Euler-Bernoulli Cantilever Beam Research Articles

Dynamic profilometry without out-of-plane conversion to measure vibration frequency of a cantilever beam

A new technique to measure oscillation frequencies and modal shapes of an Euler–Bernoulli cantilever beam using dynamic profilometry and phase extraction techniques is presented. The proposed technique does not require a fixed reference or out-of-plane conversion, and works on nonstop vibration. A binary pattern is projected on the cantilever beam surface mechanically forced to vibrate harmonically in a natural mode. The Fourier transform method is employed to obtain the phase difference between two consecutive frames, in particular it is applied to four consecutive frames so that three consecutive phase differences are available. Finally, the three-step temporal phase-shifting technique is applied to measure the vibration's eigenvalues and eigenfunctions. This paper presents the analysis of the underlying theory and the experimental results obtained.

The Characterisation of the harvesting capabilities of an electrostrictive polymer composite

Harvesting systems capable of transforming dusty environmental energy into electrical energy have attracted considerable interest throughout the last decade. Several research efforts have focused on the transformation of mechanical vibration into electrical energy. Most of these research activities refer to classical piezoelectric ceramic materials, but more recently piezoelectric polymer materials have also been considered. The goal of this paper has been to develop of a model capable of predicting the electrostrictive coefficient of EPCs. The model was developed by using the vibration transmission theory of a Euler–Bernoulli cantilever beam. After validating the model, at the macroscopic level, an empirical relationship was established predicting the value of the electrostrictive coefficient from the dielectric constants and the compliance coefficient of the material. The final results indicated that the dielectric constant was the crucial parameter, for the energy harvesting.

Crack Detection in Stepped Beam Carrying Slowly Moving Mass

A crack identification algorithm based on a mathematical model has been developed to identify crack location and depth in stepped cantilever Euler—Bernoulli beam carrying concentrated masses. In order to estimate crack location and depth in the beam the proposed algorithm utilizes the variation of the difference between the natural frequencies of cracked and intact systems versus single mass location along the beam span. The assumed mode method is used to derive the mathematical model for the system under investigation, in which the crack's effect is introduced to the system as a global effect. The advantage of the proposed algorithm is to identify the crack by monitoring a single natural frequency of the system. The algorithm can utilize the measurements of the first few system natural frequencies to check/reconfirm its identification results. Furthermore, it is efficient and can be simply implemented for any structure that has a slowly moving mass, as in the case of overhead gantry and girder cranes. The identification algorithm is tested using the first system's natural frequency obtained from an experimental work as well as from a finite element analysis. Moreover, the algorithm is tested once more using the second system's natural frequency obtained from finite element analysis. The tests' results show that the crack depth and location can be predicted with sound accuracy.

Tuneable vibration absorber design to suppress vibrations: An application in boring manufacturing process

Dynamic vibration absorbers are used to reduce the undesirable vibrations in many applications such as electrical transmission lines, helicopters, gas turbines, engines, bridges, etc. Tuneable vibration absorbers (TVA) are also used as semi-active controllers. In this paper, the application of a TVA for suppression of chatter vibrations in the boring manufacturing process is presented. The boring bar is modeled as a cantilever Euler–Bernoulli beam and the TVA is composed of mass, spring and dashpot elements. In addition, the effect of spring mass is considered in this analysis. After formulation of the problem, the optimum specifications of the absorber such as spring stiffness, absorber mass and its position are determined using an algorithm based on the mode summation method. The analog-simulated block diagram of the system is developed and the effects of various excitations such as step, ramp, etc. on the absorbed system are simulated. In addition, chatter stability is analyzed in dominant modes of boring bar. Results show that at higher modes, larger critical widths of cut and consequently more material removal rate (MRR) can be achieved. In the case of self-excited vibration, which is associated with a delay differential equation, the optimum absorber suppresses the chatter and increases the limit of stability.

