Euler-Bernoulli Cantilever Research Articles

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)].

Measurements of flow-induced vibration of a flexible splitter plate mounted on a cylinder in free stream flow

We experimentally study flow-induced vibration (FIV) of a thin, cantilevered, flexible plate attached on the lee side of a circular cylinder in a free stream airflow. The Reynolds number (ReD) based on the diameter of the cylinder was varied in the range of (4 × 103, 5×104). The plates of different lengths and the same span were tested with varying airflow velocities (U) in a wind tunnel. The reduced velocity (UR) based on the length and the second natural frequency of the plate was in the range of (1, 11). We describe the plate dynamics showing displacement, frequency, phase plane, amplitude spectral density (ASD), oscillation envelope, modal energy contribution, and flow structures. Based on the motion of the plate, three FIV regimes are observed, namely, initial excitation, transition, and lock-in. During lock-in, the plate attains self-sustained limit cycle oscillation. We observe the presence of small-amplitude higher harmonics in the ASD plot. The second mode is dominant around the onset of lock-in, while the first mode is dominant later in the lock-in regime. The dimensionless displacement increases with an increase in mass ratio (Ms) in the lock-in regime. With a decrease in Ms, the slope of the dimensionless frequency with UR increases. To explain measurements, we develop a wake oscillator model (WOM) in which the plate and fluctuating lift force are modeled as an Euler–Bernoulli cantilever beam and van der Pol oscillator, respectively. The coupled WOM qualitatively and quantitatively predict the measured amplitude and frequency response, respectively, in the lock-in regime.

Experimental Detection of Nonlinear Dynamics Using a Laser Profilometer

This paper investigates a cantilever beam nonlinear dynamic behaviour, on which the nonlinearity is introduced with permanent magnet interactions or with a non-holonomic contact. The experimental time domain responses obtained from non-zero initial conditions are measured using a laser profilometer, conventionally adopted for product shape detections in online industrial applications. The Fourier transform, Continuous Wavelet transform, and Hilbert transform are used to investigate nonlinear phenomena in the frequency content, highlighting advantages and drawbacks of the three methods in catching instantaneous phenomena. Then, a Multi-Phi approach is proposed to describe the time evolution of nonlinear systems by means of a discrete number of linearised systems. Therefore, two linearised models have been developed and tuned to describe the dynamic behaviour of different Euler–Bernoulli cantilever beam configurations. The experimental data of nonlinear systems are compared with the corresponding ones of the linear system to evaluate the effects of introduced nonlinearities on the overall dynamic properties.

Experimental Inter-Modal Targeted Energy Transfer in a cantilever beam undergoing Vibro-impacts

This work experimentally demonstrates inter-modal targeted energy transfer (IMTET) in a cantilever beam system with vibro-impacts. IMTET is known to achieve non-resonant rapid energy transfer from low-to-high frequency structural modes in mechanical systems. While previous works have computationally demonstrated the efficacy of IMTET, the theory and methodology supporting IMTET still lacks experimental verification. To address this task, an experiment is constructed whereby a steel beam is clamped at one end and excited at the other, and steel tips with clearances anchored to a host fixture serve as vibro-impacts (VIs) which induce non-resonant energy redistribution within the modal space of the beam. The system is modeled as an Euler–Bernoulli cantilever beam which is discretized by the finite element method, and the VI forces are simulated by a single-sided dissipative Hertzian contact model. The time-responses, modal dissipation ratios, and characteristic decay times measured in the experiment are compared to the computational models to confirm the validity of the theoretical predictions. It is found for both the experiment and computational model that the VIs transfer energy from low frequency structural modes to high frequency ones which, in turn, results in a substantial reduction in the characteristic decay time and thus confirms the efficacy of IMTET in practice. The diverse potential applications of IMTET in engineering practice are then emphasized.

Coupled Flexural–Torsional Forced Vibration Analysis of a Piezoelectrically Actuated Double-Cantilever Structure

Abstract In this paper, the forced coupled flexural–torsional vibration of a piezoelectrically actuated double-cantilever structure is investigated. The double-cantilever structure is composed of two uniform and identical Euler–Bernoulli cantilever beams connected by a rigid tip connection at their free ends. There is also a piezoelectric layer attached on the top surface of each cantilever beam. The characteristic equation for the coupled flexural–torsional vibrations of the structure is derived and solved to determine the natural frequencies. The time response to the forced vibrations of the structure is studied using the Galerkin approximation method. The effects of dimensional parameters, including the length of the cantilever beams and the length of the tip connection, and the piezoelectric input voltage on the coupled flexural–torsional natural frequencies and amplitude of vibrations of the structure are investigated analytically and experimentally. The results show that the coupled flexural–torsional fundamental frequency of the piezoelectrically actuated double-cantilever structure decreases as either the length of the cantilever beams or the tip connection is increased. Moreover, the amplitude of the coupled flexural–torsional vibrations of the structure is proportional to the piezoelectric input voltage; however, the slope of the curves depends on dimensional parameters. For a given voltage, the effect of either of the aforementioned dimensional parameters on the amplitude of vibrations depends on the other dimensional parameter such that there is a turning point in all the curves, whose location depends on the configuration of the structure.

