Uniform Cantilever Research Articles

Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force

The paper addresses the issue of effectively using the direct numerical method for static analysis of the flexible curved uniform cantilever beam under a tip-concentrated follower force. The angle of inclination of the follower force with respect to the deformed axis of the beam remains unchanged during deformation. After changing the variables, the original non-linear boundary value problem transforms into the initial-value problem for pendulum equation. The resulting initial value problem is solved numerically using a modified Numerov's method. In contrast to the usually used iteration methods (e.g. shooting technique), the problem is solved without iterations by direct numerical method. Some qualitative conclusions were made using Kirchhoff's kinetic analogy. It is shown that there are no critical loads in the Euler sense (divergence) for any values of the initial curvature and angle of inclination of the follower force. An extension of direct numerical method to curved spring-hinged cantilever subjected to follower force is also proposed. The paper presents some equilibrium configurations of the uniform curved fixed and spring-hinged cantilevers under normal and tangential follower force obtained by direct method.

Influence of resin-based composite restoration technique and endodontic access on cuspal deflection and cervical microleakage scores

ObjectivesTo assess cuspal deflection and cervical microleakage of mesio-occlusal-distal cavities in standardised premolar teeth restored incrementally with resin-based composite (RBC) placed horizontally or obliquely and with endodontic access cavities (with and without gutta percha and epoxy resin sealer obturation). Materials and methodsThirty-two teeth were allocated to four groups (n=8) and RBC restored in eight horizontal (Group A) or oblique increments (Groups B–D) using a quartz-tungsten-halogen light curing unit. The dependent variable for Groups B–D was endodontic access (none (Group B), obturated without (Group C) and with gutta percha and epoxy resin sealer (Group D)). Cuspal deflections were recorded post-irradiation using a twin channel deflection measuring gauge. Following restoration, the teeth were thermocycled, immersed in fuchsin dye, sectioned and examined for cervical microleakage. ResultsTukey's post hoc tests identified a significant decrease in total cuspal deflection for the horizontal (p=0.015) compared with the oblique placement technique. No significant difference in total cuspal deflection was evident between Groups B and D restored teeth (p>0.318) or in cervical microleakage score between Groups A and D (p=0.575). SignificanceDeformation is proportional to the cubed power of the length of a uniform cantilever beam and although a crude approximation of cuspal deformation, the reduction in the effective cusp length therefore significantly reduced the deflection in the MOD cavities when the horizontal rather than the oblique incremental restoration technique was employed.

A Homotopy Perturbation-Based Method for Large Deflection of a Cantilever Beam Under a Terminal Follower Force

The large deflection problem of a uniform cantilever beam subjected to a terminal concentrated follower force is investigated. The governing equations, which characterize a two-point boundary value problem, are transformed into an initial-value problem. A new algorithm based on the homotopy perturbation method is proposed and applied to the resulting problem and the characteristics of load versus displacement are obtained analytically. The convergence of this method is discussed and the details of load-deflection curves are present. Compared with other existing methods, the present scheme is shown to be highly accurate, while only lower order perturbation is required.

Dynamic stability of cantilever columns under a tip-concentrated subtangential follower force

The dynamic stability of a uniform cantilever column under a tip-concentrated subtangential follower force is examined. The governing two-point boundary value problem is converted to an initial value problem. A simple numerical iterative scheme is followed to generate a load versus frequency curve (eigencurve) in order to study the critical and post-critical behavior of the columns subjected to a tip-concentrated follower force. For the load rotation parameter greater than , the coalesce frequency and the corresponding critical load are obtained. The deformed configurations of cantilever columns are generated for different load rotation parameters. The details of the critical load parameter, tip angle and tip coordinates of the columns for the specified load rotation parameter are presented. The critical load parameter increases with the load rotation parameter.

Series solution for large deflections of a cantilever beam with variable flexural rigidity

In this paper large deflection and rotation of a nonlinear Bernoulli-Euler beam with variable flexural rigidity and subjected to a static co-planar follower loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytical technique for nonlinear problems, namely the Homotopy Analysis Method (HAM). The present solution can be used for the analysis of a wide range of loads, material/cross section properties and lengths for beams undergoing large deformations. The results obtained from HAM are compared with results reported in previous works. Finally, the load–displacement characteristics of a uniform cantilever beam with different material properties under a follower force applied normal to the deformed beam axis are presented.