On Mechanical Modeling of Cantilevered Piezoelectric Vibration Energy Harvesters

Cantilevered beams with piezoceramic (PZT) layers are the most commonly investigated type of vibration energy harvesters. A frequently used modeling approach is the single-degree-of-freedom (SDOF) modeling of the harvester beam as it allows simple expressions for the electrical outputs. In the literature, since the base excitation on the harvester beam is assumed to be harmonic, the well known SDOF relation is employed for mathematical modeling. In this study, it is shown that the commonly accepted SDOF harmonic base excitation relation may yield highly inaccurate results for predicting the motion of cantilevered beams and bars. First, the response of a cantilevered Euler—Bernoulli beam to general base excitation given in terms of translation and small rotation is reviewed where more sophisticated damping models are considered. Then, the error in the SDOF model is shown and correction factors are derived for improving the SDOF harmonic base excitation model both for transverse and longitudinal vibrations. The formal way of treating the components of mechanical damping is also discussed. After deriving simple expressions for the electrical outputs of the PZT in open-circuit conditions, relevance of the electrical outputs to vibration mode shapes and the electrode locations is investigated and the issue of strain nodes is addressed.

A model of ionic polymer–metal composite actuators in underwater operations

Ionic polymer metal composites (IPMCs) are active materials that exhibit a bi-directionalelectromechanical coupling: a voltage produces membrane bending, while by bending anIPMC membrane a voltage output is obtained.IPMCs are of increasing interest in a number of application fields. More specifically, IPMCscan work in wet environments, even in water, and this represents a valuable capability in anumber of applications fields such as underwater robotics, surveillance, and biomedicalapplications.In this work a totally new model of an active IPMC beam, solicited by a voltage signal andimmersed in water, is introduced. The model estimates the moment produced by theapplied voltage. Therefore, the classical Euler–Bernoulli cantilever beam theory and theconcept of hydrodynamic function are used to describe the interaction between the beamand the water. Knowledge of this interaction allows estimation of the IPMC active beammotion in water.

Two simple methods to suppress the residual vibrations of a translating or rotating flexible cantilever beam

In this paper, two methods for controlling the residual vibrations of a translating or rotating Euler–Bernoulli cantilever beam are considered. Although a beam has an infinite number of vibration modes, when it simply changes its position by translation or rotation the first mode is the main contributor to the total response. Thus, the problem can be reduced to the base acceleration excitation of a single-degree-of-freedom system. Two simple methods are suggested for suppressing the residual vibration of such a system without considering any control algorithms. Both methods are based on the transient response of the system—namely, the shock response spectrum (SRS). The first method is simple and can be used for lightly damped systems, while the second method can be applied to more general situations. The result of the second method is similar to that of the input shaping method; however, in the method proposed here, both position and time to move from one position to another can be controlled simultaneously.

Optimum design of structures subjected to follower forces

In this paper, shape optimization is used to optimize the critical load of an Euler–Bernoulli cantilever beam with constant volume subjected to a tangential compressive tip load and/or a tangential compressive load arbitrarily distributed along the beam. This is achieved by varying appropriately the beam cross-section, thus its stiffness and mass properties, along its length, so that the critical load reaches its maximum or a prescribed value. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds, which, together with large slenderness ratios, ensure the validity of the Euler–Bernoulli theory and the serviceability of the beam. The evaluation of the objective function requires the solution of the dynamic stability problem of a cantilever beam with variable cross-section. This problem is solved using the analog equation method (AEM) of Katsikadelis for the fourth-order hyperbolic differential equation with variable coefficients, together with a simple and direct iterative method for the evaluation of the critical load based on the eigenvalue sensitivity. Besides its accuracy, this method overcomes the shortcoming of a possible FEM solution, which would require resizing of the elements and re-computation of their stiffness properties during the optimization process. Example problems of various types of follower forces are presented, which illustrate the method and demonstrate its applicability and efficiency.

Crack identification in a stepped beam carrying a rigid disk

In this study, a simple algorithm based on a mathematical model is proposed to identify crack location and depth in a stepped cantilever Euler–Bernoulli beam carrying a rigid disk at its tip. The mathematical model that describes the lateral vibration of the beam is derived using the assumed mode method that coalesces with the Lagrange's equation. A massless torsional spring whose stiffness depend on the severity of the crack is used as a crack model. Using this crack modeling method combined with the assumed mode method, the crack effect is introduced to the system flexibility as global additional structural flexibility. For the assumed mode method, the mode shapes for two uniform beams connected by a massless torsional spring (simulating the cracked beam) are adopted as trial functions. The proposed identification algorithm utilizes the first three natural frequencies shift of the beam caused by a crack to estimate its location and depth. In addition, the proposed mathematical model is used to illustrate the effect of the crack depth and its location on the dynamic characteristics of the system. Using the commercial finite element (FE) software (ANSYS 8.0), three-dimensional finite element analysis (FEA) is carried out to show the accuracy of the derived mathematical model and to demonstrate the reliability of the proposed crack identification algorithm. The analysis showed consistency with the assumed mode results. It showed that the error in concurrent prediction of crack depth and its location using the proposed algorithm is about 10%.