Flexural–Torsional Free Vibration Analysis of a Double-Cantilever Structure

Abstract This paper aims to investigate the free coupled flexural–torsional vibrations of a double-cantilever structure. The structure consists of two identical Euler–Bernoulli cantilever beams with a piezoelectric layer on top. The beams are connected by a rigid tip connection at their free ends. The double-cantilever structure in this study vibrates in two distinct modes: flexural mode or coupled flexural–torsional mode. The flexural mode refers to the in-phase flexural vibrations of the two cantilever beams resulting in translation of the tip connection, while the coupled flexural–torsional mode refers to the coupled flexural–torsional vibrations of the cantilever beams resulting in rotation of the tip connection. The latter is the main interest of this research. The governing equations of motion and boundary conditions are developed using Hamilton’s principle. Two uncoupled equations are realized for each beam: one corresponding to the flexural vibrations and the other one corresponding to the torsional vibrations. The characteristic equations for both the flexural and the coupled flexural–torsional vibration modes are derived and solved to find the natural frequencies corresponding to each mode of vibration. The orthogonality condition among the mode shapes is derived and utilized to determine the modal coefficients corresponding to each mode of vibration. Moreover, the analytical and experimental investigations show that the coupled flexural–torsional fundamental frequency of the structure is dependent on dimensional parameters including the length of the cantilever beams and the length of the tip connection.

A strain-wavelength modelling of low-frequency cantilever fibre Bragg grating accelerometer

The accuracy of single-degree-of-freedom (SDOF) model in describing the beam motion of low-frequency cantilever fibre Bragg grating (FBG) accelerometer can be further explored, since the SDOF model is limited to fundamental vibration modes. Therefore, this paper addresses the aforementioned limitation by introducing a modal model of the cantilever Euler-Bernoulli (EB) beam into the wavelength shift equation. This modal model (FBG-MM) considered five vibration modes. The convergent series of eigenfunction for cantilevered EB beam was solved using a standard modal expansion theory. The curvature of the cantilevered beam resulted from dual differentiation of the eigenfunction (with respect to x) is then related to the strain and wavelength of the FBG. The computed wavelength shift using FBG-MM was compared with the SDOF model. The experimental results where the harmonic base excitation occurring at five different frequencies were also discussed. The simulation results showed that the wavelength shift exhibited more reasonable behaviour along the beam particularly when the excitation frequency exceeded the second bending mode (596.67 Hz). The FBG-MM and experimental wavelength shift showed convincing correlation only when the excitation frequency came close to the fundamental frequency. On the other hand, there was no agreement at low excitation frequencies due to stiffness issues of the cantilever beam and the capability of the optical spectrum analyser. In future, the improvement of this study will focus on introducing a tip mass on the cantilever beam for increasing the accelerometer sensitivity and representing the cantilever beam using Timoshenko model.

Vibration control of a piezoelectric cantilever smart beam by ℒ 1 adaptive control system

In this paper, the modelling and design of an state feedback control system applied for vibration control of a smart piezoelectric flexible Euler–Bernoulli cantilever beam is presented. For a Single-Input Single-Output (SISO) case by retaining the first two dominant vibratory modes, the dynamics of the system is presented. Classical adaptive control law, with time-varying parameters and in presence of disturbance, is employed to suppress the vibration of the beam. Three Piezoelectric patches, two as actuators and the other as a sensor, are bonded to the structure at the support of the beam and along the length of the beam. The beam structure is modelled in the state space form using the concept of piezoelectric theory, the Euler–Bernoulli beam theory and the Finite Element technique. Also, for comparing the performance of the proposed controller, PID and LQR control systems are applied. Simulation results, first for constant parameters and then for time-varying parameters and disturbance, represent the better performance of the proposed control system in comparison to the other mentioned controllers.

