Euler Bernoulli Research Articles

Nonlinear Dynamic Analysis of FG Fluid Conveying Micropipes with Initial Imperfections

This paper investigates the nonlinear dynamics of FG-FCMPs with initial imperfections. Based on the MCST, the nonlinear equations of motion and the corresponding boundary conditions are established by applying Hamilton’s principle, the Euler–Bernoulli beam theory, and von-Kármán geometric nonlinearity. To describe the initial geometric imperfection of the FG-FCMP, the first-order vibrational mode is employed. Subsequently, in cases of primary parametric resonance, 1:2 subharmonic resonance for the first-order mode, as well as primary resonance for the second-order mode, Galerkin’s method, and the multiple scale method are utilized to analyze the amplitude–frequency responses of the imperfect FG-FCMP. The numerical simulations test the influence of flow velocity, micro-scale parameter, geometrical imperfections, and external loads on the nonlinear characteristics of a coupled system with two DOFs. It is found that, as the increase of flow velocity, micro-scale effect, and external load, the amplitude of the first two modes can be increased. The hardening characteristics are converted into the softening characteristics due to the imperfect effect. Furthermore, numerical results provide a more comprehensive understanding of the nonlinear dynamics of FCMPs for both periodic and chaotic motions.

Event-triggered robust fault-tolerant control of a class of Euler–Bernoulli beam equations via sliding mode control

Event-triggered robust fault-tolerant control of a class of Euler–Bernoulli beam equations via sliding mode control

Time history nonlinear analysis for 2D modelization of an existing building using flexibility and displacement-based formulation

The object of the research is the response of a structure in time between formulations, namely force-based (flexibility-based) and displacement-based. An existing two-story reinforced concrete structure subjected to three earthquakes separately is presented in order to carry out a comparative study in terms of time history structure response between the two formulations cited above namely force based (flexibility based) and displacement based. This research aimed to compare between two formulations namely displacement based and force-based (flexibility-based), in order to show the accuracy and the reliability of the second one. In the case of nonlinear time history response of a residential two-storey reinforced concrete building, which is representative of typical residential building construction in Italy in the 1970’s and 1980’s with the aim of taking advantages of flexibility-based formulation if the comparison proves conclusive. In the course of the research, the former approach is a classical two-node, displacement-based, Euler-Bernoulli frame element with the disadvantage of discretizing the structure thus increasing the number of degrees of freedom and equation, in order to achieve the required precision. The latter is a two-node, force-based, Euler Bernoulli frame element. The main advantage of the second element is that it is «exact» within the relevant frame element theory. This implies that one element per frame member (beam or column) is used in preparing the frame mesh, thus leading to a reduction of the global number of degrees of freedom. Therefore, a reduction in number of equations to solve, resulting in a saving of time with increased precision. An existing two-story reinforced concrete structure subjected to three earthquakes separately is presented in order to carry out a comparative study in terms of time history structure response between the two formulations cited above namely force based (flexibility-based) and displacement-based. As a result of the research, it is shown that with flexibility-based approach no discretization error occurs and all governing equations are satisfied exactly, making a possibility of simpler and faster numerical modeling. In the proposed approach will be of great use in the field of numerical modeling due to its ease and accuracy. This elegant formulation is promising for future work thanks to its advantages.

Dynamic analysis of short-fiber reinforced composite nanobeams based on nonlocal strain gradient theory

This study deals with the dynamic behavior of short-fiber reinforced composite nanobeams. It is assumed that short-fibers are aligned or randomly distributed in the composite nanobeams. Nonlocal strain gradient theory is applied to composite nanobeam mechanics including Euler–Bernoulli and Timoshenko beam models. The transverse vibration of these composite nanobeams is investigated for various boundary conditions. Approximate Ritz method is used for obtaining the natural frequencies of short-fiber reinforced composite nanobeams. In addition to vibration analysis, wave propagation in short-fiber reinforced composite nanobeams is investigated and wave dispersion relations are analytically obtained for both Euler-Bernoulli and Timoshenko beam models. The vibration and wave dispersion results of short-fiber reinforced composite nanobeams are obtained for aligned and randomly distributed cases. The results obtained from this paper showed that there is no significant difference between the aligned and randomly oriented short-fiber composite nanobeams. This provides great convenience to designers where it is not possible to orient the reinforcement material in composites. The present study may be useful for the mechanical analysis and design of micro/nano-electromechanical systems (MEMS/NEMS), nanoprobes, nanosensors, nanoactuators, and atomic force microscopes.

