The analytical solutions for orthotropic cantilever beams (Ⅰ):Subjected to surface forces

This paper first gives the general solution of two-dimensional orthotropic media expressed with two harmonic displacement functions by using the governing equations. Then, based on the general solution in the case of distinct eigenvalues, a series of beam problems, including the problem of cantilever beam under uniform loads, cantilever beam with axial load and bending moment at the free end, cantilever beam under the first, second, third and fourth power ofx tangential loads, is solved by the superposition principle and the trial-and-error methods.

The analytical solutions for orthotropic cantilever beams (II):Solutions for density functionally graded beams

In this paper, the specific solutions of orthotropic plane problems with body forces are derived. Then, based on the general solution in the case of distinct eigenvalues and the specific solution for density functionally graded orthotropic media, a series of beam problem, including the problems of cantilever beam with body forces depending only on z or on x coordinate and expressed by z or x polynomial is solved by the principle of superposition and the trial-and-error method.

BIPARAMETRIC PERTURBATION SOLUTIONS OF LARGE DEFLECTION PROBLEM OF CANTILEVER BEAMS

The large deflection problem of cantilever beams was studied by means of the biparametric perturbation method and the first order derivative substitution from pseudolinear analysis approach. This kind of substitution can transform the basic equation, an integral differential equation into nonlinear algebraic ones, thus simplify computational process. Compared with present results, it indicates that the large deflection problem solved by using pseudolinear analysis can lead to simple and precise results.

ELASTIC EFFECT IN DYNAMIC RESPONSE OF PLASTIC CANTILEVER BEAMS TO IMPACT

A lumped masses-springs model is proposed to analyze the dynamic response of an elastic-plastic cantilever beam resulting from impact. The numerical results are in good agreement with those by finite-element approaches. The simplified model can catch the most essential features of elastic-plastic response of beams; in particular, it demonstrates the effect of elastic deformation on the distribution of bending moment and energy dissipation, and provides valuable quatitative results.

Analytic Static Deflection Solutions of Uniform Cantilever Beams Resting on Nonlinear Elastic Rotational Boundary

In the paper,the analytic static deflection solutions of uniform cantilever beams resting on nonlinear elastic rotational boundary are developed by the Modified Adomian Decomposition Method(MADM).If the applied force function is an analytic function,then the deflection function can be derived and expressed in Maclaurin series.A recurrence relation for the coefficients of the Maclaurin series is derived.It is shown that the proposed solution method is accurate and efficient.The solution method can be successfully applied to the uniform cantilever beam and non-linear elastic rotational boundary problem.

Supercritical carbon dioxide process for releasing stuck cantilever beams

The multi-SCCO2 （supercritical carbon dioxide） release and dry process based on our specialized SCCO2 semiconductor process equipment is investigated and the releasing mechanism is discussed. The experiment results show that stuck cantilever beams were held up again under SCCO2 high pressure treatment and the repeatability of this process is nearly 100%.

The Effects of the Longitudinal Axis of Loading upon Bending, Shear and Torsion of a Thin-Walled Cantilever Channel Beam

Three aluminium channel sections of US standard extruded dimension are mounted as cantilevers with x-axis symmetry. The flexural bending and shear that arise with applied axial torsion are each considered theoretically and numerically in terms of two longitudinal axes of loading not coincident with the shear centre. In particular, the warping displacements, stiffness and stress distributions are calculated for torsion applied to longitudinal axes passing through the section’s centroid and its web centre. The stress conversions derived from each action are superimposed to reveal a net sectional stress distribution. Therein, the influence of the axis position upon the net axial and shear stress distributions is established compared to previous results for each beam when loading is referred to a flexural axis through the shear centre. Within the net stress analysis is, it is shown how the constraint to free warping presented by the end fixing modifies the axial stress. The latter can be identified with the action of a ‘bimoment’ upon each thin-walled section.

A GaAs-Based MOEMS Tunable RCE Photodetector with Single Cantilever Beam

A GaAs-based micro-opto-electro-mechanical-systems(MOEMS) tunable resonant cavity enhanced(RCE) photodetector with a continuous tuning range of 31nm under a 6V tuning voltage is demonstrated.The single cantilever beam structure is adopted for this MOEMS tunable RCE photodetector.The maximum and minimum peak quantum efficiency during the tuning are 36.9% and 30.8%,respectively.The maximum and minimum full-width-at-half-maximum (FWHM) are 20nm and 14nm,respectively.The dark current density is 7.46A/m2 without bias.

