VIBRATION THEORY OF CONTINUOUS BEAM UNDER THE ACTION OF MOVING LOAD

This paper uses the small paramter method to investigate the dynamic calculation ofthe whole vibration process of trains passing through a continuous beam,considering theeffects of the mass and the damping as well as the masses of the moving loads.By solving aset of integral equation,we find out the general solution of continuous beam under theaction of arbitrary moving load PF(t) and calculate,the case of single moving load beingQ_i P_isin(α_it ε_i). By concluding our results,We establish the dynamic theory ofvibration of continuous beam acted by the moving load.Finally, as an example, we calculate the vibration question of two-span continuousbeam.The deflections of two midspan are shown in Fig.2 and Fig.3.     

Transverse vibrations of a single-span beam with the motion of a bending moment

Conclusions Analysis of the dynamic action of a moving bending moment on a single-span beam-type system showed that, with v=(0.2–0.8v ₁ ⁰ , taking account of the inertial forces of the load does not enter into the margin of strength of the construction, and these forces must be taken into consideration in dynamic calculations. The greatest deflections of the beam, when the mass of the load M=0.5ml, exceed the static deformations, taking account of the inertial forces of the load, by 2.5 times. The value of the velocity here is v=0.6v ₁ ⁰ .The maximal coefficient of the dynamics, calculated without taking account of the weight of the load, is equal to 1.95 and occurs with v=0.8v ₁ ⁰ . We note that, with the motion of a vertical force along the beam, the maximal value of the dynamic coefficient is equal to 1.77 and is observed with v=0.6v ₁ ⁰ [3].If v<0.6v ₁ ⁰ , where the mass of the load is not introduced into the calculations, and v<0.4v ₁ ⁰ , where account is taken of the inertial forces of the load, then the maximal deformations of the beam take place during the process of forced vibrations at the moment that the load is located in the construction. With large values of v, the greatest deflections are observed after passage of the load, during the period of free vibrations of the system.In distinction from the solution of the problem of the vibrations of a beam under the action of a moving force (load), where a sufficient degree of exactness of the computations assures taking account of the first form of the vibrations of the construction, with an analysis of dynamic deformations of a beam, brought about by the action of a moving bending moment, the higher forms of the vibrations of the system must be taken into consideration.Leningrad Institute of Railroad Engineers. Translated from Prikladnaya Mekhanika, Vol. 14, No. 1, pp. 111–115, January, 1978.     

Dual extremum principles relating to optimum beam design

Of concern is a cantilever beam resting on an elastic foundation and supporting a load at the free end. The beam is of rectangular cross section and of constant height but variable width. It is required to taper the beam for maximum strength. This means that the beam is to support a maximum vertical load W at the free end when the free end is given unit deflection. The constraint is that the weight of the beam should not exceed a given bound K. It is shown that the optimum taper should be so chosen that the curvature of the beam is constant. This yields the solution of the problem in terms of explicit formulas. For more general constraints, an inequality is found which gives upper and lower bounds for the maximum load W even though explicit formulas are not available.This paper was prepared under Research Grant DA-ARO-D-31-124-71-G17, U.S. Army Research Office (Durham).     

Investigation of two oscillation types in a beam generated plasma

High frequency oscillations are observed in a beam generated plasma subjected to an axial uniform magnetic field. The oscillation frequency is measured as a function of the cyclotron frequency. The system is in a condition just below total glow.The oscillations are classified by comparing the experimental results with theoretical analyses of beam plasma systems in literature. Two types of oscillation are found. They result from (A) the interaction between the forward plasma wave and the slow space charge wave of the beam and (B) the interaction between the backward plasma wave and the slow space charge wave of the beam. The oscillation frequency of type A is practically independent of the cyclotron frequency, whereas the frequency of oscillation B increases linearly with it. For type A oscillations to occur a minimum value of the cyclotron frequency is required.Nomenclature phase constant - B ₀ static axial magnetic induction - e electron charge - f _c cyclotron frequency - f _cc critical cyclotron frequency - f _h hybrid frequency - f _osc oscillation frequency - f _p plasma frequency - I _b beam current - I _c collector current - m electron mass - N _e electron density per unit beam length - _b beam charge density - _p plasma charge density - r _p plasma radius - S beam pervenance - V _ak anode-cathode potential - V _b beam voltage - Units SI-units are employed     

