Dynamic stability of a cantilevered Timoshenko beam on partial elastic foundations subjected to a follower force

This paper presents the dynamic stability of a cantilevered Timoshenko beam with a concentrated mass, partially attached to elastic foundations, and subjected to a follower force. Governing equations are derived from the extended Hamilton’s principle, and FEM is applied to solve the discretized equation. The influence of some parameters such as the elastic foundation parameter, the positions of partial elastic foundations, shear deformations, the rotary inertia of the beam, and the mass and the rotary inertia of the concentrated mass on the critical flutter load is investigated. Finally, the optimal attachment ratio of partial elastic foundation that maximizes the critical flutter load is presented.     

Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads

A finite-length tensioned beam on a damped elastic foundation is acted upon by an infinite series of equally spaced and steadily moving concentrated transverse loads. The deflection response of the beam is obtained by an expansion in terms of the normal modes of vibration. Numerical results are determined for various values of the load-spacing, beam tension, foundation stiffness and damping, and for a range of load-speeds. It is found that the critical velocities for repetitive loading exist at significantly lower speeds than would be expected based upon the well-known critical speed for a single moving load. An interpretation in terms of forced vibration response is given.     

Strain rate self-sensing for a cantilevered piezoelectric beam

This paper deals with the analytical modeling, and the experimental verification of the strain rate self-sensing method using a hybrid adaptive filter for a cantilevered piezoelectric beam. The piezoelectric beam consists of two laminated lead zirconium titanates (PZT) on a metal shim. A mathematical model of the beam dynamics is derived by Hamilton’s principle and the accuracy of the modeling is verified through the comparison with experimental results. For the strain rate estimation of the cantilevered piezoelectric beam, a self-sensing mechanism using a hybrid adaptive filter is considered. The discrete parts of this mechanism are realized by the DS1103 DSP board manufactured by dSPACE^TM. The efficacy of this method is investigated through the comparison of experimental results with the predictions from the derived analytical model.     

Dynamic strain response of an infinite beam and inverse calculation of impact force by numerical Laplace transform

A general method for determining the dynamic strain response of an infinite beam to impact force is presented. The method consists of formulating and solving the dynamic problem in the Laplace transform domain and obtaining the strain response by numerical inversion of the transformed solution. Also, an inverse method for estimating the impact force on an infinite beam is investigated. Once the strain is known, it is shown how the impact force can be reconstructed.     

Dynamic analysis of a spinning Rayleigh beam

The whirl speed, critical speed and mode shape of a spinning beam in six general boundary conditions are investigated analytically in this paper. The beam is in Rayleigh model with rotatory inertia and gyroscopic effects. It is shown that the whirl speeds and critical speeds can be expressed analytically by a function of the slenderness ratio (l) defined by the beam length over its radius. Contrary to common belief, only finite number of critical speeds can be found in all speed ranges. The number is functional of l, but independent of the boundary conditions. The system's unbalanced response can therefore be expressed analytically by these finite precessional modes and the corresponding generalized coordinates.     

Effect of shear flexibility in a beam on an elastic foundation

The basic equation for a beam on an elastic foundation is derived, including the effect of shear flexibility in the beam. The general solution is obtained and some particular cases examined. The practical importance of shear flexibility is examined for a ship on a slipway.     

Experimental and theoretical investigation of the characteristics of an electric response to an elastic impact excitation of piezo-containing heterogeneous materials

Experimental investigations were performed on the influence of flaws on the characteristics of an electric response of gypsum-sand specimens under impact excitation. A physico-mathematical model of an electric response to an elastic impact excitation of piezo-containing heterogeneous materials was developed. Using the apparatus of continuum mechanics, the parameters of an electric signal from a dielectric specimen exposed to a pulsed action were theoretically calculated and agreement with the experimental results was observed.     

Dynamic response of a beam carrying a lumped mass along its span

In this paper, we study the periodic solutions for the free vibration of a conservative oscillator with fifth-order nonlinearity using two new methods, the energy balance method and the variational approach. We compare the results obtained with those yielded by the traditional harmonic balance method and equivalent linearization–nonlinearization method. The periodic solutions are then verified using the Runge–Kutta approach.     

