Stepped Beam Research Articles

Gas Adsorption Response of Piezoelectrically Driven Microcantilever Beam Gas Sensors: Analytical, Numerical, and Experimental Characterizations.

This work presents an approach for the estimation of the adsorbed mass of 1,5-diaminopentane (cadaverine) on a functionalized piezoelectrically driven microcantilever (PD-MC) sensor, using a polynomial developed from the characterization of the resonance frequency response to the known added mass. This work supplements the previous studies we carried out on the development of an electronic nose for the measurement of cadaverine in meat and fish, as a determinant of its freshness. An analytical transverse vibration analysis of a chosen microcantilever beam with given dimensions and desired resonance frequency (>10 kHz) was conducted. Since the beam is considered stepped with both geometrical and material non-uniformity, a modal solution for stepped beams, extendable to clamped-free beams of any shape and structure, is derived and used for free and forced vibration analyses of the beam. The forced vibration analysis is then used for transformation to an equivalent electrical model, to address the fact that the microcantilever is both electronically actuated and read. An analytical resonance frequency response to the mass added is obtained by adding simulated masses to the free end of the beam. Experimental verification of the resonance frequency response is carried out, by applying known masses to the microcantilever while measuring the resonance frequency response using an impedance analyzer. The obtained response is then transformed into a resonance frequency to the added mass response polynomial using a polynomial fit. The resulting polynomial is then verified for performance using different masses of cantilever functionalization solution. The functionalized cantilever is then exposed to different concentrations of cadaverine while measuring the resonance frequency and mass of cadaverine adsorbed estimated using the previously obtained polynomial. The result is that there is the possibility of using this approach to estimate the mass of cadaverine gas adsorbed on a functionalized microcantilever, but the effectiveness of this approach is highly dependent on the known masses used for the development of the response polynomial model.

梁主筋に機械式定着工法を用いた梁段差を有する柱梁接合部の応力伝達機構

This paper discusses on R/C interior step beam-column joints with headed anchor for beam longitudinal bars, developed to streamline the works of reinforcing for R/C interior step joints. In general, beam longitudinal bars of step beam column joints are applied 90°-degree hook in step beam-column joints. In R/C buildings, step beam column joints are sometimes planed for architectural planning reasons. Nevertheless, interior step beam-column joints reinforcements are congested and 90°-degree hook of the left and right beams may be the case these hooks are overlap. In order to consider interior step beam-column joints design method, four specimens with beam-column strength ratio approximately 1.0 and 1.5 were tested. Regarding step beam-column joints, from the crack behavior, the stress transfer mechanism depends on positive and negative loading and is considered to consist of two struts which are formed by the balancing of the compressive forces concentrated at the ends of the column and beam and the balancing of the headed anchorage reaction forces. The stress transfer mechanism consisting of these two struts was modelled and proposed as a “double strut model”. A design method of R/C step interior beam-column joints based on the double strut model is also proposed. It provides sufficient satisfaction as to the interior step beam-column joints experimental test results.

Inverse dynamic analysis for uncertain boundary condition parameters of a stepped beam subjected to an axial force

Generally, in engineering applications, the boundary condition parameters of axially loaded beam-like components with non-uniform cross-sections are unknown. This study investigates these uncertain parameters using inverse dynamic analysis of a stepped beam with elastic supports subjected to an axial force. A dynamic coefficient method is introduced in the dynamic analysis model to fundamentally overcome the numerical instability often encountered in traditional exact methods. Subsequently, a novel approach is proposed to evaluate the measurement error of the axial forces in an actual axially loaded beam-like component through inverse analysis of uncertain boundary condition parameters using two or only one observed natural frequency. The unknown rank orders of the measured natural frequencies can also be determined using the proposed method. Numerical examples are provided to verify the exactness and numerical robustness of the proposed method. Finally, during the construction of a real tied-arch bridge, the proposed approach was successfully applied to precisely control the tension forces of the bridge hangers.

