Bernoulli beam and Timoshenko beam theories is presented. Two-dimensional mode shape plots were used to show the coupled motions between axial displacement, chordwise bending, and flapwise bending. The effects of pre-twist and double-taper in the non-rotating Timoshenko beam on the modal properties of the rotating beam were examined. Natural frequency maps were drawn to locate the frequency veering curves. Mode interactions around different veering regions were shown with variations in mode shapes. An investigation on 60° pre-twisted and 0.5 ratio double-tapered rotating Timoshenko beam is presented. [ABSTRACT FROM AUTHOR]

Dynamic and buckling of a composite beam reinforced with a combination of carbon and aramid fibers is studied in this paper. The beam is under a thermal gradient through the thickness. Timoshenko beam is made of a polymer matrix (epoxy resin) reinforced with layers of high-strength carbon and high-toughness aramid fibers in order to create a balance between stiffness and toughness and create a type of structural composite beam with excellent strength and toughness. The mechanical and thermal properties of the hybrid composite beams are obtained based on the mixed law method. The equations of motion are extracted based on the Hamilton principle and then solved by the generalized differential quadrature method (GDQ). In this study, a thermal gradient is applied to the beam and then the vibration and buckling response of this hybrid composite beam are studied. The main contribution of this paper is the vibration and buckling responses of a hybrid composite structure strengthened by carbon and aramid fibers. The effect of the hybrid ratio as well as the stacking sequence on the free vibrations and critical buckling load are presented. The fundamental frequency and critical buckling load are largely affected by the stacking sequence. The conclusions show that the use of aramid fibers in the composite beam reinforced with carbon fibers decreases the natural frequency as well as the critical buckling load of the beam. The conclusions also show that for the symmetric hybrid composite beam, despite the critical buckling temperature being the same, the critical buckling load is different and depends on the location of the fibers. [ABSTRACT FROM AUTHOR]

This study aims to determine the effect of porosity on the static response of rotating and non-rotating porous beam. Timoshenko beam theory has been used and the governing equation has been solved via B-spline collocation technique. The material distribution is a function of power law along the height of the beam, even and uneven distribution of porosity has been considered. The parameter such as power index, porosity coefficient and rotational speed have been varied. Deflection and stress variation has been plotted for even and uneven distribution of porosity for relative study. The outcome reveals that effect of even distribution of porosity is higher than uneven porosity. The study also shows that rotation of the beam has significant impact on the deflection and stress distribution of the beam and also reveals that porous beams can be used where high strength and low stiffness is required. [ABSTRACT FROM AUTHOR]

Purpose: In the present study, the nonlinear vibration analysis of a nanoscale beam with different boundary conditions named as simply supported, clamped-clamped, clamped-simple and clamped-free are investigated numerically. Methods: Nanoscale beam is considered as Euler-Bernoulli beam model having size-dependent. This non-classical nanobeam model has a size dependent incorporated with the material length scale parameter. The equation of motion of the system and the related boundary conditions are derived using the modified couple stress theory and employing Hamilton's principle. Multiple scale method is used to obtain the approximate analytical solution. Result: Numerical results by considering the effect of the ratio of beam height to the internal material length scale parameter, h/l and with and without the Poisson effect, υ are graphically presented and tabulated. Conclusion: We remark that small size effect and poisson effect have a considerable effect on the linear fundamental frequency and the vibration amplitude. In order to show the accuracy of the results obtained, comparison study is also performed with existing studies in the literature. [ABSTRACT FROM AUTHOR]

Axially loaded beam-like structures, such as beam segments and piers of cable-stayed bridges, are widely used in civil engineering. This paper presents a study on the linear and nonlinear vibration of an axially loaded Timoshenko beam with elastic supports. The governing equations of the system are derived using Timoshenko beam theory and von Kármán geometric nonlinearity within the framework of Hamilton's principle. A transition parameter is introduced to describe the direction of the axial internal force during the beam's deformation. Galerkin's method is employed to reduce the nonlinear partial differential equations, from which linear solutions can be obtained. Nonlinear discretized equations are solved using the incremental harmonic balance (IHB) method. Frequency-response curves of the beam system are tracked by pseudo arc-length continuation, and Floquet's theory is used to determine the system's stability. Numerical examples are presented to validate the proposed solutions, comparing the results with the literature and the Runge-Kutta method. The effects of the transition parameter, elastic support's stiffness, and supported position are investigated for both linear and nonlinear vibration of the axially loaded Timoshenko beam. Of particular interest is the significance of the transition parameter, especially for a Timoshenko beam subjected to extreme axial compression with a small slenderness ratio. The present analysis provides critical insights into the design and analysis of such axially loaded structures in various civil engineering applications. [ABSTRACT FROM AUTHOR]

AbstractPiezoelectric bimorph cantilevered beams are frequently employed in energy harvesting applications. The use of macro-fiber composites, which combine piezoelectric fibers with rectangular cross-sections and interdigitated electrodes, has significantly enhanced their performance by enabling operation in the 33-mode of piezoelectricity, thereby increasing conversion efficiency. In this article, the Timoshenko beam theory is applied to overcome the limitations of the Euler-Bernoulli theory in modeling transverse vibrations of harvesters with low slenderness ratios (length to thickness ratios). The mixture rule formulation is applied to determine the equivalent properties of the macro-fiber composite (MFC) structure. The electromechanical properties of a representative volume element (RVE) bounded by two successive interdigitated electrodes are coupled to the overall electro-elastic dynamic model of the structure using the Timoshenko theory. Frequency response functions for the power and tip vibration displacement of the MFC bimorph beam are determined from the model. Experimental data from the literature are used to validate the improved results predicted by the model developed in this article. A comparative analysis between the electromechanical responses predicted using a model based on Timoshenko and a model based on Euler-Bernoulli theory is conducted showing agreement, for harvesters with large slenderness ratios and discrepancies of up to 30% for low slenderness ratios. It has also been shown that the model based on Euler-Bernoulli theory overestimates both the mechanical and the electrical responses for low slenderness ratios as the shear and rotary inertia effects are not negligible in this case. [ABSTRACT FROM AUTHOR]