Dynamic response analysis of block foundations with nonlinear dry friction mounting system to impact loads

It is essential to establish a dynamic model to predict and evaluate the dynamic performance of a nonlinear dry friction mounting system during design procedure, when it is impossible to carry out the test of prototype. Unlike the conventional ideal dry friction model where the direction of dry friction force is always considered to be opposite to that of relative velocity, a new equivalent resistance model of dry friction force is proposed based on the bilinear hysteretic model by introducing a parameterg γ in this work. The equivalent resistance contains spring force and damping force, whose direction is not opposite to that of relative velocity. Then, a dynamic model of the block foundation with nonlinear dry friction mounting system is established. When the equivalent resistance is applied to the dynamic model, its dynamic responses are obtained under common practical forms of press loads: rectangular pulse, half-sine pulse, and triangular pulse. Compared to experimental results, the dynamic responses based on the equivalent resistance model are more consistent with the simulation results based on the ideal dry friction model and the validity of the equivalent resistance model for the bilinear hysteretic model in this work is verified. Furthermore, the effect of the pulse shape and pulse duration on the dynamic responses of the block foundation with nonlinear dry friction mounting system is investigated.     

Dynamic analysis of an axially moving beam subject to inner pressure using finite element method

A dynamic model of an axially moving flexible beam subject to an inner pressure is present. The coupling principle between a flexible beam and inner pressure is analyzed first, and the potential energy of the inner pressure due to the beam bending is derived using the principle of virtual work. A 1D hollow beam element contain inner pressure is established. The finite element method and Lagrange’s equation are used to derive the motion equations of the axially moving system. The dynamic responses are analyzed by Newmark-β time integration method. Based on the computed dynamic responses, the effects of inner pressure on beam dynamics are discussed. Some interesting phenomenon is observed.     

Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass

The elastodynamic response of a rectangular Mindlin plate subjected to a distributed moving mass is investigated. The set of governing characteristic partial differential equations that include the effects of shear deformation and rotary inertia is expressed in its dimensionless form. A finite difference algorithm is employed to transform the differential equations into a set of linear algebraic equations. Simply supported edge conditions were used as an illustrative example. The analysis is also valid for other edge conditions. It is found that the maximum shearing forces, bending and twisting moments occur almost the same time. Also, the values of the maximum deflections are higher for Mindlin plates than for non-Mindlin plates.     

Dynamic analysis of pseudoelastic SMA beam

The main goal of this work is to present a formulation of initial-boundary-value problem for the Bernoulli-Euler beam made of pseudoelastic shape memory alloy (SMA). The procedure of formulation of 1D constitutive relations based on the analysis of the proposed rheological model is presented in detail. The relationships to be obtained are of explicit type and were formulated within the notion of non-smooth mechanics using the so-called differential successions of constitutive equations. The system of partial differential equations is discretized with respect to spatial coordinates using the finite difference method. This procedure leads to the system of ordinary differential equations with respect to the time coordinate, which was solved using the Runge-Kutta method. The problem was coded within MATLAB system. A numerical example of a beam structure subjected to concentrated Heaviside-type loading was analysed.     

The elastic stability of a thin cantilever beam under an articulated tip force—I: Statics analysis

Previous work on the buckling of thin elastic cantilever beams when subjected to tip forces applied by means of an articulated rod is here extended to cases when the axis of the rod is directed at any angle to the beam axis. The system is typical of engineering situations in which mechanical control circuits involve a bellcrank lever connected to an articulated rod, the remote end of which is loaded by a tube directionally guided in rollers. The function of the articulated rod is to permit geometric changes to take place in the mechanical system.A numerical solution is provided to the differential equation system by finite-difference approximations. The resulting equations are of a sufficiently simple form as to permit evaluation of buckling forces by desk calculation; alternatively a computer adaptation can be made.The results of the analysis are verified by an experimental investigation.     

Effects of elastic energy of thin films on bending of a cantilevered magnetostrictive film-substrate system

In this paper, effects of elastic energy of magnetostrictive film on the deflection of a cantilevered film-substrate system are investigated. The total energy including the elastic energy of magnetostrictive film is formulated. And it is minimized to give the curvatures and the position of neutral axis of the cantilevered system. To discuss the effects of the elastic energy of film in a measured system, three magnetostrictive unimorph cantilevers and a bimorph cantilever reported elsewhere are reviewed. It is shown that the assumption, since the thickness of film is much smaller than that of substrate the film elastic energy is negligible, can cause considerable error in evaluating magnetostrictive coefficients. Not the ratio of thicknesses but elastic energies between film and substrate is also shown to play important role in making decision whether the assumption is valid or not.     