Adaptive Control of a Flexible Robot Using Fuzzy Logic

Operational problems with robot manipulators in space relate to several factors, one most importantly being structural flexibility and subsequently significant difficulties with the control systems, especially, for endpoint position control. Elastic vibrations of the links coupled with their large rotations and nonlinear dynamics is the primary cause. This paper presents a control scheme for tracking the endpoint of a two-link flexible robot. The dominant assumed modes of vibration for Euler-Bernoulli cantilever and pinned-pinned beam boundary conditions are coupled with the nonlinear dynamics for rigid links to form an Euler-Lagrange inverse flexible dynamics robot model. A Jacobian transpose control law actuating the robot joints is adapted by a fuzzy logic system (FLS) with link deformation inputs and a single variable output. Results obtained with an FLS adaptive control strategy show significantly diminished vibration amplitudes for both cantilever and pinned-pinned link dynamics and greatly improved control performance compared to the nonadaptive strategy.

Lyapunov-Based Vibration Control of Translational Euler-Bernoulli Beams Using the Stabilizing Effect of Beam Damping Mechanisms

A translational cantilevered Euler-Bernoulli beam with tip mass dynamics at its free end is used to study the effect of several damping mechanisms on the stabilization of the beam displacement. Specifically, a Lyapunov-based controller utilizing a partial differential equation model of the translational beam is developed to exponentially stabilize the beam displacement while the beam support is regulated to a desired set-point position. Depending on the composition of the tip mass dynamics assumption (i.e. body-mass, point-mass, or massless). it is shown that proper combination of different damping mechanisms (i.e. strain-rate, structural, or viscous damping) guarantees exponential stability of the beam displacement. This novel Lyapunov-based approach. which is based on the energy dissipation mechanism in the beam, brings new dimensions to the stabilization problem of translational beams with tip mass dynamics. The stability analysis utilizes relatively simple mathematical tools to illustrate the exponential and asymptotic stability results. The numerical results are presented to show the effectiveness of the controller.

Damage detection in vibrating bodies using genetic algorithms

This paper proposes a method of locating and quantifying the damage in structural members using the concept of residual forces. To describe the damage in a structure, finite element (FE) models are parameterised by structural stiffness reduction parameters. The damage parameters are determined by minimising a global error derived from dynamic residual vectors, which are obtained by introducing a simulated “experimental” data into the eigenproblem. An eigenvalue prediction algorithm along with normalised residual function is employed to formulate the objective function. Two-point crossover binary coded genetic algorithm (GA) with tournament selection approach is adopted in minimising the objective and optimum set of stiffness reduction parameters are predicted. Current structural defect-identification scheme is verified and assessed using an analytically derived plane truss, a cantilever Euler–Bernoulli beam and a portal truss. Results are presented in the form of tables and graphs.

Repetitive learning with fuzzy logic adaptive control of a flexible robot manipulator

Operational problems with robot manipulators in space relate to several factors, one most importantly being structural flexibility and subsequently significant difficulties with the control systems, especially, position control. A control strategy is devised for positioning the endpoint of a two-link robot manipulator modeled with assumed modes flexible dynamics repetitively tracking a square trajectory. The dominant assumed modes of vibration are determined for Euler-Bernoulli cantilever beam boundary conditions then, coupled with the nonlinear dynamics for rigid links to fonn an Euler-Lagrange inverse flexible dynamics robot model. A Jacobian transpose control law actuates the robot links. While repetitive tracking alone achieves no improvement in control precision, adapting the control law by a fuzzy logic system achieves consistent tracking precision.