Longwall mining “cutting cantilever beam theory” and 110 mining method in China——The third mining science innovation

With the third innovation in science and technology worldwide, China has also experienced thismarvelous progress. Concerning the longwall mining in China, the ＂masonry beam theory＂ （MBT） wasfirst proposed in the 1960s, illustrating that the transmission and equilibrium method of overburdenpressure using reserved coal pillar in mined-out areas can be realized. This forms the so-called ＂121mining method＂, which lays a solid foundation for development of mining science and technology inChina. The ＂transfer rock beam theory＂ （TRBT） proposed in the 1980s gives a further understanding forthe transmission path of stope overburden pressure and pressure distribution in high-stress areas. In thisregard, the advanced 121 mining method was proposed with smaller coal pillar for excavation design,making significant contributions to improvement of the coal recovery rate in that era. In the 21st century,the traditional mining technologies faced great challenges and, under the theoretical developmentspioneered by Profs. Minggao Qian and Zhenqi Song, the ＂cutting cantilever beam theory＂ （CCBT） wasproposed in 2008. After that the 110 mining method is formulated subsequently, namely one stope face,after the first mining cycle, needs one advanced gateway excavation, while the other one is automaticallyformed during the last mining cycle without coal pillars left in the mining area. This method can beimplemented using the CCBT by incorporating the key technologies, including the directional presplittingroof cutting, constant resistance and large deformation （CRLD） bolt/anchor supporting systemwith negative Poisson＇s ratio （NPR） effect material, and remote real-time monitoring technology. TheCCBT and 110 mining method will provide the theoretical and technical basis for the development ofmining industry in China.

Suppression of multiple modal resonances of a cantilever beam by an impact damper

Impact dampers are usually used to suppress single mode resonance. The goal of this paper is to clarify the difference when the impact damper suppresses the resonances of different modes. A cantilever beam equipped with the impact damper is modeled. The elastic contact of the ball and the cantilever beam is described by using the Hertz contact model. The viscous damper between the ball and the cantilever beam is modeled to consume the vibrational energy of the cantilever beam. A piecewise ordinary differential-partial differential equation of the cantilever beam is established, including equations with and without the impact damper. The vibration responses of the cantilever beam with and without the impact damper are numerically calculated. The effects of the impact absorber parameters on the vibration reduction are examined. The results show that multiple resonance peaks of the cantilever beam can be effectively suppressed by the impact damper. Specifically, all resonance amplitudes can be reduced by a larger weight ball. Moreover, the impacting gap is very effective in suppressing the vibration of the cantilever beam. More importantly, there is an optimal impacting gap for each resonance mode of the cantilever beam, but the optimal gap for each mode is different.

Analytical solution for functionally graded anisotropic cantilever beam under thermal and uniformly distributed load

The bending problem of a functionally graded anisotropic cantilever beam subjected to thermal and uniformly dis-tributed load is investigated,with material parameters being arbitrary functions of the thickness coordinate. The heat conduction problem is treated as a 1D problem through the thickness. Based on the elementary formulations for plane stress problem,the stress function is assumed to be in the form of polynomial of the longitudinal coordinate variable,from which the stresses can be derived. The stress function is then determined completely with the compatibility equation and boundary conditions. A practical example is presented to show the application of the method.

Analysis of chaos behaviors of a bistable piezoelectric cantilever power generation system by the second-order Melnikov function

By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.

Four-Beam Model for Vibration Analysis of a Cantilever Beam with an Embedded Horizontal Crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton＇s principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.

ACTIVE VIBRATION CONTROL OF A CANTILEVER BEAM USING PIEZOELECTRIC SENSORS AND ACTUATORS

The topic of vibration control with distributed actuators has been the subject of many researches. This paper is concerned with the vibration control of a cantilever beam equipped with piezoelectric ceramics as sensors and actuators. One piezoelectric ceramic is bonded to the structure and provides control input for the structure, while the other piezoelectric ceramic provides the feedback signal. An approach to identification and control is presented. Observation spillover is eliminated by prefiltering the sensor data. A procedure used to determine actuator and sensor location, is discussed based on the modes to be controlled. Finally, the experimental results are presented to verify the proposed method.