The plastic extension of a chain of rings due to an axial impact load

The plastic response of a chain of circular rings due to an axial tensile impact load was investigated both experimentally and in part, analytically. Chains were built-up from circular aluminium rings and subjected to axial impact loading at one end. High-speed photography was employed to record the development of the deformation process. It was found that plastic collapse was progressively transmitted to neighbouring rings in the manner of a plastic wave. To investigate the situation analytically, a simplified approach was adopted which assumed rigid-perfectly plastic behaviour of the ring material and arrived at an equation of the same form as the classical one-dimensional elastic wave equation.     

Detachment of an elastic beam from a viscoelastic support: A variational approach

The paper is devoted to the analysis of failure in multi-layer composite materials. For the experimental study of the failure mechanism, the peel tests are used. The peel test is modeled as the detachment of an elastic beam attached to a rigid adherend by a viscoelastic adhesive layer.For solving this problem, we use the principle of minimum free energy and derive a new energy criterion of the detachment. We propose a model of an aging viscoelastic material describing the viscoelastic behavior in terms of elastic objects. Using this model we propose an expression for the specific potential energy of an aging viscoelastic medium and derive a new formulation of the principle of minimum free energy for a viscoelastic system.A numerical method for solving nonlinear integro-differential equations describing the detachment process is developed, and its convergence is established. We analyze numerically the dependence of the detachment on the construction and material parameters, and on the loading history. We demonstrate the existence of a limited training effect, which allows considerable reduction of the detachment zone by a special preliminary loading program (“training”).     

The elastic stability of a thin cantilever beam under an articulated tip force—II: Kinetic analysis

Previous work on the buckling of thin elastic cantilever beams when subjected to tip forces applied by means of an articulated rod is here extended to cases when the axis of the rod is directed at any angle to the beam axis. The system is typical of engineering situations in which mechanical control circuits involve a bellcrank lever connected to an articulated rod, the remote end of which is loaded by a tube directionally guided in rollers. The function of the articulated rod is to permit geometric changes to take place in the mechanical system.A numerical solution is provided to the differential equation system by finite-difference approximations. The resulting equations are of a sufficiently simple form as to permit evaluation of buckling forces by desk calculation; alternatively a computer adaptation can be made.The results of the analysis are verified by an experimental investigation.     

Spectral analysis of dynamic response of a thin beam subjected to a varying speed moving mass

A parametric survey was conducted to capture the dynamic response of a thin beam subjected to a varying speed moving mass. The existing literature lacks a comprehensive study on the beam dynamic behavior under a varying speed moving mass with arbitrary constant acceleration and mass ratio. The current work represents midpoint response spectra for a thin beam acted upon by a varying speed moving mass for a wide range of design parameters. Findings show that for a given mass ratio, higher response amplitudes are observed in decelerating motion compared to accelerating one. Moreover, increasing the mass ratio of the moving mass generally leads to higher beam dynamic response. Among the methods that can be utilized to calculate beam response, the Eigenfunction expansion method (EFM) and Orthonormal polynomial series expansion method (OPSEM) were used. Then an improvement technique was applied on both aforementioned methods and computational efficiency and convergence rate of all utilized methods was compared.     

Dynamic behaviors of an elastically restrained beam carrying a moving mass

Dynamic responses of a simply supported beam with a translational spring carrying a moving mass are studied. Governing equations of motion including all the inertia effects of a moving mass are derived by employing the Galerkin’s mode summation method, and solved by using the Runge-Kutta integral method. Numerical solutions for dynamic responses of a beam are obtained for various cases by changing parameters of the spring stiffness, the spring position, the mass ratio and the velocity ratio of a moving mass. Some experiments are conducted to verify the numerical results obtained. Experimental results for the dynamic responses of the test beam have a good agreement with numerical ones.     

On the equivalent mass-spring parameters and assumed mode of a cantilevered beam with a tip mass

In the Rayleigh-Ritz approach to the mathematical model of a cantilevered beam with a tip mass, the proper selection of basis functions is critical in representing the original system by an equivalent mass-spring system. Although the fundamental bending mode shape of a beam varies for a different tip mass magnitude, the numerical values of 33/140 and 3 have been conventionally employed as those of the normalized dimensionless equivalent mass and spring constants, respectively, which correspondingly yield errors in its calculated natural frequencies. This work firstly proposes a method to evaluate more accurate values of the equivalent mass and spring for a wide range of the tip mass-to-beam mass ratio by direct use of a fundamental mode, and then proposes a new basis function as a linear combination of two polynomials, which represent static deflection shapes of a beam under a tip force and a uniformly distributed force, respectively, yielding natural frequencies fairly close to those by the continuous beam equation.