Bernoulli Beam Theory Research Articles

Geometrically nonlinear analysis of planar beam and frame structures made of functionally graded material

Geometrically nonlinear analysis of planar beam and frame structures made of functionally graded material (FGM) by using the finite element method is presented. The material property of the structures is assumed to be graded in the thickness direction by a power law distribution. A nonlinear beam element based on Bernoulli beam theory, taking the shift of the neutral axis position into account, is formulated in the context of the co-rotational formulation. The nonlinear equilibrium equations are solved by using the incremental/iterative procedure in a combination with the arc-length control method. Numerical examples show that the formulated element is capable to give accurate results by using just several elements. the influence of the material inhomogeneity in the geometrically nonlinear behavior of the FGM beam and frame structures is examined and highlighted.

Improving voltage output with PZT beam array for MEMS-based vibration energy harvester: theory and experiment

Piezoelectric vibration energy harvesters as an autonomous power source for various types of sensors, actuators and MEMS devices have attracted increasing attention in recent years. Traditional MEMS-based PZT single beam vibration energy harvester has low voltage output (usually <1 V) which causes rectifier circuit to consume most of power or turn off in practical applications. In this paper, in order to improve the voltage output of MEMS-based PZT vibration energy harvester as well as ensure high power output, a PZT beam array configuration for MEMS-based vibration energy harvester is proposed. Based on Hamilton's principle and Euler---Bernoulli beam theory, mathematical model for the proposed vibration energy harvester is established. Two different dimensions MEMS-based PZT beam array vibration energy harvesters are designed and fabricated through micro-fabrication techniques. The performances of the two types of fabricated vibration energy harvesters are characterized and compared with traditional vibration energy harvester in experiment. Experimental results agree well with theoretical analysis, and the experimental results show that the better performance fabricated vibration energy harvester can generate 121.4 μW and 3.5 V respectively at 1 g acceleration.

Variational formulation of curved beams in global coordinates

In this paper we derive a variational formulation for a linear curved beam which is natively expressed in global Cartesian coordinates. During derivation the beam midline is assumed to be implicitly described by a vector distance function which eliminates the need for local coordinates. The only geometrical information appearing in the final expressions for the governing equations is the tangential direction, and thus there is no need to introduce normal directions along the curve. As a consequence zero or discontinuous curvature, for example at inflection points, pose no difficulty in this formulation. Kinematic assumptions encompassing both Timoshenko and Euler--Bernoulli beam theories are considered. With the exception of truly three dimensional formulations, models for curved beams found in literature are typically derived in the Frenet frame defined by the geometry of the beam midline. While it is intuitive to formulate curved beam models in these local coordinates, the Frenet frame suffers from ambiguity and sudden changes of orientation in straight sections of the beam. Based on the variational formulation we implement finite element models using global Cartesian degrees of freedom and discuss curvature coupling effects and locking. Numerical comparisons with classical solutions for both straight and curved cantilever beams under a tip load are given, as well as numerical examples illustrating curvature coupling effects.

Damping in beams using viscoelastic layers

Various types of sandwich beams with viscoelastic cores are currently used in aerospace and automotive industries, indicating the need for simple methods describing the dynamics of these complex structures. In order to understand the effectiveness of the sandwich structures, the dynamics of bare beam with unconstrained and constrained viscoelastic layers are investigated in this study. The viscoelastic layer is bonded uniformly on the beam. The effects of distributed viscoelastic layer treatment on the loss factors are studied. From the experiments it is observed that beams with constrained viscoelastic layer provide higher loss factors than those with unconstrained layer. The dynamics of sandwich beams is modeled using Euler and Bernoulli beam theory. Frequency-dependent Young’s modulus and loss factors are considered in the model of viscoelastic material. The predicted Eigen frequencies obtained from the model are compared with the experimental results for two viscoelastic materials with aluminum base material. Frequency response functions are obtained from the finite element model and compared with experimental results for harmonic input. Reductions in vibration amplitudes for two viscoelastic materials (EAP-2 and EAP-43) are also compared. Based on the experimental results, it has been observed that the loss factors of EAP-43 are higher than that of EAP-2.