Linear matrix inequality-based vibration control of three-dimensional Euler–Bernoulli beam with external disturbance

In this paper, we address the LMI-based vibration control problem of a three-dimensional Euler–Bernoulli beam subject to external disturbance. First, the dynamics of the system is characterized by several partial differential equations (PDEs). Second, in order to realize the disturbance rejection, we propose the disturbance observers. Based on these, the boundary control protocols are designed via linear matrix inequality (LMI) method for suppressing elastic vibration. A significant feature of LMI is that the control gains can be efficiently solved by this method. Finally, some simulation examples illustrate the effectiveness of the proposed theoretical results.

Static bending analysis of a transversely cracked strip tapered footing on a two-parameter soil using a new beam finite element

In this paper, a new beam Euler–Bernoulli finite element for the transverse static bending analysis of cracked slender strip tapered footings on an elastic two-parameter soil is presented. Standard Hermitian cubic interpolation functions are selected to derive the closed-form expressions of complete stiffness matrix and the load vector. The efficiency of the proposed finite element is verified on an example with several width tapering variations of a simple cracked footing with the results of governing differential equation. Another novelty of this study is improved bending moment functions with included discontinuity conditions at the crack location. These functions now accurately describe the bending moments in the vicinity of the crack of the finite element.

A modified couple stress model to analyze the effect of size-dependent on thermal interactions in rotating nanobeams whose properties change with temperature

Understanding the behavior of rotating materials and structures on small scales is crucial for many scientific and engineering fields, and such studies play an important role in this regard. This paper aims to propose a novel paradigm for analyzing the vibrational characteristics of thermoelastic nanobeams with diverse physical attributes. The incorporation of size effects in the structural and thermal constitutive relationships involves the consideration of nonlocal elasticity theory (NET) and the modified couple stress (MCS) model, together with the utilization of the Euler–Bernoulli assumptions for thin beams. The work also involves the development of a new non-Fourier thermoelasticity model that incorporates the Moore–Gibson–Thompson (MGT) equation. Furthermore, it was taken into account that the thermal conductivity of the flexible materials is not consistent but rather changes with temperature. Periodic pulse heating was applied to rotating nanobeams, and the behavior of the nanobeams was investigated with respect to thermal, rotational, and length-scale effects. To demonstrate the impact of the distinctive characteristics of the MCS and MGT thermoelastic models on the physical fields, a range of numerical data are presented. The study also investigated the propagation characteristics of thermo-mechanical waves, taking into account aspects such as thermal relaxation time and the influence of temperature change on physical properties. Based on the observed results, including the size impact in the structural and thermal equations can lead to significant disparities when compared to conventional models. The inclusion of the length-scale component in the MCS theory, which increases the rigidity and hardiness of the nanobeam structure, may help to explain the observed effect.

Finite Element Analysis of a Double Beam connected with Elastic Springs

This paper develops a finite element method for double beams subjected to static loading. The double beam consists of two Euler–Bernoulli beams connected continuously by an elastic spring connection. The finite element for a double beam is formulated with eight degrees of freedom based on the Euler-Bernoulli beam theory. The finite element method is implemented in MATLAB software to analyze the behavior of the double beams. The MATLAB code calculates the displacements of both the upper and lower beams. Numerical examples are compared with the analytical solution to demonstrate the high accuracy of the proposed method.

