Coupled Beam System Research Articles

Structures with varying discontinuities and curvatures: A dynamic analysis approach by the p-version finite element method

The present paper investigates the geometrically nonlinear time domain dynamic analysis of a coupled Timoshenko beam-beam or beam-arch mechanical system. The coupled structure is modelled with a variable discontinuity in an elastic layer, which represents a real case from technical practice where there is no continuous distribution of the elastic layer or the stiffness of the layer is changed by other influences at different locations. A modified p-version finite element method is developed for the vibrations of a shear deformable coupled beam system with a discontinuity in an elastic layer. The main contribution of this work is the discovery of coupled effects and phenomena in the simultaneous vibration analysis of varying discontinuity and varying curvature of the newly modelled coupled mechanical system. New general mode shapes are presented, and the forced vibrations in the time domain are analyzed using the Newmark method. The study compares the results for various locations and types (size) of discontinuity in an elastic coupling layer and various curvatures of the lower supported beam. It explains the cases in which the influence of curvature takes precedence over discontinuity and vice versa in achieving a steady-state regime of vibrations. The analysis results are valuable and have broader applications in the field of solids and structures.

Non-stability of a bogie moving along a specific infinite complex flexibly beam-layer structure

A comprehensive investigation was conducted to analyze the vibration stability of a bogie that moves uniformly along a complexly modeled flexibly supported infinite Reddy-Bickford coupled beam system situated on a viscoelastic base. The bogie was simulated as a non-deformable body of finite length, connected through a specific system of two springs and two dampers with identical stiffness and damping, interacting with masses within a complex beam-layer system commonly encountered in technical practice. These masses, through which the bogie interacts with the high-order shear deformable beam-layer system, form part of a mechanical system consisting of a double oscillator in constant contact with the infinitely long flexible structure. As it is well-known that when the velocity of masses, such as the bogie or any other complex oscillator, exceeds the minimum velocity at which waves propagate in the beam, the system's oscillation may become unstable, it is crucial to determine the conditions and stable regions of motion that involve specific combinations of system parameters. The region of instability is identified within the parameter space of the system using the D-decomposition technique and the argument principle. A comprehensive analysis was conducted to investigate the impacts of the viscous damping in the supports of the bogie, the mass of the bogie bar, and the wheelbase on the stability of the model. The derived conclusions, focused on determining the stable regions of motion based on the considered parameters, represent a significant novelty and contribution to the research previously unknown for the motion system of a bogie along a complexly modeled flexibly supported infinite high-order shear deformable coupled beam-layer system on a viscoelastic base.

Dynamics of moving coupled objects with stabilizers and unconventional couplings

A comprehensive investigation was carried out to analyze the vibration stability of a coupled bogie system moving uniformly along a complexly modeled flexibly supported infinite high-order shear deformable coupled beam system on a viscoelastic base. The main contribution of this study involves a comparative analysis of the specific coupling of the bogie system with and without the proposed additional stabilizer, in comparison to conventional cases. The study demonstrates significant benefits of the newly proposed options of dual coupling and additional stabilizers in terms of stability. Another contribution of the study is the feasibility of the proposed unconventional models in technical practice. A phenomenon has been discovered that increasing the viscous damping in a special coupling leads to the occurrence of motion instability in the mechanical system. The paper also presents a novel technical solution for connecting moving objects at high speeds. The benefits of the additional oscillator as a stabilizer are extensively illustrated through numerous different examples. It is well-known that when the velocity of masses, such as the bogie or any other complex oscillator, exceeds the minimum velocity at which waves propagate in the beam, the oscillation of the system may become unstable. Therefore, it is crucial to determine the conditions and stable regions of motion that involve specific combinations of system parameters. The region of instability is identified within the parameter space of the system using the D-decomposition technique and the argument principle. The bogie is specifically connected to another bogie in two new ways: with an additional stabilizer allowing double transverse displacement and providing an additional contact with the base, or with a specific coupling that enables simultaneous rotational and transverse displacement at the contact point. The analysis reveals two key findings. First, the additional mechanical stabilizer allows for the widest range of permissible suspension stiffness (largest stability region) when the connection is conventional. Second, enabling transverse displacement in the couplings results in a stable regime of motion at much lower suspension stiffness compared to other cases.