Optimal Trajectory of Flexible Manipulators Using Genetic Algorithms

The applications of flexible manipulators are increasing and due to the high demand on fuel consumption there is a need to optimize the energy consumption for stable and durable operation of the flexible manipulators. In the present work the Genetic Algorithm (GA) is employed to optimize the total torque and the torque of the first link of a two-link flexible manipulator with a fourth order polynomial trajectory. The mathematical model of the manipulator is obtained using the extended Hamilton's Principle where the flexible links are treated as Euler- Bernoulli's beam theory. A fifth order polynomial trajectory undergoes a rest-to rest maneuvering is proposed as a bench mark for validation.

Mode I Fracture Toughness of a Tri-Layered Structure with Interfacial Crack

Analytical model based on the Bernoulli beam theory and fracture mechanics have been developed to predict the fracture toughness of a tri-layered beam with interfacial crack. In this work, a double cantilever beam test is conducted to determine the fracture toughness of mode I. The strain energy release rate for the double cantilever beam is expressed as a function of the material properties and thickness of the tri-layer beam. The analytical solution is validated with the finite element result. Good agreement demonstrates that the proposed approach is able to provide an efficient way for the calculation of the fracture toughness of a multi-layered structure. The effects of material properties and thickness between the adjacent layers of the interfacial crack are examined through a parametric study.

Thermal Stress Analysis of a Multi-Layered Structure

Analytical model based on the Bernoulli beam theory and strain compatibility conditions at the interfaces between the two layers have been developed to predict the distribution of thermal stresses within the multi-layered structure due to the mismatch of thermal expansion. The closed-form solution of thermal stresses related to the material properties and geometry were obtained. It is useful to provide a simple and efficient analytical model, so that the stress level in the layers can be accurately estimated. The analytical results are compared with finite element results. Good agreement demonstrates that the proposed approach is able to provide an efficient way for the calculation of the thermal stresses.

Thermal Stress Analysis of a Bi-Material Layered Structure

Thermal stress induced by the mismatch of the thermal expansion coefficients between dissimilar materials becomes an important issue in many bi-layered systems, such as composites and micro-electronic devices. It is useful to provide a simple and efficient analytical model, so that the stress level in the layers can be accurately estimated. Basing on the Bernoulli beam theory, a simple but accurate analytical formulation is proposed to evaluate the thermal stresses in a bi-material beam. The analytical results are compared with finite element results. Good agreement demonstrates that the proposed approach is able to provide an efficient way for the calculation of the thermal stresses. It is shown that thermal stresses are linear proportion to the ratio of thermal expansion coefficients between the two materials. Parametric studies reveal that thermal stresses in each layer are decreasing with the increase of thickness, and are increasing with the increase of Young’s modulus ratio between the two materials.

Inverse dynamic analysis and trajectory planning for flexible manipulator

In this study, dynamic analysis and trajectory planning for a rigid-flexible manipulator is considered. The manipulator under consideration consists of two links–the first link is rigid while the second link is flexible. Euler–Bernoulli's beam theory is utilized to model the elastic link, where shear deformation and rotary inertia can be neglected if the cross-sectional area is small compared to the length of the beam. The tangential coordinate frame is employed to describe the elastic deflection of the flexible link. The equations of motion of the manipulator are derived using the extended Hamilton's principle. The joints trajectory will be designed based on four different trajectories and the joints torques required to move the end effector according to a prescribed trajectory will be obtained through the solution of the inverse dynamics problem. To validate the applicability of soft motion trajectory for flexible manipulators, a comparison with other standard trajectories is carried out for the first three mode shapes of vibration of the flexible link.

Integrated monitoring scheme for a maglev guideway using multiplexed FBG sensor arrays

In the structural monitoring of maglev guideways, electromagnetic interference (EMI) can be a significant problem, as the maglev train is powered by high-voltage electric feeding systems. Recently, researchers have successfully applied fiber optic sensors to modern railway structures, mainly because they offer EMI immunity. This study presents an integrated monitoring scheme for a maglev guideway using wavelength division multiplexing (WDM)-based fiber Bragg grating (FBG) sensors. The physical quantities such as strains, curvatures, and vertical deflections are measured in field tests. The strains are directly measured from multiplexed FBG sensors at various locations on a test bridge followed by curvature calculations based on the plane section assumption. Vertical deflections are then estimated using the Bernoulli beam theory and regression analyses. Frequency information obtained from the proposed method is compared with those from a conventional accelerometer. Verification tests were conducted on a newly developed Korean maglev test track at different vehicle speeds.