Study on added damping due to resistive shunting on cantilever, simply supported beam, and effect on the variation of amplitude at resonance

The current work determines the piezoelectric damping of resistively shunted beams induced by resistors through joule heating. To predict this damping, a piezoelectric patch [Formula: see text] is attached to the surface of a cantilever beam. The current work emphasizes the effect of various parameters on damping added to the cantilever beam through the energy dissipation approach. This work analyzes different methods for predicting added damping. Results from different approaches rely on the base structure modeshape. Using the energy dissipation method from voltage generation, estimated added damping, included a stepped beam model to account for the curvature of the beam with PZT. Further, the same is validated with existing literature and found to be consistent. With the energy dissipation approach, including the stepped beam model compared with experimental literature data, the accuracy of the adopted beam model is validated. Few critical observations were made, such as the location of the PZT patch plays a vital role in determining the required added damping for a particular mode. This optimal location was the nodal location of the other modes. The location of the piezoelectric patch is an important parameter to achieve the required added damping for a particular mode. The optimal location of PZT to achieve maximum added damping for a particular mode may lie at the nodal location of other modes. This optimal location is addressed by rigorous experiments on a simply-supported beam initially and further correlated to a cantilever beam with the help of an energy dissipation approach. The energy dissipation model is correlated and validated for cantilever beam straight PZT on cantilever beams and a simply supported beam. In practical situations, in ambient conditions, environmental effects may result in a change of resonant frequency and thus amplitude. This variation may alter the base and shunt damping performance attributed to air damping. We have experimentally investigated the influence of air damping for base damping (inherent) and added damping (PZT).

Reformulation for the Solution of the Dynamic Response of Co-Axial Segmented Beams

Abstract Stepped beams constitute an important class of engineering structures whose vibration response has been widely studied. Many of the existing methods for studying stepped beams manifest serious numerical difficulties as the number of segments or the frequency of excitation increase. In this article, we focus on the transfer matrix method (TMM), which provides a simple and elegant formulation for multistep beams. The main idea in the TMM is to model each step in the beam as a uniform element whose vibration configurations are spanned by the segment’s local eigenfunctions. Utilizing these local expressions, the boundary conditions at the ends of the multistep beam as well as the continuity and compatibility conditions across each step are used to obtain the nonlinear eigenvalue problem. Also, and perhaps more importantly, we provide a reformulation for multistep Euler–Bernoulli beams that avoids much of the numerical singularity problems that have plagued most of the earlier efforts. When this reformulation is extended to multisegment Timoshenko beams, the numerical difficulties appear to be mitigated, but not solved.

Vibration of Complex Euler–Bernoulli and Timoshenko–Ehrenfest Beams Through Affine GPSFs

Abstract Vibrations of thin and thick beams containing internal complexities are analyzed through generalized bases made of global piecewise-smooth functions (GPSFs). Such functional bases allow us to globally analyze multiple domains as if these latter were only one, such that a unified formulation can be used for different mechanical systems. Such bases were initially introduced to model a specific part of stress and displacement components through the thickness of multi-layered plates; subsequent extensions were introduced in the literature to allow the modeling of thin-walled beams and plates. However, in these latter cases, certain analytical difficulties were experienced when inner boundary conditions needed to be englobed into the GPSFs; in this work, such mentioned difficulties are successfully overcome through certain affine transformations which allow the analyses of vibrating complex beam systems through a straightforward analytical procedure. The complex mechanical components under investigation are Euler and/or Timoshenko models containing inner complexities (stepped beams, concentrated mass or stiffness, internal constraints, etc.). The ability of the models herein analyzed is shown through the comparison of the resulting solutions to exact counterparts, if existing, or to finite elements solutions.

An analytical model of the electromechanical coupling for a piezoelectric stepped buckled beam for energy harvesting applications

In this article we intend to present an experimentally validated complex mathematical framework of electromechanical coupling for a piezoelectric stepped Bernoulli–Euler beam for energy harvesting applications, valid for both the pre- and postbuckled states. The proposed model, takes into account the predeformed buckled configuration due to the out-of-plane displacements caused by an axial compressive load higher than the critical one. Moreover, several sources of nonlinearities are also considered such as: inertia, quadratic damping, geometrical and constitutive relations for the piezoelectric material. In addition, an axial elastic restraint as, a non-ideal boundary condition, is assumed at one end of the beam which allows treating intermediate cases between perfectly restricted and unrestricted axial displacements commonly found in experimental realization of axially loaded beams. The spatial discretization of the problem is carried out according to Galerkin Method. Based on a single-mode assumption, the Method of Multiple Scales (MMS) is used to obtain an approximated analytic solution for the harvester. Several experimental tests, based both in the static and dynamic response of the beam, were carried out for a sample formed by a MFC-8504 piezoelectric sheet and bearing steel beam evidencing not only the usefulness of the model here developed, but also the benefits associated with the bistable regime for energy harvesting applications.