Output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance

This paper is concerned with performance output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance. An infinite‐dimensional disturbance estimator is designed to estimate the total disturbance. By compensating the total disturbance, a servomechanism corresponding to the reference signal and servomechanism‐based output feedback control law are designed. It is proved that under such control law, the performance output tracks exponentially the reference signal and the involved states of closed‐loop system are bounded. The most important contribution is to deal with the shear boundary term stemmed from the error system between the disturbance estimator and the original system. The admissibility does not hold for such shear boundary term, while the corresponding boundary terms in the existing literature was proved to be admissible. Two key steps are presented to cope with such problem: First, the semigroup generation and exponential stability for a coupled beam system are verified by Riesz basis approach; second, the admissibility of a control operator for semigroup governed by such coupled beam system is proved. Moreover, Sobolev embedding theorem is introduced to simplify the proof of the boundedness of the closed‐loop systems with respect to the available literature. Some numerical simulations are presented to illustrate the effectiveness.

Nonlinear analysis and effectiveness of weakly coupled microbeams for mass sensing applications

In this work, we develop a general model of a mass sensor made of N weakly mechanically coupled microbeams subject to electric actuation. The developed model is verified by comparing the simulated pull-in voltages, natural frequencies, and frequency response of a two-weakly coupled beam system against their experimental counterparts reported in the literature. The sensitivity of the mass sensor in terms of frequency shift is observed to significantly increase when enlarging the size of the beams array. The simulation results reveal a clear transformation of the frequency response from a nearly linear to nonlinear behavior as result of the deposition of a small mass on the coupled system. As such, we show the potential use of bifurcations that result in an abrupt jump to a large-amplitude motion for sensing purposes. Furthermore, by exploiting the mode localization effect, the nonlinear response can be triggered on one of the beams when an added masses is introduced, allowing for an amplitude-based mass detection mechanism of the device. The proposed sensing method has the possibility to operate in bifurcation mode for mass threshold detection that can be tuned using the AC actuation or in continuous mode based on extracting the added mass from the amplitude of the sensing beam’s oscillations.

New architectures for micromechanical coupled beam array filters

Coupled resonator filters implemented as microelectromechanical systems (MEMS) offer performance advantages as band-pass filters at MHz frequencies. Here new designs based on resonant cavities for acoustic slow waves are developed to allow alternative frequency responses. Derivation of the lumped element model for coupled beam systems with in-plane motion from Rayleigh–Ritz perturbation theory is first reviewed. Departures from ideal behaviour caused by mechanical and electrostatic detuning are resolved. Slow wave theory is then used to develop linear array topologies with novel responses including band-stop and comb filtering with controlled filter roll-off. A systematic procedure is developed to allow rapid identification of design parameters without the need for lengthy numerical simulation, using the lumped element, stiffness matrix and finite element methods to investigate the layout parameters of initial design concepts, detailed mechanical effects and detailed electrostatic effects, respectively. High performance is demonstrated, with good agreement between the models.

Free and Forced Vibration of Coupled Beam Systems Resting on Variable Viscoelastic Foundations

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

Stability Analysis of Initially Curved Beams Mechanically Coupled in a Parallel Arrangement

This paper investigates the stability behavior of a mechanically coupled bi-stable mechanism made of two parallel and initially curved microbeams with focus on the influence of the coupled beams’ initial curvatures on the system. First, the nonlinear and coupled force–displacement equation is derived. Then, a parametric study of the coupled beams system is studied with the system categorized into different structural compartment types according to the initial curvature of the coupled beams. It is concluded that the snap-through of such a coupled bi-stable system is governed by the beams’ initial curvatures difference. The simulation results showed that these two parameters (the beams’ initial curvatures) essentially govern the structural behavior of the coupled system in satisfying the necessary structural stability condition. It is found that the smallest (critical) value of the minimum force amplitude occurs only when both initial beams’ mid-point elevations are equal to each other. Furthermore, it is shown that any probability to increase or decrease the curvature of any beam will alter the nonlinear behavior of the coupled beam system from a simple regular snap-through to a constrained-snap-through, and even to the disappearance of the snap-through. Finally, a finite element method is conducted to investigate the stability of the coupled mechanism, of which the results show a good agreement with the analytical results of this paper.

Axial–bending coupling vibration of mass eccentric double-beam system with discrete elastic connections

The existence of mass eccentricity will lead to the energy transfer between axial and flexural vibrations of a beam. To study the coupling properties of a double-Timoshenko beam system, a non-uniform coupled double-beam system is modeled in which the upper beam is typical and the lower beam is mass eccentric simulated by a non-uniform two-layer Timoshenko beam. By incorporating Hamilton’s principle and spectral element method, the axial–bending coupled governing equations of the system are derived and the approach can also be easily used to analyze the influences of the parameters and other coupled beam systems. Both the free and forced vibration results of a double-beam system by this method are consistent with the corresponding finite element model’s and thus this method is validated. The coupled properties and their mechanism are revealed. The influences of axial and transverse flexible connection on the coupling properties including free and forced vibration are investigated. A systematic matching principle of reducing the vibration of the coupled system is proposed.