LQR-Based Suspension for Heavy Vehicles Considering the Time-Varying Characteristics of Vehicle–Road Interaction

The rapidly increasing demand for heavy vehicles in the transportation sector has led to more severe road damage and significantly higher road maintenance costs. Investigation of active suspension system incorporating vehicle–road interaction (VRI) can provide effective means to reduce the road damage and improve the ride comfort of vehicles. In this paper, an active vehicle suspension controller based on the strongly stable and robust linear–quadratic regulator (LQR) principle is developed for the time-varying VRI system. The coupled system is modeled by a quarter-car traveling on the road modeled by an Euler–Bernoulli beam resting on a viscoelastic foundation. Two types of road irregularities, deterministic sine wave and random, are considered in the numerical studies. A time-frozen technique, by which the linear time-varying system is converted to a time sequence of time-invariant systems, is applied to solve for the coupled responses. Three variables, the dynamic tire load, vehicle body acceleration, and suspension relative displacement, are used to assess the effectiveness of the LQR-based controller in comparison to the passive suspension system. The performance of the active control is investigated for different vehicle speeds, vehicle loads, and road profiles. Numerical studies show that the controller can effectively reduce the road damage and improve the ride comfort, with a slight increase in the suspension relative displacement. This work lays a useful foundation in the problem formulation and system parameter influences for future vehicle suspension design and road damage mitigation including VRI.

Existence and uniqueness of s-curve segments of tensioned elastica satisfying geometric Hermite interpolation conditions

It has been recently proved that every proper restricted elastic spline is a stable nonlinear spline, and this yields a broad existence proof for stable nonlinear splines. When tension is included in the setup, stable nonlinear splines under tension always exist, but they do not always have the property that each piece (connecting one interpolation point to the next) is an s-curve. Being correlated with the fairness of an interpolating curve, this property is desirable and we conjecture that the framework employed successfully with restricted elastic splines will also work well with nonlinear splines under tension. Our purpose is to prove the following foundational result: Given points P1≠P2, in the plane, along with corresponding unit directions d1,d2 that satisfy d1⋅(P2−P1)≥0 and d2⋅(P2−P1)≥0, there exists a unique s-curve segment of Euler–Bernoulli elastica under tension λ>0 that connects P1 to P2 with initial direction d1 and terminal direction d2.

Some decay properties in extensible thermoelastic beam with Gurtin–Pipkin’s law

In this article, we derive the equations that constitute the mathematical model of extensible thermoelastic beam in the context of second sound model which turns to the Gurtin–Pipkin’s one. These nonlinear governing equations are derived according to the von Kármán theory and simplified by the Euler–Bernoulli approximation in the context of Gurtin–Pipkin’s law. Even more so, the case of Fourier’s law can be recovered from the derived system one through a proper singular limit procedure, where the memory kernel collapses into the Dirac mass at zero. Then, we show that the Gurtin–Pipkin’s derived model is globally well-posed using the semigroup theory and the corresponding solutions decay exponentially under a condition on physical coefficients of the model. Finally, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we prove that the linear (without extensibility) associated semigroup is not analytic. Finally, we show that Cattaneo’s law is a particular example of Gurtin–Pipkin’s law for suitable choice of the memory kernel.

Research on Mechanical Model and Torsional Stiffness Properties of Leaf Spring Torsional Vibration Dampers for Marine Diesel Engines

The torsional stiffness parameter significantly influences the natural frequency of a leaf spring torsional vibration damper and its proper match with a diesel engine, and the nonlinear characteristics of torsional stiffness avoid reduced reliability due to the excessive torsion angle of the damper. An efficient mechanical model for the damper with nonlinear characteristics is established by integrating the Euler–Bernoulli beam theory and accounting for the geometric nonlinearity of leaf spring deformation during operation. The model’s validity is confirmed through finite element analysis. This study then explores the influence of design parameters on the mechanical characteristics of the damper. The results reveal a gradual increase in the torsional stiffness of the damper with the expanding arc radius of the clamping groove. Simultaneously, the torsional stiffness curve exhibits more pronounced nonlinear characteristics. In contrast, an elongation of the leaf spring leads to a sharp decline in torsional stiffness, accompanied by a diminishing prominence of nonlinear traits. Thus, both the arc radius of the clamping groove and the spring length significantly impact the torsional stiffness and nonlinear features of the leaf spring torsional vibration damper. The nonlinear characteristics intensify with an enlarged arc radius of the clamping groove and a reduced leaf spring length. Additionally, the damper’s torsional stiffness is influenced by the leaf spring thickness and the red copper gasket length. Future damper designs should comprehensively consider these relevant parameters.