Analytical and experimental investigation of bend-twist coupling on the vibrational response of multi-layered stepped composite beams

This paper investigates the bend-twist coupling analysis of multi-layered stepped generally orthotropic composite beams subjected to mixed end-of-beam and mid-span supports. Specifically, an analytical closed-form model was developed based on first-order shear deformation theory (FSDT), which discretizes the domain into elements based on the step change of geometry, laminate configuration, or mid-span boundary supports. Hamilton's principle was used to derive the governing equations within each element and connection and boundary equations. The state-Space approach was then utilized to provide an analytical solution. Moreover, an experimental investigation was conducted to validate the mode shapes and natural frequencies of the beam subjected to several mixed boundary conditions. The results are also validated with the literature and a finite element model developed using ANSYS. Comparisons demonstrate the reliability and accuracy of the analytical model.

A numerical-experimental dynamic analysis of high-efficiency and broadband bistable energy harvester with self-decreasing potential barrier effect

It is difficult for a conventional bistable energy harvester (CBEH) to form inter-well motion under a weak excitation, resulting in a narrow output bandwidth. In this paper, a bistable energy harvester with a self-decreasing potential barrier effect (BEH-DB) is proposed. It adopts a spring-magnetic oscillator to rectify the cantilever beam bi-directionally causing the potential barrier to be dynamically lowered when returning to the trivial position; while the potential energy is enhanced when moving away from the trivial position. The theoretical model is established by using Euler-Bernoulli's stepped beam and magnetic dipole theories. The influence of system parameters on the performance of the self-decreasing potential barrier is studied by tracking the trajectories on the potential energy surface. The frequency and amplitude sweeps, phase portraits, Poincare maps, and bifurcation analysis are used numerically to investigate the dynamics of the system. The simulation results present a series of complex dynamic behaviors such as intra-well, inter-well harmonic and sub-harmonic orbits, and chaotic motion. The BEH-DB is more easily to be excited to the inter-well orbit to form a large oscillation in an ultra-wide bandwidth, even at the low frequency. Its effective bandwidth and maximum average power are approximately 16.17 Hz and 9.02mW, respectively. Finally, the experiment was carried out, which illustrates that the power generation capacity and bandwidth of the BEH-DB are improved several times that of the CBEH. The experimental results are agreed qualitatively with the numerical simulation. The proposed BEH-DB can be used to harvest energy efficiently and broadly from a low or ultra-low frequency ambient vibration.

Analytical and numerical investigation of free vibration for stepped beam with different materials

Abstract This work describes the application of classical Rayleigh method (CRM), modified Rayleigh method (MRM), and ANSYS finite element method (FEM) to calculate the natural frequency of non-homogenous cantilever beam. Two-step cantilever stepped beam was investigated through six studied cases. Each step has different material properties and the same cross section area. Results showed that the combination of materials is useful in order to increase the natural frequencies and reduce the weight of the beam at the same time when the cantilever beam is fixed by the side of the stronger material. There is a good agreement between the CRM and FEM for the region with length larger than half length of beam, on the other hand, there is an excellent agreement between the MRM and FEM for the region with length smaller than half length of beam.

The Mixed-Body Model: A Method for Predicting Large Deflections in Stepped Cantilever Beams

AbstractThis paper presents a method for predicting endpoint coordinates, stress, and force to deflect stepped cantilever beams under large deflections. This method, the mixed-body model or MBM, combines small deflection theory and the pseudo-rigid-body model for large deflections. To analyze the efficacy of the model, the MBM is compared to a model that assumes the first step in the beam to be rigid, to finite element analysis, and to the numerical boundary value solution over a large sample set of loading conditions, geometries, and material properties. The model was also compared to physical prototypes. In all cases, the MBM agrees well with expected values. Optimization of the MBM parameters yielded increased agreement, leading to average errors of <0.01 to 3%. The model provides a simple, quick solution with minimal error that can be particularly helpful in design.