Cantilever arrays with self-aligned nanotips of uniform height

Cantilever arrays are employed to increase the throughput of imaging and manipulation at the nanoscale. We present a fabrication process to construct cantilever arrays with nanotips that show a uniform tip–sample distance. Such uniformity is crucial, because in many applications the cantilevers do not feature individual tip–sample spacing control. Uniform cantilever arrays lead to very similar tip–sample interaction within an array, enable non-contact modes for arrays and give better control over the load force in contact modes. The developed process flow uses a single mask to define both tips and cantilevers. An additional mask is required for the back side etch. The tips are self-aligned in the convex corner at the free end of each cantilever. Although we use standard optical contact lithography, we show that the convex corner can be sharpened to a nanometre scale radius by an isotropic underetch step. The process is robust and wafer-scale. The resonance frequencies of the cantilevers within an array are shown to be highly uniform with a relative standard error of 0.26% or lower. The tip–sample distance within an array of up to ten cantilevers is measured to have a standard error around 10 nm. An imaging demonstration using the AFM shows that all cantilevers in the array have a sharp tip with a radius below 10 nm. The process flow for the cantilever arrays finds application in probe-based nanolithography, probe-based data storage, nanomanufacturing and parallel scanning probe microscopy.

Authors' response to the comments on, “Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction”

Authors' response to the comments on, “Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction”

Comments on “Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction”

Using a power series expansion for the roots of the characteristic equation, the natural frequencies of a cantilevered Euler–Bernoulli beam with an end rigid mass are computed. Results are obtained for a range of center of mass offsets from the end of the beam. The results of current study corrects significant errors in the previous work by Rama Bhat and Wagner (1976).

Large deformation analysis of Euler-Bernoulli beamshell under own weight based on HAM

Abstract Large deformation of a horizontally uniform cantilevered Euler-Bernoulli beamshell subjected to its own weight is studied. The governing equation for the first time is analytically solved using the analytical technique for nonlinear problems Homotopy Analysis Method (HAM). A series solution is presented that can be used for the analysis of beamshells undergoing large deformations with wide range of weight, material/cross section properties and lengths. The results obtained by HAM are compared with the results reported in the previous works, and good agreement is found. Finally, the load-displacement characteristics of a uniform cantilever beamshell under different load conditions, dimensionless weight parameter are presented.

Optimization of Non-Uniform Optical Waveguide in MOEMS Accelerometer

Miniaturization is the rule of the day. This has even extended to optical domain where entire sensor arrays complete with laser or led source and pin photodiode are fabricated within a single chip. The device discussed and analyzed in this paper is a MOEMS accelerometer which consists of uniform micro cantilever on one arm of MZI structure. The refractive index is assumed to vary even along the direction of pulse propagation along with the frequently analyzed refractive index profile perpendicular to the light pulse path. The efficiency of device is increased by numerous optimization measures. A dual approach of optimal curved waveguide with increased difference is found to maximize power in waveguide core with minimal leakage to the cladding. This effectively translates to increased sensitivity as greater power is available for accurate detection process. The analysis is supported by appropriate simulations.

Large deflection of flexible tapered functionally graded beam

In this paper the semi-analytical analyses of the flexible cantilever tapered functionally graded beam under combined inclined end loading and intermediate loading are studied. In order to derive the fully non-linear equations governing the non-linear deformation, a curvilinear coordinate system is introduced. A general non-linear second order differential equation that governs the shape of a deflected beam is derived based on the geometric nonlinearities, infinitesimal local displacements and local rotation concepts with remarkable physical properties of functionally graded materials. The solutions obtained from semi-analytical methods are numerically compared with the existing elliptic integral solution for the case of a flexible uniform cantilever functionally graded beam. The effects of taper ratio, inclined end load angle and material property gradient on large deflection of the beam are evaluated. The Adomian decomposition method will be useful toward the design of tapered functionally graded compliant mechanisms driven by smart actuators.

Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: Analytical solutions

A mathematical model describing the process of large amplitude vibration of a uniform cantilever beam arising in the structural engineering is proposed. Six different analytical methods are applied to solve the dynamic model of the large amplitude non-linear oscillation equation. Periodic solutions are analytically verified, and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. Comparison of the present solutions is made with the exact solution and excellent agreement is noted.

Error in dynamic spring constant calibration of atomic force microscope probes due tononuniform cantilevers

Many common atomic force microscope (AFM) spring constant calibration methods regardthe AFM probe as a uniform cantilever, neglecting the tip mass and any nonuniformity inthe thickness of the probe along its length. This work quantifies the error in the springconstant estimated by the Sader and thermal calibration methods due to nonuniformity inthe thickness of the cantilever and the influence of the mass loading effect of the probe tip.Formulae are presented that can be used to compute the uncertainty in cantilevercalibration for an arbitrary thickness nonuniformity, or to correct the calibration methods ifthe thickness nonuniformity is known. The results show that both methods arequite sensitive to nonuniformity. When the first dynamic mode is used in thecalibration, the error in the spring constant estimated by either method is between − 4% and 9% for a cantilever whose thickness increases or decreases linearly by 30% along itslength. The errors are several times larger if the second or higher dynamic modes areused. To illustrate the proposed methods, a commercial AFM probe that hassignificant nonuniformity is considered and the error in calibrating this probe isquantified and discussed. For this particular probe, variations in the thickness ofthe probe over the last 15% of its length are found to significantly reduce theaccuracy of the calibration when the thermal method is used, since that method issensitive to changes in the shape of the eigenmode of the probe near its free end.