Non-linear energy harvesting from coupled impacting beams

Energy harvesting has many potential applications for structures with broadband excitation, such as aircraft noise and low frequency vibrations from human motion. The advantage with a vibro-impacting system is the capability of converting low frequency response to high frequencies. A coupled beam system is base excited and the influence of different system parameters are studied. Exciting the system at a single resonant frequency highlights the influence of clearance and base excitation amplitude on the beam responses. The frequency sweep study shows the sensitivity of the power generated to the contact stiffness, damping and clearance between the beams. The power generated by the coupled system from the non-linear impact is sensitive to the thickness ratio of the beams and the clearance. The variation in thickness ratio alters the spacing of the natural frequencies of the system which causes modes to interact. This study shows that higher power is produced than the linear system, depending on the dissimilarity in the mode shapes of the interacting close modes.

Identification of Clearance and Contact Stiffness in a Simplified Barrel-Cradle Structure of Artillery System

In gun barrel-cradle structure, the presence of clearance usually changes the dynamic response of muzzle and results in shooting dispersion (under continuous firing condition). The parameter estimation of such clearance nonlinear system is the prerequisite for establishing quantitative relation between the clearance and muzzle disturbance. In this paper, the restoring force surface (RFS) method and the nonlinear identification through feedback of outputs (NIFO) method are first combined for parameter identification in a simplified barrel-cradle structure. With the RFS method, clearance value can be obtained by analyzing the restoring force plot. Then the contact stiffness can be identified by using NIFO method. This identification process is verified in a single-degree-of-freedom (SDOF) system with clearance. To adapt to the rigid-flexible coupled beam system with clearances which is simplified from the barrel-cradle structure, a modification for the combined method mentioned above is proposed. The core idea of the modification is reducing the continuous system to multiple-degree-of-freedom (MDOF) system to reserve the nonlinear characteristics through modal transformation matrix. The advantage of this transformation is that the linear parts of the MDOF systems are decoupled, which greatly reduces the difficulty of identification. The simulation results have shown the effectiveness of current method.

Analyze on a Coupled Beam Vibration System by FEM (II): Case Study

From the article of Analyze on a coupled beam vibration system by FEM (I): theory, equation of motion of the beam system with attachments is established by conventional finite element method, and some parameters, such as natural frequencies, associated mode shapes, are obtained. The mode localization and frequency loci veering phenomena of a weakly coupled beam system with multiple spring-mass systems are investigated. Studies show that for weakly coupled beam system with attachments, the mode localization and loci veering will occur once there is a disorder in the system.

Shock transmission in a coupled beam system

This paper investigates the circumstances under which high peak acceleration can occur in the internal parts of a system when subjected to impulsive driving on the outside. Motivating examples include the design of packaging for transportation of fragile items. The system is modelled in an idealised form using two beams coupled with point connections. A Rayleigh–Ritz model of such coupled beams was validated against measurements on a particular beam system, then the model was used to explore the acceleration response to impulsive driving in the time, frequency and spatial domains. This study is restricted to linear vibration response and additional mechanisms for high internal acceleration due to nonlinear effects such as internal impacts are not considered. Using Monte Carlo simulation in which the indirectly driven beam was perturbed by randomly placed point masses a wide range of system behaviour was explored. This facilitates identification of vulnerable configurations that can lead to high internal acceleration. The results from the study indicate the possibility of curve veering influencing the peak acceleration amplification. The possibility of veering within an ensemble was found to be dependent on the relative coupling strength of the modes. Understanding of the mechanism may help to avoid vulnerable cases, either by design or by preparatory vibration testing.

On mode localization of a weakly coupled beam system with spring-mass attachments

There are a large number of papers in the literature dealing with the free vibration analysis of single/multi-span uniform beam with multiple spring-mass systems, but that of coupled multi-span beams carrying spring-mass attachments is rare. In this note, free vibration analysis of a weakly coupled beam system with spring-mass attachments is conducted. The mode localization and frequency loci veering phenomena of the coupled beam system are investigated. Studies show that for weakly coupled beam system with spring-mass attachments, the mode localization and frequency loci veering will occur once there is a disorder in the system.

Vibration Analysis of a Coupled Beam System Carrying Any Number of Sprung Masses

There are a large number of papers in the literature dealing with the free vibration analysis of single/multi-span uniform beam with multiple spring-mass systems, but that of coupled multi-span beams with any number of spring-mass systems is rare. In this paper, an approach based on finite element analysis is presented to analyze the free vibration of coupled beams with spring-mass attachments. Firstly, the equation of motion of the beam system with attachments is established by conventional finite element method. Secondly, the natural frequencies and the associated mode shapes are obtained from the generalized eigenvalue problem. It is found that the results from the proposed method agree well with those from analytical methods in the literature. Finally, the mode localization and frequency loci veering phenomena of a weakly coupled beam system with multiple spring-mass systems are investigated. Studies show that for weakly coupled beam system with attachments, the mode localization and loci veering will occur once there is a disorder in the system.