Effect of gravity on nonlinear oscillations of a horizontal,immovable-end beam

A critical problem in designing large structures for space applications, such as space stations and parabolic antennas, is the limitation of testing these structures and their substructures on earth. These structures will exhibit very high flexibilities due to the small loads expected to be encountered in orbit. It has been reported in the literature that the gravitational sag effect under dead weight is of extreme importance during ground tests of space-station structural components [1–4]. An investigation of a horizontal, pinned-pinned beam with complete axial restraint and undergoing large-amplitude oscillations about the statically deflected position is presented here. This paper presents a solution for the frequency-amplitude relationship of the nonlinear free oscillations of a horizontal, immovable-end beam under the influence of gravity.The governing equation of motion used for the analysis is the Bernoulli-Euler type modified to include the effects of mid-plane stretching and gravity. Boundary conditions are simply supported such that at both ends there is no bending moment and no transverse and axial displacements. These boundary conditions give rise to an initial tension in the statically deflected shape. The displacement function consists of an assumed space mode using a simple sine function and unknown amplitude which is a function of time. This assumption provides for satisfaction of the boundary conditions and leads to an ordinary differential equation which is nonlinear, containing both quadratic and cubic functions of the amplitude. The perturbation method of multiple scales is used to provide an approximate solution for the fundamental frequency-amplitude relationship.Since the beam is initially deflected the small-amplitude fundamental natural frequency always increases relative to the free vibration situation provided in zero gravity. The nonlinear equation provides for interactions between frequency and amplitude in that both hardening and softening effects arise. The coefficient of the quadratic term in the nonlinear equation arises from the static (dead load) portion of the deflection. This quadratic term, depending upon its magnitude, introduces a softening effect that overcomes the hardening term (due to initial axial tension developed by deflection) for large slenderness ratios.For very large slender, immovable-end beams, the fundamental natural frequency is greater than that of beams without axial constraints undergoing small amplitude oscillations. This phenomenon is attributed to the stiffening effect of the statically-induced axial tension. However, the stiffening effect of axial tension in beams with slenderness ratios greater than approximately 392 undergoing large-amplitude symmetric-mode oscillations is overpowered by the presence of gravitational loading.Nomenclature A amplitude of the first harmonic - A ₁ cross-sectional area of beam - a(t) vibratory amplitude - E Young's modulus - g acceleration due to gravity - g ₁,g ₂,g ₃ constants defined in equations (8) - I area moment of inertia of cross-section - L length of beam - N axial tension force induced by gravitational loading     

Three-dimensional effects in beams: Isodyne assessment of a plane solution

The stress distribution in a homogeneous beam subjected to three-point bending is investigated using the method of optical isodynes. The three stress components _xx,_yy and _xy acting in the planes formed by the longitudinal and vertical axes of the beam are determined in three planes situated at different through the thickness locations with respect to the beam's midplane. The experimental results are subsequently correlated with the two-dimensional elasticity solution. It is illustrated that at locations sufficiently removed from the centrally applied concentrated load, good correlation between theory and experiment is obtained. In the regions where high stress gradients exist however, differences are observed in the in-plane stress distributions in the different planes. These differences are explained by the presence of the out of plane normal stress _zz using the relations of optical isodynes. Greatest differences between theory and experiment are obtained for the in-plane shear stress component _xy.     

A photoelastic study of friction at multipoint contacts

A new method of measuring the normal and sliding loads associated with multiple-point contact is introduced. A multiple-point contact is modeled with a steel die with a profile that simulates a rough surface. A very large scale factor is used in modeling this surface. The steel die is placed in contact with a photoelastic model of a half plane and is subjected to a normal load. This normal load is partitioned over the multiple points of contact producing an isochromatic fringe pattern that describes the stress distribution in the local neighborhood of the contact points. A sliding load is then imposed on the model which destroys the symmetry of this fringe pattern. The fringe data in this pattern are sufficient to determine the local loadsP _i andQ _i and the local coefficient of frictionf _i =Q _i /P _i at each contact point. An overdeterministic method is introduced which gives the solution forP _i ,Q _i andf _i using many data points taken from the isochromatic pattern in the local neighborhood of the contacts.Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WJ on June 9–13.     

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.     

Structural optimization with parameter-transfer finite element

A mathematical method to solve structural problems, using parameter-transfer finite elements (P-TFE) was recently proposed by the authors [1] [2] [3]. The proposed transfer finite element approach is able to create a mathematical model of a structure, taking into account directly the whole behaviour of the structure under dynamic, aerodynamic, and thermal actions, and not by assembling, in a separate fashion, the stiffness and the mass matrix on one side and the external load vector as performed by the classical finite element procedure.The purpose of this paper is to apply the above methodology to optimization problems, in particular to obtain the minimum structural weight for a beam, under primary constraints on buckling load or natural frequencies.The use of P-TFE in the field of structural optimization overcomes most difficulties of the usual techniques of solution and the element is particularly useful in the evaluation of the sensitivity matrix.The formulation of the optimization problem based on P-TFE is presented and some applications are studied. The numerical results obtained are compared with other existing methodologies and briefly discussed.