Investigation on the thermo‐piezo‐flexoelectric energy harvesting performance of self‐powered microbeam devices considering strain gradient and dual‐phase‐lag effects

AbstractAchieving renewable power for self‐powered electronic devices such as micro/nano‐electro‐mechanical systems is attractive in the fields of energy resources and environment, in this regard, dielectric material structures with excellent energy storage and electromechanical responses have stimulated a surge of research interest. However, in the micro/nanoscale filed, the existing thermo‐electromechanical continuum theories are not capable of fully and effectively characterizing the arising effects, for example, size‐dependent effect and heat lagging effect, etc. To further develop the existing theories, a theoretical prototype for piezo‐flexoelectricity by incorporating strain gradient effect and dual‐phase‐lag heat conduction model is proposed for the first time in this study, and then this proposed model is applied to investigating the energy harvesting performances of a piezo‐flexoelectric micro‐beam under piezoelectric‐thermoelastic coupling. In terms of the differential form of two‐phase elasticity and Hamilton's principle, the governing equations and the boundary conditions for the transverse vibration of a micro‐beam based on the Euler–Bernoulli beam model are derived. By means of the assumed mode method, the influences of the flexoelectricity, the load resistance and the strain gradient on voltage and output power are examined. From the obtained results, it can be aware of that this study not only presents the piezoelectric‐thermoelastic coupling effect of piezo‐flexoelectric configuration in microscale field, but also presents that a piezo‐flexoelectric system at microscale can generate higher power than that reported in previous works. It is hoped that this study could provide some guidelines in designing and optimizing dielectric energy harvesters of micro/nano electronic devices.

Multi-parametric optimization for controlling bifurcation structures

Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( > 10 2 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

Modeling and vibration suppression based on Nussbaum function for beam bridge with unknown control directions and unknown nonlinear time-varying actuator faults

In this paper, the modeling of the beam bridge system with in-domain control and the vibration suppression under unknown control directions, unknown nonlinear time-varying actuator faults and external disturbances are studied. The beam bridge with fixed ends can be regarded as Euler Bernoulli beam, which is a typical distributed parameter system. Firstly, the partial differential equation (PDE) model of the beam bridge is established according to Hamilton principle. Then, an adaptive in-domain control scheme based on Nussbaum function is designed on the PDE model to eliminate the vibration of the beam bridge. It is proved theoretically that all closed-loop signals are uniformly bounded under this control law, and the vibration of the system asymptotically converges to zero. The simulation results also prove the effectiveness of the proposed control scheme. What’ more, the modeling and control scheme presented in this paper can be applied to a class of systems similar to beam bridges, such as vibration suppression of the main beam of an overhead crane bridge.

Inverting mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation

Abstract We propose an inverse problem of determining the mechanical and variable-order parameters of the Euler–Bernoulli beam on viscoelastic foundation. For this goal, we develop a fully-discrete Hermite finite element scheme for this model and analyze the corresponding error estimates. The Levenberg–Marquardt method is then applied to determine the multiple parameters. Extensive numerical experiments are performed under practical settings to demonstrate the behavior of the proposed model and the efficiency of the algorithm.