Measurement of length-scale and solution of cantilever beam in couple stress elasto-plasticity

Owing to the absence of proper analytical solution of cantilever beams for couple stress/strain gradient elasto-plastic theory, experimental studies of the cantilever beam in the micro-scale are not suitable for the determination of material length-scale. Based on the couple stress elasto-plasticity, an analytical solution of thin cantilever beams is firstly presented, and the solution can be regarded as an extension of the elastic and rigid-plastic solutions of pure bending beam. A comparison with numerical results shows that the current analytical solution is reliable for the case of σ0 〈〈 H 〈〈 E, where σ0 is the initial yield strength, H is the hardening modulus and E is the elastic modulus. Fortunately, the above mentioned condition can be satisfied for many metal materials, and thus the solution can be used to determine the material length-scale of micro-structures in conjunction with the experiment of cantilever beams in the micro-scale.

Approximate analytical solutions and experimental analysis for transient response of constrained damping cantilever beam

Vibration mode of the constrained damping cantilever is built up according to the mode superposition of the elastic cantilever beam. The control equation of the constrained damping cantilever beam is then derived using Lagrange＇s equation. Dynamic response of the constrained damping cantilever beam is obtained according to the principle of virtual work, when the concentrated force is suddenly unloaded. Frequencies and transient response of a series of constrained damping cantilever beams are calculated and tested. Influence of parameters of the damping layer on the response time is analyzed. Analyitcal and experimental approaches are used for verification. The results show that the method is reliable.

Dynamic analysis of a deployable/retractable damped cantilever beam

Deployable/retractable damped cantilever beams are a class of time-varying parametric structures which have attracted considerable research interest due to their many potential applications in the intelligent robot field and aerospace.In the present work,the dynamic characteristics of a deployable/retractable damped cantilever beam are investigated experimentally and theoretically.The time-varying damping,as a function of the beam length,is obtained by both the enveloped fitting method and the period decrement method.Furthermore,the governing equation of the deployable/retractable damped cantilever beam is derived by introducing the time-varying damping parameter,and the corresponding closed-form solution and vibration principles are investigated based on the averaged method.The theoretical predictions for transient dynamic responses are in good agreement with the experimental results.The dynamic mechanism analysis on time-varying damping offers flexible technology in mechanical and aerospace fields.

CRACK IDENTIFICATION IN STEPPED CANTILEVER BEAM COMBINING WAVELET ANALYSIS WITH TRANSFORM MATRIX

This paper illustrates the crack identification method combining wavelet analysis with transform matrix. Firstly, the fundamental vibration mode was applied to wavelet analysis. The crack location was found by the peaks of the wavelet coefficients. Secondly, based on the identified crack locations, a simple transform matrix method requiring only the first two tested natural frequencies was used to further identify the crack depth. The present method can be used for crack identification in a complex structure. Numerical results of crack identification of a stepped cantilever beam show that the suggested method is feasible.

The theoretical analysis of dynamic response on cantilever beam of variable stiffness

The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable co- efficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last, the study obtains the solution of dy- namic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied, the dynamic characteristics are analyzed under several cases. The results dem- onstrate that the natural angular frequency reduces as the fiexural rigidity reduces. When the rigidity of clamped end is higher than that of free end, low-level mode contributes the larger displacement response to the total re- sponse. On the contrary, the contribution of low-level mode is lesser than that of hi^h-level mode.

Study on adhesively-bonded surface of tapered double cantilever specimen made of aluminum foam affected with shear force

Aluminum foam is widely used in diverse areas to minimize the weight and maximize the absorption of shock energy in lightweight structures and various bio-materials.It presents a number of advantages,such as low density,incombustibility,non-rigidity,excellent energy absorptivity,sound absorptivity and low heat conductivity.The aluminum foam with an air cell structure was placed under the TDCB Mode II tensile load by using Landmark equipment manufactured by MTS to examine the shear failure behavior.The angle of the tapered adhesively-bonded surfaces of specimens was designated as a variable,and three models were developed with the inclined angles differing from one another at 6°,8° and 10°.The specimens with the inclined angles of 6°,8° and 10° have the maximum reaction forces of 168 N,194 N when the forced displacements are 6,5 and 4.2 mm respectively.There are three specimens with the inclined angles of 10°,8° and 6° in the order of maximum reaction force.As the analysis result,the maximum equivalent stresses of 0.813 MPa and 0.895 MPa happened when the forced displacements of 6 mm and 5 mm proceeded at the models of 6° and 8°,respectively.A simulation was carried out on the basis of finite element method and the experimental design.The results of the experiment and the simulation analysis are shown not different from each other significantly.Thus,only a simulation could be confirmed to be performed in substitution of an experiment,which is costly and time-consuming in order to determine the shearing properties of materials made of aluminum foam with artificial data.