The impact-rubbing dynamic characteristics of the power turbine rotor with the hollow shaft and offset discs for aircraft engine are investigated, and the impact-rubbing analytical method for the complex rotor based on MDOP Timoshenko beam theory is proposed in this paper. Compared with the traditional approach, the novel method can obtain more data to satisfy the need of engineering. The Lagrange equation is adopted to derive the equations of motion for the rotor system, and the Newmark-β method is applied to solve the equations. The diagrams such as the bifurcation, axis trajectory, spectrum, and Poincaré map are obtained to research on the effect of the rotating speed, gap, and eccentricity on the vibration response. The finite element analysis was carried out to validate the correctness of the theoretical modeling method. The research results indicate that the power turbine rotor with the hollow shaft on operation shows the various nonlinear dynamic behaviors including the multiperiod, quasi-period, jumping phenomenon, and chaotic motions; there exists an optimal gap between the rotor and the stator from the perspective of the efficiency and the dynamics; the optimal gap should make system avoid the resulting chaos or the quasi-period motion for the stability and safety of the machinery. [ABSTRACT FROM AUTHOR]

In this study, the Kevin–Voigt viscoelastic constitutive relationship is used to investigate the vibration characteristics and stability of a functionally graded viscoelastic(FGV) fluid-conveying pipe with initial geometric defects under thermal–magnetic coupling fields. First, the nonlinear dimensionless differential equations of motion are derived by applying Timoshenko beam theory. Second, by solving the equilibrium position of the system, the nonlinear term in the differential equations of motion is approximated as the sum of the longitudinal displacement at the current time and longitudinal displacement relative to the position, and the equations are linearized. Third, these equations are discretized using the Galerkin method and are numerically solved under simply supported conditions. Finally, the effects of dimensionless temperature field parameters, dimensionless magnetic field parameters, thermal–magnetic coupling, initial geometric defect types, and the power-law exponent on the complex frequency of the pipe are examined. Results show that increasing the magnetic field intensity enhances the critical velocity of first-order mode instability, whereas a heightened temperature variation reduces the critical velocity of first-order diverge instability. Under thermal–magnetic fields, when the magnetic field intensity and temperature difference are simultaneously increased, their effects on the complex frequency can partially offset each other. Increasing the initial geometric defect amplitude increases the imaginary parts of the complex frequencies; however, for different types of initial geometric defect tubes, it exhibits the most distinct influence only on a certain order. [ABSTRACT FROM AUTHOR]

For vibration energy harvesters, broadband power spectral characteristics are often desirable. One way of achieving broadband spectrum is to employ curved-beam hinges, utilizing their nonlinear spring characteristics. In our previous study, electret-based vibration energy harvesters employing curved-beam hinges were investigated via numerical analysis based on stochastic differential equations and colored-noise inputs. It showed that the harvesters with curved-beam hinges can produce higher power than the ones with ordinary straight beams when the external acceleration is between 0.02g and 0.05g. It was also learned that the straight-beam device, a Duffing oscillator, performs better than the curved-beam device, a Duffing–Holmes oscillator, at higher acceleration ( ≥ 0.1 g ). Since the energy harvesting efficiency is one of the most important performance indicators, continuous search for novel configurations with improved efficiency is needed. For that purpose, a symmetric configuration of curved-beam hinges (a Duffing oscillator) is proposed in this work, in contrast to the previously reported one with an asymmetric configuration (a Duffing–Holmes oscillator). This study shows, via numerical analysis, that the symmetric configuration can produce higher (up to 7.3% more) power outputs for the external acceleration magnitude higher than 0.1g, when compared to an asymmetric configuration. The study results also show that it can produce higher power outputs (up to 4.5 times) than the harvesters employing ordinary straight-beam hinges. [ABSTRACT FROM AUTHOR]

A long transmission shaft is typically constructed by connecting multiple shafts through couplings. Joining two non-collinear shafts causes misalignment, which is one of the primary faults in the shaft system. The flexible coupling can compensate for the misalignment up to some extent. However, in this process, the system dynamics may be affected by the coupling characteristics. The present research proposes flexible disk coupling in a misaligned propeller shaft. The coupling is modeled by segmenting the disk into multiple viscoelastic links exhibiting restoring and dissipating behavior. A combination of parallel and angular misalignment is taken to obtain coupling characteristics. Finite element formulation considering the Timoshenko beam theory is applied to obtain the laminated composite shaft equation, and coupling characteristics are integrated. Operator based constituted relationship is utilized to incorporate material damping of both the shaft and coupling. The stiffness and damping of the coupling rotate with the spin synchronized rotor frame and modify the overall system matrices. Therefore, the coupling influences the modal characteristics and response of the rotor shaft system. A comparative study between the coupled and the uncoupled shafts is performed based on the eigenvalue analysis to show the effect of coupling behavior. [ABSTRACT FROM AUTHOR]