Structural Stiffness Matching Modeling and Active Design Approach for Multiple Stepped Cantilever Beam

Aiming at the problem that it is difficult to realize the optimal design due to the fuzzy mapping relationship for the structural stiffness of multiple stepped cantilever beam; a stiffness matching modeling and active stiffness design approach was proposed. Firstly, by deriving out the continuous coordination conditions and the load extrapolation expressions of the cantilever joint, the stiffness analytical model and the recursive model were established for multiple cantilever beam segments, and the stiffness influence coefficient of those composition parameters were obtained by the sensitivity analysis. Then, the active stiffness optimization design process was constructed according to the stiffness design level of the stepped cantilever beam, and those implementation procedures were clearly figured out. Finally, the comparison and verification of the stiffness design of the stepped cantilever beam was carried out through numerical simulations, finite element analysis and bench test. The obtained results showed that the established models and the active stiffness design method are reasonable and effective. The stiffness match parameters are easy to meet the stiffness index requirements, and the safety factor is greater than 1; when the number of steps is not more than 5. The relative error between the match stiffness and the test stiffness is less than 15%, which can be reduced to less than 5% by adding redundancy coefficient (1.05, 1.15).

An analytical approach for free vibration analysis of Euler-Bernoulli stepped beams with axial-bending coupling effect

Free vibration of eccentrically stepped beams with one step change in cross-section is considered. It is assumed that the longitudinal symmetry axes of the beam segments are translationally shifted along the vertical direction with respect to each other. The effect of that arrangement of the segments on the coupling of axial and bending vibrations of the stepped beam is analyzed. The beam segments are modeled in the frame of the Euler-Bernoulli theory of elastic beams. Two numerical examples are presented.

Primena Dingo optimizacionog algoritma (DOA) na rešavanje kontinualnih problema u inženjerstvu

In the current research, problems in engineering are becoming more and more prominent. One of the classes of engineering problems in engineering design problems, where a set of variables is calibrated in order for the optimization function to have a minimal or maximal value. This function considers energy efficiency, cost efficiency, and production efficiency in engineering design. One of the ways such problems are solved is metaheuristics. In this paper, we demonstrate how Dingo Optimization Algorithm (DOA) can be used to solve certain optimization problems in mechanical engineering. Firstly, a brief review of the DOA and its biological inspiration is given, along with the most important formulae. The pseudo-code for this algorithm was written using MATLAB R2020a software suite. Dingo Optimization Algorithm (DOA) was used to optimize engineering problems, such as pressure vessel optimization, stepped cantilever beam, car side-impact, and cone clutch optimization. The results presented in this paper show that the DOA can produce relevant results in engineering design problems.

An Accurate Estimation of the Eigenfrequency of an Offshore Wind Turbine Considered as the Stepped Euler-Bernoulli Beam in Three-Spring Flexible Foundation Using the Power Series Method

The present study employs the power series method (PSM) to accurately predict the natural frequencies of eleven offshore wind turbines (OWT). This prediction is very important as it helps in the quick verification of experimental or finite element results. This study idealizes the OWT as a stepped Euler-Bernoulli beam carrying a top mass and connected at its bottom to a flexible foundation. The first part of the beam represents a monopile and the transition piece while its second part is a tower. The foundation is modeled using three springs (lateral, rotational, and cross-coupling springs). This work’s aim is at improving therefore the previous researches, in which the whole wind turbine was taken as a single beam, with a tower being tapered and its wall thickness being negligible compared to its diameter. In order to be closer to real-life OWT, three profiles of the tapered tower are explored: case 1 considers a tower with constant thickness along its height. Case 2 assumes a tower’s thickness being negligible compared to its mean diameter, while case 3 describes the tower as a tapered beam with varying thickness along its height. Next, the calculated natural frequencies are compared to those obtained from measurements. Results reveal that case 2, used by previous researches, was only accurate for OWT with tower wall thickness lower than 15 mm. Frequencies produced in case 3 are the most accurate as the relative error is up to 0.01%, especially for the OWT with thicknesses higher or equal to 15 mm. This case appears to be more realistic as, practically, wall thickness of a wind tower varies with its height. The tower-to-pile thickness ratio is an important design parameter as it highly has impact on the natural frequency of OWT, and must therefore be taken into account during the design as well as lateral and rotational coupling springs.