Dynamic modelling, simulation and experiments of a micro-cutter with applications to cell perforation

The nonlinear three-dimensional dynamic equations of motion of a micro-cutter driven longitudinally at ultrasonic operating frequencies are derived using Kane's method. The micro-cutter is assumed to be immersed in a fluid, and in contact with an oocyte, which is to be perforated by the micro-cutter. The micro-cutter is modelled as an Euler-Bernoulli cantilever beam attached to a moving base. The shear and rotary inertia effects are neglected by considering a slender-shaped beam with homogeneous and isotropic material properties. It is assumed that there is no slip between the micro-cutter tip and the embryo membrane. The model presented demonstrates that the longitudinal excitation input of a micropipette results in excitation of out-of-plane, lateral motion due to the nonlinear dynamic coupling of the dynamics. Experimental results of membrane perforation are presented in the work supporting oocyte micro-cutter observations regarding suitable frequencies of excitation for effective oocyte membrane perforation.

Multi-objective robust design optimization (MORDO) of an aeroelastic high-aspect-ratio wing

In this paper, a new approach for multi-objective robust optimization of flutter velocity and maximum displacement of the wing tip are investigated. The wing is under the influence of bending–torsion coupling and its design variables have different levels of uncertainty. In designing and optimizing wings with a high aspect ratio, the optimization process can be done in such a way to increase the flutter velocity, but this can increase the amplitude of the wing tip displacement to a point that leads to the wings damage and structural failure. Therefore, single-objective design optimization may lead to infeasible designs. Thus, for multi-objective optimization, modeling is based on the Euler–Bernoulli cantilever beam model in quasi-steady aerodynamic condition. Using the Galerkin’s techniques, the aeroelastic equations are converted to ODE equations. After validating the results, the system time response is obtained by the numerical solution of the governing equations using 4th Runge–Kutta method and the flutter velocity of the wing is obtained using the theory of eigenvalues. Subsequently, by choosing bending and torsional rigidity and mass per unit wing length as the optimization variables, using Monte Carlo–Latin hypercube (MC-LH) simulation and 4th polynomial chaos expansion (PCE), the effect of uncertainty on these variables is modeled in modeFRONTIER™ software coupled with MATLAB™ and optimization is performed by genetic algorithm. Finally, by plotting the Pareto front, it is observed that with an acceptable increase in flutter velocity, the maximum wing displacement amplitude is reduced as much as possible. The results of the multi-objective robust optimization show more feasible results compared with deterministic optimization.

Estimation of Reduced Stiffness under the Influence of Crack in a Beam

Machine components with cantilever boundary conditions are most prominently used in mechanical engineering applications. When such components are subjected to fatigue loading, crack may get initiated and failure may occur. In order to prevent catastrophic failure and safe guard these components one has to condition monitor the dynamic behavior under fatigue loading condition. Vibration based condition monitoring is one of the most effective method to assess the fatigue failure of the component. In this paper, a cantilever beam is analyzed for its dynamic behaviour under the influence of crack. The cantilever beam considered for the analysis is of rectangular cross section and is uniform throughout its length. The characteristic equation was derived for the Euler-Bernoulli cantilever beam to obtain the relationship between the location of the crack and stiffness of the beam because the stiffness of the beam influences the natural frequency. It is found that the stiffness of the cantilever beam is varying for varying locations of the crack in different modes of vibration. It is clearly understood from the analysis that the vibration response of the cracked cantilever beam in its modes of vibration is affected by the corresponding stiffness reduction based upon the location of the crack. So, it can be inferred that the natural frequency of the cracked cantilever beam may have different values for different locations of the crack.

Vibratory Response Characteristics of High-Frequency Shape Memory Alloy Actuators

Abstract This paper presents a shape memory alloy actuator design using a bimorph structure capable of high-speed actuation and low power consumption. Two active layers of shape memory alloy wires are separated by a passive layer of thermoplastic polyurethane. This structure results in a bending actuator when current is alternated between the two active shape memory alloy layers. Actuators of lengths 20, 25, 30, 35, and 40 mm were tested at peak current input of 110, 120, 130, and 140 mA. The 40-mm actuator was shown to have a natural frequency of 11.4 Hz with a bending displacement of 26.4 mm perpendicular to the neutral position and a power input of 0.78 W (140 mA peak current input). A relationship between the input current and the resulting vibratory characteristics was found. As the current increases, the natural frequency decreases and the damping ratio increases. The experimental results are compared with a finite element method (FEM) vibration analysis and an Euler–Bernoulli cantilever beam equations.

