Auto-parametric resonance of a continuous-beam-bridge model under two-point periodic excitation:an experimental investigation and stability analysis

The auto-parametric resonance of a continuous-beam bridge model subjected to a two-point periodic excitation is experimentally and numerically investigated in this study.An auto-parametric resonance experiment of the test model is conducted to observe and measure the auto-parametric resonance of a continuous beam under a two-point excitation on columns.The parametric vibration equation is established for the test model using the finite-element method.The auto-parametric resonance stability of the structure is analyzed by using Newmark's method and the energy-growth exponent method.The effects of the phase difference of the two-point excitation on the stability boundaries of auto-parametric resonance are studied for the test model.Compared with the experiment,the numerical instability predictions of auto-parametric resonance are consistent with the test phenomena,and the numerical stability boundaries of auto-parametric resonance agree with the experimental ones.For a continuous beam bridge,when the ratio of multipoint excitation frequency(applied to the columns)to natural frequency of the continuous girder is approximately equal to 2,the continuous beam may undergo a strong auto-parametric resonance.Combined with the present experiment and analysis,a hypothesis of Volgograd Bridge's serpentine vibration is discussed.

The quaternion beam model for hard-magnetic flexible cantilevers

The recently developed hard-magnetic soft(HMS)materials manufactured by embedding high-coercivity micro-particles into soft matrices have received considerable attention from researchers in diverse fields,e.g.,soft robotics,flexible electronics,and biomedicine.Theoretical investigations on large deformations of HMS structures are significant foundations of their applications.This work is devoted to developing a powerful theoretical tool for modeling and computing the complicated nonplanar deformations of flexible beams.A so-called quaternion beam model is proposed to break the singularity limitation of the existing geometrically exact(GE)beam model.The singularity-free governing equations for the three-dimensional(3D)large deformations of an HMS beam are first derived,and then solved with the Galerkin discretization method and the trustregion-dogleg iterative algorithm.The correctness of this new model and the utilized algorithms is verified by comparing the present results with the previous ones.The superiority of a quaternion beam model in calculating the complicated large deformations of a flexible beam is shown through several benchmark examples.It is found that the purpose of the HMS beam deformation is to eliminate the direction deviation between the residual magnetization and the applied magnetic field.The proposed new model and the revealed mechanism are supposed to be useful for guiding the engineering applications of flexible structures.

Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

We present a study on the dynamic stability of porous functionally graded(PFG)beams under hygro-thermal loading.The variations of the properties of the beams across the beam thicknesses are described by the power-law model.Unlike most studies on this topic,we consider both the bending deformation of the beams and the hygro-thermal load as size-dependent,simultaneously,by adopting the equivalent differential forms of the well-posed nonlocal strain gradient integral theory(NSGIT)which are strictly equipped with a set of constitutive boundary conditions(CBCs),and through which both the stiffness-hardening and stiffness-softening effects of the structures can be observed with the length-scale parameters changed.All the variables presented in the differential problem formulation are discretized.The numerical solution of the dynamic instability region(DIR)of various bounded beams is then developed via the generalized differential quadrature method(GDQM).After verifying the present formulation and results,we examine the effects of different parameters such as the nonlocal/gradient length-scale parameters,the static force factor,the functionally graded(FG)parameter,and the porosity parameter on the DIR.Furthermore,the influence of considering the size-dependent hygro-thermal load is also presented.

Stiffness and Shear Stress Distribution of Glulam Beams in Elastic-Plastic Stage:Theory,Experiments and Numerical Modelling

Traditional methods focus on the ultimate bending moment of glulam beams and the fracture failure of materials with defects,which usually depends on empirical parameters.There is no systematic theoretical method to predict the stiffness and shear distribution of glulam beams in elastic-plastic stage,and consequently,the failure of such glulam beams cannot be predicted effectively.To address these issues,an analytical method considering material nonlinearity was proposed for glulam beams,and the calculating equations of deflection and shear stress distribution for different failure modes were established.The proposed method was verified by experiments and numerical models under the corresponding conditions.Results showed that the theoretical calculations were in good agreement with experimental and numerical results,indicating that the equations proposed in this paper were reliable and accurate for such glulam beams with wood material in the elastic-plastic stage ignoring the influence of mechanic properties in radial and tangential directions of wood.Furthermore,the experimental results reported by the previous studies indicated that the method was applicable and could be used as a theoretical reference for predicting the failure of glulam beams.