Carbon nanotube-reinforced composites: frequency analysis theories based on the matrix stiffness

Strong and versatile carbon nanotubes are finding new applications in improving conventional polymer-based fibers and films. This paper studies the influence of matrix stiffness and the intertube radial displacements on free vibration of an individual double-walled carbon nanotube (DWNT). For this, a double elastic beam model is presented for frequency analysis in a DWNT embedded in an elastic matrix. The analysis is based on both Euler–Bernoulli and Timoshenko beam theories which considers shear deformation and rotary inertia and for both concentric and non-concentric assumptions considering intertube radial displacements and the related internal degrees of freedom. New intertube resonant frequencies and the associated non-coaxial vibrational modes are calculated. Detailed results are demonstrated for the dependence of resonant frequencies and mode shapes on the matrix stiffness. The results indicate that internal radial displacement and surrounding matrix stiffness could substantially affect resonant frequencies especially for longer double-walled carbon nanotubes of larger innermost radius at higher resonant frequencies, and thus the latter does not keep the otherwise concentric structure at ultrahigh frequencies. Therefore, depending on the matrix stiffness, for carbon nanotubes reinforced composites, different analysis techniques should be used while the aspect ratio of carbon nanotubes has a little effect on the analysis theory which should be selected.

Surface Effect on the Elastic Behavior of Static Bending Nanowires

The surface effect from surface stress and surface elasticity on the elastic behavior of nanowires in static bending is incorporated into Euler-Bernoulli beam theory via the Young-Laplace equation. Explicit solutions are presented to study the dependence of the surface effect on the overall Young's modulus of nanowires for three different boundary conditions: cantilever, simply supported, and fixed-fixed. The solutions indicate that the cantilever nanowires behave as softer materials when deflected while the other structures behave like stiffer materials as the nanowire cross-sectional size decreases for positive surface stresses. These solutions agree with size dependent nanowire overall Young's moduli observed from static bending tests by other researchers. This study also discusses possible reasons for variations of nanowire overall Young's moduli observed.

Stability Issues of Welded Pipe Containing Pulsatile Flows

This paper deals with the dynamics and stability behavior of a welded pipe containing flowingfluid having a small harmonic component superposed. The equation of motion was derived torepresent the motion of a welded pipe conveying a pulsatile flow using a tensioned Euler- Bernoullibeam theory. The finite element analysis was used to simulate the harmonic motion of a welded pipeconveying fluid. It was shown that welded pipes with clamped-clamped and clamped-pinned supportsare subject to a multitude of parametric instabilities in all their modes. Stability maps are presented forparametric instabilities of welded pipe with clamped-clamped and clamped-pinned ends. It is foundthat the extent of the instability regions increases with flow velocity for clamped-clamped andclamped-pinned welded pipes. The most important consideration from a practical point of view is toavoid the onset of parametric resonance.

Vibration of beams with general boundary conditions due to a moving random load

The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail.

Identifiability of distributed parameters in beam-type systems

Abstract - Necessary and sufficient conditions are established for the identifiability, or non-identifiability, of coefficients, e. g. modulus of elasticity, moment of inertia, conductivity, etc., from prescribed deflection and load acting on a beam associated to the Euler -Bernoulli beam theory.

On the Use of Consistent Approximations for the Optimal Design of Beams

This paper presents a discretization strategy, based on the concept of consistent approximations, for certain optimal beam design problems, where the beam is modeled using Euler--Bernoulli beam theory. It is shown that any accumulation point of the sequence of the stationary points of the family of resulting approximating problems is a stationary point of the original, infinite-dimensional problem. The construction of approximating problems requires the development of a relaxation of constraints to ensure existence of solutions. The numerical solution of the approximating problems, by means of nonlinear programming algorithms that are not scale invariant, must be preceded by a change of variables to guard against deterioration of performance. The use of such approximating problems, in conjunction with a diagonalization strategy, is illustrated by a numerical example.