Identification of an unknown shear force in the Euler–Bernoulli cantilever beam from measured boundary deflection

In this paper, a novel mathematical model and new approach is proposed for identification of an unknown shear force in a system governed by the general form Euler–Bernoulli beam equation , subject to the boundary conditions u(0,t) = ux(0,t) = 0, , , from available boundary observation (measured output data), namely, the measured deflection at x = l. The approach is based on weak solution theory for PDEs, Tikhonov regularization combined with the adjoint method. A uniqueness result for the problem under consideration is proved. The Neumann-to-Dirichlet operator corresponding to the inverse problem is introduced. It is shown that this operator is injective, compact and Lipschitz continuous. The last property allows us to prove an existence of a quasi-solution of the inverse problem. Fréchet differentiability of the Tikhonov functional is also proved. In the case when Tr = 0, an implicit formula for the Fréchet gradient of this functional is derived by making use of the unique solution to corresponding adjoint problem. Furthermore, a class of admissible shear forces in which the Fréchet gradient of the Tikhonov functional is Lipschitz continuous, is derived. Numerical examples with random noisy measured outputs are presented to illustrate the validity and effectiveness of the proposed approach.

Comparison of machine learning methods for crack localization

In this paper, the Haar wavelet discrete transform, the artificial neural networks (ANNs), and the random forests (RFs) are applied to predict the location and severity of a crack in an Euler–Bernoulli cantilever subjected to the transverse free vibration. An extensive investigation into two data collection sets and machine learning methods showed that the depth of a crack is more difficult to predict than its location. The data set of eight natural frequency parameters produces more accurate predictions on the crack depth; meanwhile, the data set of eight Haar wavelet coefficients produces more precise predictions on the crack location. Furthermore, the analysis of the results showed that the ensemble of 50 ANN trained by Bayesian regularization and Levenberg–Marquardt algorithms slightly outperforms RF.

Hybrid reference-free total displacement for railroad bridge campaign monitoring

According to railroad managers in North America, measuring bridge displacements under service traffic can assist in quantifying the reliability and safety of an aged railroad network. Railroad bridge campaign monitoring consists of collecting as much information as possible annually to inform decisions related to safety and for maintenance, repair, and replacement (MRR). However, acquiring displacement data without a stationary reference frame in the field is often challenging during railroad bridge campaign monitoring. This research proposes a hybrid reference-free sensing technology to enable the collection of bridge displacements under service traffic during inspections. This technology fuses two sensing systems: the estimated pseudostatic displacement from an accelerometer pair and the measured direct dynamic displacement from a passive-servo-electro-magnetic-induction (PSEMI) sensing system. Researchers designed a cantilever column in the laboratory representing a timber bridge pier model in order to validate the effectiveness of the hybrid sensing system. This technique used an Euler–Bernoulli cantilever beam model to translate the pier tilt from the accelerometer pair to pseudostatic displacement. Researchers used a shake table to excite the laboratory model with real bridge displacement collected in the field under a train crossing. The pseudostatic displacement was obtained by applying a low-pass filter to the accelerometer data, and the dynamic displacement was obtained by using a high-pass filter on the PSEMI data. The total reference-free transverse displacements acquired through the fusion of the pseudostatic and dynamic components were compared to the LVDT measurements. The comparison results showed that the proposed hybrid sensing system measures the reference-free total bridge displacements accurately and reliably without the need for postprocessing dynamic displacement. The authors conducted this comprehensive laboratory testing to support the suitability of this system for railroad bridge campaign monitoring in railroad bridge environments.

Reduced-order modeling and usefulness of non-uniform beams for flexoelectric energy harvesting applications

Energy harvesting at micro- and nanoscales has recently seen a renewed interest that the flexoelectric effect can counter the inability of piezoelectric energy harvesters to generate enough energy at small scales. Almost all small-scale energy harvesters use uniform rectangular geometries, whereas at the macroscale energy harvesters use a wide array of geometries including tapered rectangular geometries. The incorporation of non-uniform effects into a piezoelectric system considering the flexoelectric effect should give insight into how these systems can benefit from different geometries. A non-uniform flexoelectric Euler–Bernoulli cantilever energy harvester is modeled using classical continuum theories and is examined at the microscale. The non-uniformity of the energy harvester is governed by linear and nonlinear tapering effects, with the nonlinearities represented by high-order polynomials. The system is assumed to be linear, only undergoing harmonic base excitation. The varied tapering ratios and powers of the geometric tapering, considering that only the thickness and the width of the beam are tapered, are compared with uniform systems. The results show that non-uniform beams exhibit more harvested power than their uniform counterparts and also increase the range of resonant frequencies where significant power can be generated. Nonlinear tapering increases the amount of power that could be harvested compared to linear tapering; however, the nonlinearity of the tapering effects is limited to cubic and quadratic forms. It is demonstrated that higher-order tapering effects reduce the amount of harvested power compared to the linear taper counterpart. Non-uniform beams prove to be more effective than their rectangular counterparts within a linear system, whereas optimal resistive loads decrease as the tapering effects increase.

Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation

In the paper, transient vibrations of a Bernoulli-Euler cantilever beam with a rigid mass attached at the end and subjected to base motion are presented. The tip mass centre of gravity coincides with the endpoint of the beam. The viscoelastic properties of the beam material are described using a fractional Kelvin-Voigt model. The Riemann–Liouville fractional derivative of an order of 0 < γ ≤ 1 is used. Exact relationships for the natural frequencies and mode shapes of the beam are derived. Moreover, a method of calculating the damped natural frequencies of the analyzed beam is presented. The forced-vibration solution of the beam is derived using the mode superposition method. Transient movement of the base is described by an oscillating function with a linearly time-varying frequency. A convolution integral of the fractional Green's and forcing functions is used to achieve the beam response. The Green's function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second describes the drift of the equilibrium position. The dynamic responses are calculated numerically. The impact of a fractional order derivative on transient responses when passing through resonance regions of the analyzed beam is shown. The beam responses obtained using a fractional viscoelastic material model are compared with those obtained using an integer viscoelastic material model. In the system analyzed, the influence of the term describing the drift of the equilibrium position on the beam deflection is relatively low and may be neglected. The results obtained indicate that an increase in the derivative order causes a decrease in the vibration amplitudes of the beam. The calculations reveal that the vibration amplitudes in the region of the second resonance are larger than in the region of the first resonance for some parameters of the system under consideration, which is not the case when using integer order derivatives in the viscoelastic material model.

Analytical Solution for Large Deflection of Multilayered Composite Cantilever Beams with Interlayer Slip

The solution methodologies for nonlinear differential equations play a significant role in the area of theoretical mechanics, in particular large deflection of beams. In this paper, Euler–Bernoulli’s equation is adapted along with R. Da Silva method for solving the problem of large deflection on multilayered cantilever beams with interlayer slip analytically. Interlayer slip, horizontal displacement and large deflection nonlinear differential equations of multilayered Euler–Bernoulli cantilever beams were derived. The equations are applied to the typical partially interacted multilayered cantilever beam problem with point and uniformly distributed load. Solutions of the geometrically linear theory and geometrically nonlinear theory are compared.

Projection methods for stochastic dynamic systems: A frequency domain approach

A collection of hybrid projection approaches are proposed for approximating the response of stochastic partial differential equations which describe structural dynamic systems. In this study, an optimal basis for the approximation of the response of a stochastically parametrised structural dynamic system has been computed from its generalised eigenmodes. By applying appropriate approximations in conjunction with a reduced set of modal basis functions, a collection of hybrid projection methods are obtained. These methods have been further improved by the implementation of a sample based Galerkin error minimisation approach. In total six methods are presented and compared for numerical accuracy and computational efficiency. Expressions for the lower order statistical moments of the hybrid projection methods have been derived and discussed. The proposed methods have been implemented to solve two numerical examples: the bending of a Euler–Bernoulli cantilever beam and the bending of a Kirchhoff–Love plate where both structures have stochastic elastic parameters. The response and accuracy of the proposed methods are subsequently discussed and compared with the benchmark solution obtained using an expensive Monte Carlo method.

Frequency lock-in during vortex induced vibration of a rotating blade

The vortex-induced vibration (VIV) of a rotating blade is studied in this paper. Uniform Euler–Bernoulli cantilever beam equation is used to model the rotating blade, while the nonlinear oscillator satisfying Van der Pol equation is used to represent the vortex shedding. The fluctuating lift force on the blade due to the wake dynamics is expressed such that it is proportional to the square of the magnitude of the relative velocity at which the blade interacts with the fluid. An unsteady lift due to vortex shedding acts on the blade, where as the blade is coupled with the fluid through a linear inertial coupling, resulting in a fluid–structure interaction problem. The coupled equations are discretized by using modal functions which satisfy the Eigenvalue problem. The work attempts to understand the interaction between the blade modes and the wake modes. The influence of the mass ratio and the reduced flow velocity on the coupling of the modes is explored. It is concluded form the study that the first mode is dominant and the effect of higher modes is less significant. Therefore, the single mode approximation is considered to study the instabilities associated with the frequency lock-in phenomenon of a rotating blade. The influence of fluid and blade damping, mass ratio and the coupling parameter on the lock-in phenomenon is investigated in this study.