A novel experimental technique for determining node location in resonant mode cantilevers

We report on a new technique for measuring transverse mode node locations in cantilever sensors using a simple experimental arrangement. The technique relies on measuring resonant frequency response as liquid immersion depth is varied at a constant rate. When the air–liquid interface traverses a node, resonant frequency change decreases considerably and is discernable by a simple correlation analysis. The accuracy of the approach is verified using uniform cross-section cantilevers, whose node locations of the first three modes were measured and the values were within 5.1% of theory. For further testing of the technique, node locations of non-uniform cross-section cantilevers were measured, and the results compared favorably with three-dimensional (3D) finite element models were within 6.6%.

Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

This paper is concerned with analytical treatment of non-linear oscillations of planar, flexural large amplitude free vibrations of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. An analytic approximate technique, namely Optimal Homotopy Asymptotic Method (OHAM) is employed for this purpose. It is proved that OHAM provide accurate solutions for large amplitudes and large modal constants in the considered nonlinear equations, when other classical methods fail. Our procedure provides us with a convenient way to optimally control the convergence of solution, such that the accuracy is always guaranteed. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. Two examples are given and the results reveal that this procedure is very effective, simple and accurate. This paper demonstrates the general validity and the great potential of the OHAM for solving strongly nonlinear problems.

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh–Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys ©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.

On the uniqueness of large deflections of a uniform cantilever beam under a tip-concentrated rotational load

The problem of a uniform cantilever beam under a tip-concentrated load, which rotates in relation with the tip-rotation of the beam, is studied in this paper. The formulation of the problem results in non-linear ordinary differential equations amenable to numerical integration. A relation is obtained for the applied tip-concentrated load in terms of the tip-angle of the beam. When the tip-concentrated load acts always normal to the undeformed axis of the beam (the rotation parameter, β = 0 ) there is a possibility of obtaining non-unique solution for the applied load. This phenomenon is also observed for other rotation parameters less than unity. When the tip-concentrated load is acting normal to the deformed axis of the beam ( β = 1 ) , many load parameters are obtained for a tip-angle with different deformed configurations of the beam. However, each load parameter corresponds to a tip-angle, which confirms the uniqueness on the solution of non-linear differential equations.

О задаче устойчивости составного консольного стержня при одновремeнном действии “следящей” и неизменного направления сил, приложенных в пролёте.

О задаче устойчивости составного консольного стержня при одновремeнном действии “следящей” и неизменного направления сил, приложенных в пролёте.

Analytical solution for large deflections of a cantilever beam under nonconservative load based on homotopy analysis method

In this article, large deflection and rotation of a nonlinear beam subjected to a coplanar follower static loading is studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam remains unchanged during deformation. The governing equation of this problem is solved analytically for the first time using a new kind of analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). The present solution can be used in wide range of load and length for beams under large deformations. The results obtained from HAM are compared with those results obtained by fourth order Range Kutta method. Finally, the load-displacement characteristics of a uniform cantilever under a follower force normal to the deformed beam axis are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:541–553, 2011

Coupled lateral–torsional frequencies of asymmetric, three-dimensional structures comprising shear-wall and core assemblies with stepwise variable cross-section

A simple and accurate model for asymmetric, three-dimensional wall-core structures is developed that enables any desired natural frequency to be determined by a method which guarantees that no natural frequencies can be missed. The model assumes that the primary walls and cores run in two orthogonal directions and that their properties may vary in a stepwise fashion at one or more storey levels. A vectorial approach is used to generate the governing differential equations for coupled flexural-torsional motion that are finally incorporated into an exact dynamic stiffness matrix (exact finite element) that can model any uniform segment of the structure. A model of the original structure can then be assembled in the usual way. Since the mass of each segment is assumed to be uniformly distributed, it is necessary to solve a transcendental eigenvalue problem, which is accomplished using the Wittrick–Williams algorithm. When the structure can be represented realistically by a uniform cantilever, solutions can be found easily, by hand. A parametric study comprising five, three-dimensional, asymmetric wall-core structures is given to compare the accuracy of the current approach with that of a full finite element analysis.