Analysis of the Motion of a Flexible Beam Fixed on a Moving Cart and Carrying a Concentrated Mass

In this paper, the equations of motion of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying a lumped mass concentrated at an arbitrary position along the beam is derived. The motion of the beam-mass-cart system is analyzed through unconstrained modal analysis, and a unified characteristic equation for calculating the natural frequencies of the system is obtained. The changes of natural frequencies and the corresponding mode shapes with respect to the changes in mass ratios of the system and to the concentrated position of the lumped mass are investigated with the frequency equation, which can be generally applied to this kind of systems. The exact and assumed-mode solutions including the dynamics of the base cart are obtained, and the open-loop responses of the system by arbitrarily designed forcing function are given by numerical simulations. The results match well with physical phenomena even at the extreme cases where the concentrated mass is attached to the bottom and to the top of the beam.

Boundary control of a cantilevered flexible beam with point-mass dynamics at the free end

In this paper, we develop boundary controllers for a cantilevered Euler-Bernoulli beam with point-mass dynamics at the free end. Specifically, we develop an exact model knowledge controller which exponentially stabilizes the beam displacement and an adaptive controller which asymptotically stabilizes the beam displacement. Experimental results are utilized to validate the performance of the controllers.

Eigen-analysis using optimization with Rayleigh's quotient

An eigen-analysis is usually required if it is desired to obtain the natural frequencies and mode shapes (i.e. eigen-parameters) from a finite element mass and stiffness matrix. In certain instances, however, only the lowest and/or largest natural frequencies are of interest. It is shown that determining the lowest and largest eigen-parameters can be formulated as an unconstrained optimization with Rayleigh's quotient as the objective function. The Powell method and a heuristic search technique, called a genetic algorithm, are used in the optimization. In addition, four initial starting point estimators are discussed for the Powell method. The effectiveness of the proposed optimization is shown using a 6 and 8 degree-of-freedom system, and a 42 degree-of-freedom finite element model of a cantilevered Euler-Bernoulli beam. The lowest and largest eigen-parameters were shown to be extremely well identified. Furthermore, Powell's Method produced better results than the genetic algorithm in terms of efficiency and accuracy.

Improvement in model reduction schemes using the system equivalent reduction expansion process

The SEREP produces an exact reduced model. However, this method requires knowledge of the full system eigenvectors. An extension of this method was presented that uses only the mode shapes of a reduced system. The reduced system mode shapes can be obtained from any reduction procedure, including Guyan, for example. Results from a simulation of a cantilevered Euler Bernoulli beam validated the technique. It was shown that the method improved upon the Guyan and the IRS reduction methods provided that a proper set of master DOF were chosen

Aerodynamic stability of a beam in steady flow

The galloping response of a cantilever Euler-Bernoulli beam of an arbitrary cross-section in a steady flow is investigated. A simple closed form analytical solution and stability criteria as well as the numerical results are presented. The boundary value problem, describing mutually coupled screw bending and torsion, is analytically approximately solved by the Boobnov-Galerkin method. The resulting initial value problem is then solved by the Krylov-Bogoliuboff asymptotic technique at the critical angle flow in accordance with Den Hartog's criterion. For a particular beam with a right-angle cross-section, for which aerodynamic properties and its dynamic response were experimentally determined elsewhere, the results compare favourably. The analysis of the right-angle section response shows that the overall influence of the cross-sectional shear centre eccentricity may play an important role in substantially diminishing the critical flow velocity threshold value and therefore should not be disregarded.

Minimum stiffness of optimally located supports for maximum value of beam eigenfrequencies

The application of Courant's maximum-minimum principle to the problem of maximum raising of the lowest undamped eigenfrequency in flexural vibration of beams is reviewed. A detailed numerical study of the influence on eigenfrequencies of varying locations and stiffnesses of one or two additional lateral supports is performed for a cantilever Euler-Bernoulli beam. Sensitivities and cost optima are discussed.

Lumping of Mass in Calculating Vibration Response

Analytical results are presented in which the vibration response of a uniform Bernoulli-Euler cantilever beam carrying a tip mass is compared with that of a weightless beam supporting sixteen concentrated masses. Natural frequencies and effective masses are presented for the impedance or steady-state behavior. For the shock problem, effective modal masses, as well as modal bending moment factors, are examined. Results show that the lumped mass beam depicts the shock response parameters more accurately than those associated with steady velocity at a driven point. For both shock and steady loading, the lumped mass model becomes more accurate as the ratio of tip mass to beam mass increases.