Non-linear modeling of a bi-layer magnetostrictive cantilever considering [formula omitted]E effect

Magnetostrictive cantilever beams are reliable and straightforward energy harvesting devices for powering wireless sensors using environmental vibrations. However, despite their widespread use, a comprehensive analytical model encompassing both mechanical and electromagnetic main features is still lacking. The mechanical-to-magnetic coupling of a magnetostrictive beam, with its various working conditions and nonlinear characteristics, can be effectively approximated by the Euler–Bernoulli (E–B) equation with damping (viscous and Kelvin–Voigt) coupled with a non-linear magnetostrictive model. In this study, we propose a practical and convenient model for a general concept device designed to convert vibrations into electrical energy. The model is thoroughly compared with experimental measurements aimed to prove the goodness of the approach rather than energy harvesting highest performance.

Numerical and experimental study of impact dynamics of bistable buckled beams

Bistable systems have been of great research interest due to their rich dynamics arising from a variety of excitations. This study presents an investigation into the nonlinear dynamics of a continuous bistable beam subjected to a form of vibro-impact forcing. A fixed–fixed Euler–Bernoulli beam, precompressed to achieve bistability, is considered, while a shaker with sinusoidally-prescribed velocity applies transverse impact forcing. By varying excitation parameters such as frequency and amplitude, the system response can be tuned. The study begins by employing a Galerkin expansion to the continuous equation of motion, using the buckling modes to remove buckling level dependence and ensure equilibrium symmetry. A Hertzian contact law is chosen to model the collision between the shaker and the buckled beam. The rich dynamics of this system are then explored through numerical and experimental analyses. The focus lies on investigating the emergence of intrawell and interwell dynamics that arise with varying excitation frequency, amplitude, and location. Moreover, we evaluate the significance of including higher-order modes in the modeling of bistable beam dynamics, particularly near snapthrough. This research provides valuable insights into the behavior of continuous bistable structures under vibro-impact forcing, contributing to the understanding and control of such systems in engineering applications.

Accurate vertical nanoelectromechanical measurements

Piezoresponse Force Microscopy (PFM) is capable of detecting strains in piezoelectric materials down to the picometer range. Driven by diverse application areas, numerous weaker electromechanical materials have emerged. The smaller signals associated with them have uncovered ubiquitous crosstalk challenges that limit the accuracy of measurements and that can even mask them entirely. Previously, using an interferometric displacement sensor (IDS), we demonstrated the existence of a special spot position immediately above the tip of the cantilever, where the signal due to body-electrostatic (BES) forces is nullified. Placing the IDS detection spot at this location allows sensitive and BES artifact-free electromechanical measurements. We denote this position as xIDS/L=1, where xIDS is the spot position along the cantilever and L is the distance between the base and tip. Recently, a similar approach has been proposed for BES nullification for the more commonly used optical beam deflection (OBD) technique, with a different null position at xOBD/L≈0.6. In the present study, a large number of automated, sub-resonance spot position dependent measurements were conducted on periodically poled lithium niobate. In this work, both IDS and OBD responses were measured simultaneously, allowing direct comparisons of the two approaches. In these extensive measurements, for the IDS, we routinely observed xIDS/L≈1. In contrast, the OBD null position ranged over a significant fraction of the cantilever length. Worryingly, the magnitudes of the amplitudes measured at the respective null positions were typically different, often by as much as 100%. Theoretically, we explain these results by invoking the presence of both BES and in-plane forces electromechanical forces acting on the tip using an Euler–Bernoulli cantilever beam model. Notably, the IDS measurements support the electromechanical response of lithium niobate predicted with a rigorous electro-elastic model of a sharp PFM tip in the strong indentation contact limit [deff≈12pm/V, Kalinin et al., Phys. Rev. B 70, 184101 (2004)].

On wave-like differential equations in general Hilbert space with application to Euler–Bernoulli bending vibrations of a beam

Wave-like differential equations occur in many engineering applications. Here the engineering setup is embedded into the framework of functional analysis of modern mathematical physics. After an overview, the L2–Hilbert space approach to free Euler–Bernoulli bending vibrations of a beam in one spatial dimension is investigated. We analyze in detail the corresponding positive, selfadjoint differential operators of 4-th order associated to the boundary conditions in statics. A comparison with free string wave swinging is outlined.