Calibration of quantitative rescattering model for simulating vortex high-order harmonic generation driven by Laguerre–Gaussian beam with nonzero orbital angular momentum

We calibrate the macroscopic vortex high-order harmonic generation(HHG)obtained by the quantitative rescattering(QRS)model to compute single-atom induced dipoles against that by solving the time-dependent Schr?dinger equation(TDSE).We show that the QRS perfectly agrees with the TDSE under the favorable phase-matching condition,and the QRS can accurately predict the main features in the spatial profiles of vortex HHG if the phase-matching condition is not good.We uncover that harmonic emissions from short and long trajectories are adjusted by the phase-matching condition through the time-frequency analysis and the QRS can simulate the vortex HHG accurately only when the interference between two trajectories is absent.This work confirms that it is an efficient way to employ the QRS model in the single-atom response for precisely simulating the macroscopic vortex HHG.

Propagation of Gaussian Schell-Model Array Beams through a Jet Engine Exhaust

Here a Gaussian Shell Model Array (GSMA) beam is used to investigate the propagation characteristics in the jet engine exhaust region. It has great significance to improve various optical systems for wide application in trapping cold atoms, creating gratings, and atmospheric optical communication. We calculate analytical formulas for the spectral density (SD) and the propagation factors Mx2 and My2 of a GSMA beam. The influence of inner scale of turbulence in the jet engine exhaust region on its power spectrum has been also analyzed. According to these results, the influence of turbulence in a jet engine exhaust on a GSMA beam has been reduced by changing the parameters of light source and turbulence. For example, it is an excellent tool for mitigation of the jet engine exhaust-induced anisotropy of turbulence to increase the source coherence length, the root-mean-squared (rms) beam width, the wavelength or reduce the outer scale of turbulence.

Dynamic and electrical responses of a curved sandwich beam with glass reinforced laminate layers and a pliable core in the presence of a piezoelectric layer under low-velocity impact

The dynamic responses and generated voltage in a curved sandwich beam with glass reinforced laminate(GRL)layers and a pliable core in the presence of a piezoelectric layer under low-velocity impact(LVI)are investigated.The current study aims to carry out a dynamic analysis on the sandwich beam when the impactor hits the top face sheet with an initial velocity.For the layer analysis,the high-order shear deformation theory(HSDT)and Frostig's second model for the displacement fields of the core layer are used.The classical non-adhesive elastic contact theory and Hunter's principle are used to calculate the dynamic responses in terms of time.In order to validate the analytical method,the outcomes of the current investigation are compared with those gained by the experimental tests carried out by other researchers for a rectangular composite plate subject to the LVI.Finite element(FE)simulations are conducted by means of the ABAQUS software.The effects of the parameters such as foam modulus,layer material,fiber angle,impactor mass,and its velocity on the generated voltage are reviewed.

Fluid Flow and Solidification Simulation in Beam Blank Continuous Casting Process With 3D Coupled Model

Based on turbulent theory, a 3D coupled model of fluid flow and solidification was built using finite difference method and used to study the influence of superheating degree and casting speed on fluid flow and solidification, analyze the interaction between shell and molten steel, and compare the temperature distribution under different technological conditions. The results indicate that high superheating degree can lengthen the liquid-core depth and make the crack and breakout possible, so suitable superheating should be controlled within 35℃ according to the simulation results. Casting speed which is one of the most important technological parameters of improving production rate, should be controlled between 0. 85 m/min and 1.05 m/min and the caster has great potential in the improvement of blank quality.

Dynamic response of clamped sandwich beams: analytical modeling

An improved analytical model is developed to predict the dynamic response of clamped lightweight sandwich beams with cellular cores subjected to shock loading over the entire span.The clamped face sheets are simplified as a single-degree-of-freedom(SDOF)system,and the core is idealized using the rigid-perfectly-plastic-locking(RPPL)model.Reflection of incident shock wave is considered by incorporating the bending/stretching resistance of the front face sheet and compaction of the core.The model is validated with existing analytical predictions and FE simulation results,with good agreement achieved.Compared with existing analytical models,the proposed model exhibits superiority in two aspects:the deformation resistance of front face sheet during shock wave reflection is taken into account;the effect of pulse shape is considered.The practical application range of the proposed model is therefore wider.

Verification of Beam Models for Ionic Polymer-Metal Composite Actuator

Ionic Polymer-Metal Composite (IPMC) can work as an actuator by applying a few voltages.A thick IPMC actuator,where Nafion-117 membrane was synthesized with polypyrrole/alumina composite filler,was analyzed to verify the equivalent beam and equivalent bimorph beam models.The blocking force and tip displacement of the IPMC actuator were measured with a DC power supply and Young's modulus of the IPMC strip was measured by bending and tensile tests respectively.The calculated maximum tip displacement and the Young's modulus by the equivalent beam model were almost identical to the corresponding measured data.Finite element analysis with thermal analogy technique was utilized in the equivalent bimorph beam model to numerically reproduce the force-displacement relationship of the IPMC actuator.The results by the equivalent bimorph beam model agreed well with the force-displacement relationship acquired by the measured data.It is confirmed that the equivalent beam and equivalent bimorph beam models are practically and effectively suitable for predicting the tip displacement,blocking force and Young's modulus of IPMC actuators with different thickness and different composite of ionic polymer membrane.

Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model

This paper deals with the free vibration analysis of circular alumina （Al2O3） nanobeams in the presence of surface and thermal effects resting on a Pasternak foun- dation. The system of motion equations is derived using Hamilton＇s principle under the assumptions of the classical Timoshenko beam theory. The effects of the transverse shear deformation and rotary inertia are also considered within the framework of the mentioned theory. The separation of variables approach is employed to discretize the governing equa- tions which are then solved by an analytical method to obtain the natural frequencies of the alumina nanobeams. The results show that the surface effects lead to an increase in the natural frequency of nanobeams as compared with the classical Timoshenko beam model. In addition, for nanobeams with large diameters, the surface effects may increase the natural frequencies by increasing the thermal effects. Moreover, with regard to the Pasternak elastic foundation, the natural frequencies are increased slightly. The results of the present model are compared with the literature, showing that the present model can capture correctly the surface effects in thermal vibration of nanobeams.

Bending of Euler-Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach

Eringen's nonlocal elasticity theory is extensively employed for the analysis of nanostructures because it is able to capture nanoscale effects.Previous studies have revealed that using the differential form of the strain-driven version of this theory leads to paradoxical results in some cases,such as bending analysis of cantilevers,and recourse must be made to the integral version.In this article,a novel numerical approach is developed for the bending analysis of Euler-Bernoulli nanobeams in the context of strain-and stress-driven integral nonlocal models.This numerical approach is proposed for the direct solution to bypass the difficulties related to converting the integral governing equation into a differential equation.First,the governing equation is derived based on both strain-driven and stress-driven nonlocal models by means of the minimum total potential energy.Also,in each case,the governing equation is obtained in both strong and weak forms.To solve numerically the derived equations,matrix differential and integral operators are constructed based upon the finite difference technique and trapezoidal integration rule.It is shown that the proposed numerical approach can be efficiently applied to the strain-driven nonlocal model with the aim of resolving the mentioned paradoxes.Also,it is able to solve the problem based on the strain-driven model without inconsistencies of the application of this model that are reported in the literature.

Timoshenko beam model for chiral materials

Natural and artificial chiral materials such as deoxyribonucleic acid （DNA）, chromatin fibers, flagellar filaments, chiral nanotubes, and chiral lattice materials widely exist. Due to the chirality of intricately helical or twisted microstructures, such materials hold great promise for use in diverse applications in smart sensors and actuators, force probes in biomedical engineering, structural elements for absorption of microwaves and elastic waves, etc. In this paper, a Timoshenko beam model for chiral materials is developed based on noncentrosymmetric micropolar elasticity theory. The governing equations and boundary conditions for a chiral beam problem are derived using the variational method and Hamilton＇s principle. The static bending and free vibration problem of a chiral beam are investigated using the proposed model. It is found that chirality can significantly affect the mechanical behavior of beams, making materials more flexible compared with nonchiral counterparts, inducing coupled twisting deformation, relatively larger deflection, and lower natural frequency. This study is helpful not only for understanding the mechanical behavior of chiral materials such as DNA and chromatin fibers and characterizing their mechanical properties, but also for the design of hierarchically structured chiral materials.

Equivalent continuum modeling of beam-like truss structures with flexible joints

The paper investigated the equivalent continuum modeling of beam-like repetitive truss structures considering the flexibility of joints,which models the contact between the truss member and joint by spring-damper with six directional stiffnesses and dampings.Firstly,a two-node hybrid joint-beam element was derived for modeling the truss member with flexible end joints,and a condensed model for the repeating element with flexible joints was obtained.Then,the energy equivalence method was adopted to equivalently model the truss structure with flexible joints and material damping as a spatial viscoelastic anisotropic beam model.Afterwards,the equations of motion for the equivalent beam model were derived and solved analytically in the frequency domain.In the numerical studies,the correctness of the presented method was verified by comparisons of the natural frequencies and frequency responses evaluated by the equivalent beam model with the results of the finite element method model.

Four-Beam Model for Vibration Analysis of a Cantilever Beam with an Embedded Horizontal Crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton＇s principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.