Hybridization of Chaotic Maps and Gravitational Search Algorithm for Constrained Mechanical and Civil Engineering Design Frameworks

The Chaotic Gravitational Search Algorithm (CGSA) is a physics-based heuristic algorithm inspired by Newton's law of universal gravitation. It uses 10 chaotic maps for optimal global search and fast convergence rate. The advantages of CGSA has been incorporated in various Mechanical and Civil engineering design frameworks which include Speed Reducer Design (SRD), Gear Train Design (GTD), Three Bar Truss Design (TBTD), Stepped Cantilever Beam Design (SCBD), Multiple Disc Clutch Brake Design (MDCBD), and Hydrodynamic Thrust Bearing Design (HTBD). The CGSA has been compared with eleven state of the art stochastic algorithms. In addition, a non-parametric statistical test namely the Signed Wilcoxon Rank-Sum test has been carried out at a 5% significance level to statistically validate the results. The simulation results indicate that CGSA shows efficient performance in terms of high convergence speed and minimization of the design parameter values as compared to other heuristic algorithms. The source codes are publicly available on Github i.e. https://github.com/SajadAHMAD1.

Spatial evolution leading to multiple filament formation of higher order super Gaussian beam in bulk medium at input power P >>Pcr

Multiple filamentation in higher order super Gaussian laser beam propagating in chalcogenide glass is investigated by numerically solving nonlinear Schrödinger equation with split step beam propagation method at input power range of 40 – 160 Pcr and compared to the process in super Gaussian beam. The effect of raising the beam order on the deterministic nature of multiple filaments, threshold power of filamentation, length of filamentation and the number of filaments formed are analysed. The results indicate that within a range of beam order and input power, the process of multiple filamentation in super Gaussian beam may be controlled through beam order. The study further exemplifies the importance of deterministic filaments and the ‘role of beam order as a tool’ in control of self-guided laser beam properties in filamentation process.

Vibration source identification using an energy density method

The localization of shaking forces acting on an operating machine is an important step to identify vibration and noise sources. The forced vibration response of a linearly vibrating structure is assumed to be linear. However, the energy distribution of a linearly vibrating structure contains “coupled terms” in the modal decomposition of the vibration energy density function. These coupled energy terms represent the cross-modal energy density associated with the exciting force of a dynamic structure under forced vibration. In this research, it is proved analytically that the high-order cross-modal energy densities of a linear dynamic structure are highly correlated to the location of the external exciting force. Using this finding, a new force localization index based on the high-order cross-modal energy densities of a dynamic structure is proposed and tested. Numerical tests on uniform and step beam structures under force excitation with different frequencies and locations have been carried out to test the effectiveness of the proposed force localization method. It is found that the proposed force localization method works well on vibrating beam structures. Experiments are carried out to verify the proposed force localization method.

A Multimode Approach to Geometrically Nonlinear Free and Forced Vibrations of Multistepped Beams

The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.

An analytical solution for the vibrational response of stepped smart cross-ply laminated composite beams with experimental validation

This paper investigates the vibrational behaviour of smart orthotropic cross-ply laminated stepped beams. Specifically, an analytical closed-form model based on the first-order shear deformation theory (FSDT) is developed to consider a step change in laminate configuration along the beam axis due to integration of a piezoelectric patch for actuation applications. The 6th order governing equation within each element, along with appropriate continuity, equilibrium, and boundary equations, were formulated. State-space approach was utilised to find the beam's response to piezoelectric sinusoidal actuation. Results of the natural frequency analysis were compared with results in the literature and finite element modelling. An experimental investigation examined the accuracy of the analytical method to find the natural frequencies and forced vibrational response of a smart carbon-fibre/polyetheretherketone (PEEK) cantilever beam. Comparisons demonstrate that the provided closed-form method is capable of efficiently predicting the free and forced vibrational behaviour of smart stepped laminated composite beams with high accuracy.