Modeling and Control of the Vibration of Two Beams Coupled with Fluid and Active Links

Investigated are modeling and control approaches for vibration analysis of two identical beams which are coupled with fluid and active mechanical links. In the modeling of the coupled beam system, orthogonal functions are used to represent vibration of the beams and the fluid-structure interaction is considered. Frequency Response Functions (FRFs) are derived from the coupled governing equations and the superposition principle for linear vibration systems. In the control of vibration of the beams, impulse response functions corresponding to the FRFs and an adaptive control algorithm are employed to attenuate vibration transmission between the two beams. Natural frequencies, mode shapes as well as the pressure distribution in the fluid are computed. The results obtained by the proposed modeling method are in good consistency with those obtained by the finite element analysis. Moreover, it is demonstrated that the active mechanical link is able to reduce vibration transmission and change the deformation of beams as well as the distribution of fluid pressure.

Modeling and Control of the Vibration of Two Beams Coupled with Fluid and Active Links

Investigated are modeling and control approaches for vibration analysis of two identical beams which are coupled with fluid and active mechanical links. In the modeling of the coupled beam system, orthogonal functions are used to represent vibration of the beams and the fluid-structure interaction is considered. Frequency Response Functions (FRFs) are derived from the coupled governing equations and the superposition principle for linear vibration systems. In the control of vibration of the beams, impulse response functions corresponding to the FRFs and an adaptive control algorithm are employed to attenuate vibration transmission between the two beams. Natural frequencies, mode shapes as well as the pressure distribution in the fluid are computed. The results obtained by the proposed modeling method are in good consistency with those obtained by the finite element analysis. Moreover, it is demonstrated that the active mechanical link is able to reduce vibration transmission and change the deformation of beams as well as the distribution of fluid pressure.

Effects of nearfield waves and phase information on the vibration analysis of curved beams

At high frequencies, energy methods such as the statistical energy analysis and the power flow analysis have been popularly used to predict the averaged responses of vibro-acoustic subsystems. Usually, these energy methods ignore flexural nearfield components and phase information, mainly for simplicity. Such assumptions sometimes lead to an erroneous conclusion, in particular for complex structures and at medium frequencies around the Schroeder cutoff frequency. This paper deals with the effects of nearfield waves and phase information at medium to high frequencies by using the ray tracing method (RTM). A curved beam and a coupled beam system were chosen as test examples, which exhibit the typical mode conversion between various types of travelling waves. Propagation of longitudinal, flexural, and torsional waves was studied based on the Euler-Bernoulli beam theory. Analyses of the spatial distribution of vibrational energy quantities revealed that the conventional RTM could mimic the overall trend of the traveling wave solution. However, the results varied smoothly in space due to the neglect of wave interference. By considering the phase information, local fluctuations of vibration energy could be correctly described. It was confirmed that the flexural nearfield plays a significant role near boundaries and junctions. It was also shown that the accuracy of the analysis depends mainly on the modal overlap factor. Similar to other high frequency methods, the results become close to the traveling wave solutions as the modal overlap factor increases.

Identification of local damages in coupled beam systems from measured dynamic responses

This paper deals with local damages identification for strongly and weakly coupled beam systems with close and repeated natural frequencies based on the measured dynamic responses of the systems under external moving forces. The dynamic responses of the coupled beam systems are calculated from Newmark integration method and they are used for structural damage detection. The mode localization phenomenon due to local damage(s) in the weakly coupled beam system is studied. In the inverse analysis, a dynamic response sensitivity-based finite element model updating method is employed for the detection of local damage(s) in the coupled beam systems. Numerical simulation shows that the proposed method is effective in identifying the damages in the coupled beam systems with good accuracy from several measurements. It is found the proposed method is insensitive to artificial measurement noise and has the potential for practical applications.

Vibrations and Power Flows in a Coupled Beam System

Vibrations of and power flow between coupled beams have been the subject of many investigations, and various techniques have been developed over the years. However, most of the existing methods will require a certain level of modifications or adaptations to account for the variations in the coupling and∕or boundary conditions. In this study, a general analytical method is developed for predicting the vibrations of and power flow between two-coupled beams. The coupling between the beams is generically represented by two (translational and rotational) springs of arbitrary stiffnesses. Thus, many rigid and nonrigid connectors of practical interest can be directly taken into account. In addition, because the beams are elastically restrained at each end, the current method can be universally applied to different boundary conditions by simply varying the stiffnesses of the boundary springs. Numerical results are presented to show the excellent accuracy of the proposed approach.