Influence of moderate-to-strong anisotropic non-Kolmogorov turbulence on intensity fluctuations of a Gaussian–Schell model beam in marine atmosphere

The scintillation index（SI） of a Gaussian–Schell model（GSM） beam in a moderate-to-strong anisotropic nonKolmogorov turbulent atmosphere is developed based on the extended Rytov theory. The on-axis SI in a marine atmosphere is higher than that in a terrestrial atmosphere, but the off-axis SI exhibits the opposite trend. The on-axis SI first increases and then begins to decrease and saturate as the turbulence strength increases. Turbulence inner and outer scales have different effects on the on-axis SI in different turbulent fluctuation regions. The anisotropy characteristic of atmospheric turbulence leads to the decline in the on-axis SI, and the rise in the off-axis SI. The on-axis SI can be lowered by increasing the anisotropy of turbulence, wavelength, and source partial coherence before entering the saturation attenuation region. The developed model may be useful for evaluating ship-to-ship/shore free-space optical communication system performance.

Beam Approximation for Dynamic Analysis of Launch Vehicles Modelled as Stiffened Cylindrical Shells

A beam approximation method for dynamic analysis of launch vehicles modelled as stiffened cylindrical shells is proposed.Firstly,an initial beam model of the stiffened cylindrical shell is established based on the cross-sectional area equivalence principle that represents the shell skin and its longitudinal ribs as a beam with annular cross-section,and the circumferential ribs as lumped masses at the nodes of the beam elements.Then,a fine finite element model(FE model)of the stiffened cylindrical shell is constructed and a modal analysis is carried out.Finally,the initial beam model is improved through model updating against the natural frequencies and mode shapes of the fine FE model of the shell.To facilitate the comparison between the mode shapes of the fine FE model of the stiffened shell and the equivalent beam model,a weighted nodal displacement coupling relationship is introduced.To prevent the design parameters used in model updating from converging to incorrect values,a pre-model updating procedure is added before the proper model updating.The results of two examples demonstrate that the beam approximation method presented in this paper can build equivalent beam models of stiffened cylindrical shells which can reflect the global longitudinal,lateral and torsional vibration characteristics very well in terms of the natural frequencies.

Analytical modeling for rapid design of bistable buckled beams

Double-clamped bistable buckled beams demonstrate great versatility in various fields such as robotics,energy harvesting,and microelectromechanical system(MEMS).However,their design often requires time-consuming and expensive computations.In this work,we present a method to easily and rapidly design bistable buckled beams subjected to a transverse point force.Based on the Euler–Bernoulli beam theory,we establish a theoretical model of bistable buckled beams to characterize their snapthrough properties.This model is verified against the results from a finite element analysis(FEA)model,with maximum discrepancy less than 7%.By analyzing and simplifying our theoretical model,we derive explicit analytical expressions for critical behavioral values on the force-displacement curve of the beam.These behavioral values include critical force,critical displacement,and travel,which are generally sufficient for characterizing the snapthrough properties of a bistable buckled beam.Based on these analytical formulas,we investigate the influence of a bistable buckled beam's key design parameters,including its actuation position and precompression,on its critical behavioral values,with our results validated by FEA simulations.Our analytical method enables fast and computationally inexpensive design of bistable buckled beams and can guide the design of complicated systems that incorporate bistable mechanisms.

Conversion between solid and beam element solutions of finite element method based on meta-modeling theory:development and application to a ramp tunnel structure

In this study, a new method for conversion of solid finite element solution to beam finite element solution is developed based on the meta-modeling theory which constructs a model consistent with continuum mechanics. The proposed method is rigorous and efficient compared to a typical conversion method which merely computes surface integration of solid element nodal stresses to obtain cross-sectional forces. The meta-modeling theory ensures the rigorousness of proposed method by defining a proper distance between beam element and solid element solutions in a function space of continuum mechanics. Results of numerical verification test that is conducted with a simple cantilever beam are used to find the proper distance function for this conversion. Time history analysis of the main tunnel structure of a real ramp tunnel is considered as a numerical example for the proposed conversion method. It is shown that cross-sectional forces are readily computed for solid element solution of the main tunnel structure when it is converted to a beam element solution using the proposed method. Further, envelopes of resultant forces which are of primary importance for the purpose of design, are developed for a given ground motion at the end.

Analytical modeling of sandwich beam for piezoelectric bender elements

Piezoelectric bender elements are widely used as electromechanical sensors and actuators, An analytical sandwich beam model for piezoelectric bender elements was developed based on the first-order shear deformation theory （FSDT）, which assumes a single rotation angle for the whole cross-section and a quadratic distribution function for coupled electric potential in piezoelectric layers, and corrects the effect of transverse shear strain on the electric displacement integration. Free vibration analysis of simplysupported bender elements was carried out and the numerical results showed that, solutions of the present model for various thickness-to-length ratios are compared well with the exact two-dimensional solutions, which presents an efficient and accurate model for analyzing dynamic electromechanical responses of bender elements.