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Sommario Gli autori hanno già proposto un metodo per studiare problemi strutturali [1] [2] [3], introducendo una nuova metodologia di discretizzazione, basata sull'impiego di elementi finiti di trasferimento, funzioni esplicite di un parametro, indicati come P-TFE. Tali elementi sono in grado di rappresentare, in similitudine alla funzione di trasferimento, il comportamento completo dell'elemento strutturale in esame, soggetto ad azioni dinamiche, aerodinamiche e termiche; sono parimenti in grado di produrre, in similitudine al metodo degli elementi finiti, un modello matematico discreto di un continuo.Scopo del presente lavoro è di applicare detta metodologia a problemi di ottimizzazione, in particolare alla ricerca del minimo peso per una trave che mantenga inalterate le sue caratteristiche di carico critico o le frequenze naturali di vibrazione.Vengono quindi presentati alcuni risultati numerici dei casi esaminati e confrontati con quelli ottenuti da altri autori con l'impiego di altre metodologie.

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List of Symbols {B} _m vector of the generalized state variables - {C} _m vector of integration constants - [I] unit matrix - EI bending stiffness - A cross-sectional area - u adimensional thickness - l beam length - M,M bending moment - [N] _m shape function ofm-th order - [N*] shape function atx ₀ - P axial load - [R] _i transfer matrix of thei-th element - T,T shear force - w transverse displacement - x adimensional independent variable - x ₀ value ofx at the left of the element - {Y} vector of state variables - {Y*} imposed condition atx ₀ - _0m Kronecker delta with the first pedix always set equal to zero - normalized eigenfrequency - normalized buckling load - mass density     

纳米颗粒增强镍基MEMS器件材料的蠕变性能研究

利用同步辐射LIGA微铸复合工艺,将纳米氧化物增强颗粒复合到微电子机械系统(MEMS)结构材料中。制作了专用夹具,采用微力材料试验机测量了纳米Al2O3颗粒增强镍基复合材料的强度为1GPa;将恒加载速率/载荷法和恒载荷法相结合,利用纳米压痕仪测量了该材料的室温蠕变速率敏感指数m。结果表明,LIGA复合技术得到的纳米颗粒增强镍基复合材料具有较高的强度;MEMS器件材料在室温下会发生蠕变;材料在相同压深下最大载荷不随加载速率而改变,加载段粘弹性和粘塑性变形极少;主要由局部高应力导致压痕蠕变;材料的蠕变速率敏感指数m值为0.004,说明纳米Al2O3颗粒可有效增强基体材料的抗蠕变能力;且不同恒.P/P下获得的m值基本相同,表示此种材料对加载速率不敏感。     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

Influence of Horizontal Excitations on Dynamic Stability of a Slender Beam Under Vertical Excitation

Experimental studies have been conducted to clarify the influence of horizontal harmonic excitations on the dynamic stability of a slender cantilever beam under vertical harmonic excitation. Three kinds of aluminum test beams with rectangular cross section have been used. The test beam being clamped at one end and free at the other end, was vertically stood, and was harmonically excited to both vertical and horizontal directions simultaneously. The direction of the horizontal excitation was taken parallel to one of the beam side faces, i.e. two directions were considered as X and Y directions which have the largest and smallest flexural rigidity, respectively. By varying the horizontal excitation amplitude, keeping the amplitude of excitation in the vertical direction, the influence of the horizontal excitation has been investigated on the principal instability regions in which unstable vibration of the fundamental vibration mode occurs. The excitation frequency in the vertical excitation was taken around twice the fundamental natural frequency 2f _{Y 1} in smallest rigidity direction, while that in the horizontal direction was taken around both the fundamental natural frequency f _{Y 1} and twice of it 2f _{Y 1}. Obtained experimental results present useful fundamental data for aseismatic design of structures under earthquake containing both vertical and horizontal excitation components.     

Thermoelastic investigations for fatigue life assessment

An investigation is presented on the suitability and accuracy of a thermoelastic technique for the analysis of fatigue cracks. The stress intensity factor ranges ΔK _I and ΔK _II are determined from thermoelastic data recorded from around the tip of a sharp slot in a steel specimen under biaxial load, in order to assess the accuracy of the technique. ΔK _I and ΔK _II are determined to within 4% and 9% of a theoretical prediction, respectively. The results from a similar test on a fatigue crack under biaxial load are also presented. These show that thermoelastic stress analysis is a rapid and accurate way of analyzing mixed-mode fatigue cracks. A discussion is given on the potential of thermoelastic stress analysis of propagating cracks.     

Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m^′ are more important for the high-frequency compliance. An important parameter is the elastic length l=(EI/G)^1/4 of the beam–soil system. Around the corresponding frequency ω_l=v_S/l, the wave velocity of the combined beam–soil system changes from the Rayleigh wave v_R≈v_S to the bending wave velocity v_B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω_l is found not far from the characteristic frequency ω₀=(G/m^′)^1/2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u/P(ρv_Paiω)^−3/4 in case of the dominating damping and u/P(−m^′ω²)^−3/4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u/P1/Gl, but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u/P₀=f(a/l). The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.     

A Simplified Model for Calculating Hydrodynamic Lubrication Film Thickness in Elastoplastic Line Contacts

The present paper proposes a simplified model for calculating hydrodynamic lubrication film thickness in elastoplastic line contacts. According to the Saint-Venant’s principle, the pressure in the contact is taken as uniformly distributed, this gives the contact surface elastic deformations in the inlet zone far away from the contact center close to real ones while gives those close to the contact center greater than real ones. This treatment is validated for hydrodynamic lubricated elastic contacts for relatively light loads and high rolling speeds. It gives the film thickness at the contact center a little higher than that calculated based on the real elastic model. The treatment is extended to a hydrodynamic lubricated elastoplastic line contact. The contact surfaces in the inlet zone are assumed as elastic and their deformations are calculated based on the uniform pressure distribution in the elastoplastic contact area. An inlet zone analysis is taken for obtaining the calculating equation of the hydrodynamic film thickness at the contact center. The equation overestimates the central film thickness but gives a satisfactory film thickness prediction for the heavy load which gives significant plastic deformations in the elastoplastic contact. It is found that when the load is lighter than 0.6 w _pc , the contact can be taken as elastic when calculating the central film thickness, while when the load is heavier than 0.6 w _pc , the contact can be taken as fully plastic; Here w _pc is the critical load for the contact fully plastic deformation. The plastic deformation in an elastoplastic line contact is found to reduce the hydrodynamic lubrication film thickness in the contact. This reduction is greater for higher rolling speeds and heavier loads. However, it is significantly dropped with increasing surface hardness.     

Observations on Mode I and III fracture of steel and PMMA

Considered in this work are the Mode I and III fractures of W18Cr4V steel, 60Si2Mn steel and PMMA specimens. The mixed mode critical stress intensity factor, denoted by K_1f, is shown to be greater than K_1c for Mode I. The ratio K_1f/K_1c depends on the ways in which K₁ and K₃ interact and is affected by the position of the load with reference to the crack position. Analytical and experimental results are presented and discussed in connection with the microfracture surface observed experimentally.     

Closed-form solution for a mode-III interfacial edge crack between two bonded dissimilar elastic strips

A closed-form solution is obtained for the problem of a mode-III interfacial edge crack between two bonded semi-infinite dissimilar elastic strips. A general out-of-plane displacement potential for the crack interacting with a screw dislocation or a line force is constructed using conformal mapping technique and existing dislocation solutions. Based on this displacement potential, the stress intensity factor (SIF, K_III) and the energy release rate (ERR, G_III) for the interfacial edge crack are obtained explicitly. It is shown that, in the limiting special cases, the obtained results coincide with the results available in the literature. The present solution can be used as the Green’s function to analyze interfacial edge cracks subjected to arbitrary anti-plane loadings. As an example, a formula is derived correcting the beam theory used in evaluation of SIF (K_III) and ERR (G_III) of bimaterials in the double cantilever beam (DCB) test configuration.     

Transverse vibration of a multiple-Timoshenko beam system with intermediate elastic connections due to a moving load

Based on Timoshenko beam theory, the dynamic response of an elastically connected multiple-beam system is investigated. The identical prismatic beams are assumed to be parallel and connected by a finite number of springs. Assuming n parallel Timoshenko beams, the motion of the system is described by a coupled set of 2n partial differential equations. The method involves a change of variables and modal analysis to decouple and to solve the governing differential equations, respectively. A case study is solved in detail to demonstrate the methodology and several plots of the midpoint deflections of beams are given and investigated for different values of moving load velocity and the stiffness of elastic connections. From the numerical results it is observed that the maximum deflection of the multiple Timoshenko beam system is always smaller than one of a single beam.