Vibration and stability of a non-uniform Timoshenko beam subjected to a follower force

An analysis is presented for the vibration and stability of a non-uniform Timoshenko beam subjected to a tangential follower force distributed over the center line by use of the transfer matrix approach. For this purpose, the governing equations of a beam are written in a coupled set of first-order differential equations by using the transfer matrix of the beam. Once the matrix has been determined by numerical integration of the equations, the eigenvalues of vibration and the critical flutter loads are obtained. The method is applied to beams with linearly, parabolically and exponentially varying depths, subjected to a concentrated, uniformly distributed or linearly distributed follower force, and the natural frequencies and flutter loads are calculated numerically, from which the effects of the varying cross-section, slenderness ration, follower force and the stiffness of the supports on them are studied.     

On the vibration and stability of a free-free beam subjected to end-loads

In this paper the vibration and stability of a free-free beam subjected to direction-controlled axial loads at its ends are investigated. The eigencurves and mode shapes of the beam are presented for various values of the directional control parameter. It is found that the behaviour of the free-free beam subjected to compressive axial loads is unstable for any direction parameter—except for the follower loading case. However, the same beam subjected to tensile loads is stable.     

On prandtl's cantilever beam subjected to a bending moment

The stability behaviour of a cantilever beam subjected to the bending moment is investigated. It is found that the beam has divergence and flutter instability loads depending on the type of the loading. Moreover, it is shown that a beam subjected to a follower moment and a beam subjected to a bending moment which keeps its direction in the course of the motion behave in exactly the same way, according to the numerical calculations.     

Combination resonance of beams under traveling follower load systems

The regions of simple parametric and combination resonances of a thin walled beam under a sequence of equidistant follower loads moving at constant speed are estimated by using the stability criterion with the characteristic exponent, and the effects on the combination resonance of load mass, speed and frequency of load are examined.     

The vibration characteristics of a beam with an axial force

Equations of motion are found for a non-uniform damped Timoshenko beam with a distributed axial force. Principal modes may be extracted by numerical means when the boundary conditions are specified, and the appropriate orthogonality conditions are given. The theory of linear forced vibration can thus be derived. It is an implicit requirement that all axial forces are conservative. That is to say, tangential, follower and partial follower axial forces (whether applied at an extremity or distributed along the beam) are excluded.     

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.     

Dynamic stability of Timoshenko beams resting on an elastic foundation

A finite element model is developed for the stability analysis of a Timoshenko beam resting on an elastic foundation and subjected to periodic axial loads. The effect of an elastic foundation on the natural frequencies and static buckling loads of hinged-hinged and fixed-free Timoshenko beams is investigated. The results obtained for a Bernoulli-Euler beam which is a special case of the present analysis show excellent agreement with the available results obtained by other analytical methods. The regions of dynamic instability are determined for different values of the elastic foundation constant. As the elastic foundation constant increases the regions of dynamic instability are shifted away from the vertical axis and the width of these regions is decreased, thus making the beam less sensitive to periodic forces.     

Stability of a pretwisted tapered cantilever beam subjected to dissipative and follower forces

Stability of a pretwisted tapered cantilever beam of rectangular cross-section subjected to a follower force at its free end is investigated. The effects of internal and external damping are included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli theory and an associated adjoint boundary value problem is introduced. Approximate values of the critical load are calculated on the basis of a suitable adjoint variational principle for several values of the geometric and material parameters of the beam. The results are shown in graphs.     

A CONCENTRATED MASS ON THE SPINNING UNCONSTRAINED BEAM SUBJECTED TO A THRUST

The dynamic stability of a spinning unconstrained beam subjected to a pulsating follower forceP₀ +P₁cos Ωt is analyzed. A concentrated mass is located at an arbitrary location on the beam, and the stability of the beam is studied with the mass at various locations. The beam is analyzed using the Timoshenko-type shear deformation theory with the rotary inertia. Hamilton's principle is used to derive the equations of motion, and the spinning speed of the beam with various non-dimensional parameters subjected to a pulsating follower force is investigated. The finite element method is applied to analyze the spinning beam model, and the method of multiple scales is used to investigate the dynamic stability characteristics. A pulsating follower force is applied, and then the stability regions are changed with the transitions of the stability area in many regions. The results show that the concentrated mass increases the dynamic stability of the spinning unconstrained beam subjected to a thrust. As the spinning speed of the beam is increased, the instability regions are reduced, but various slight instability regions are additionally developed.     

Transverse Vibration and Stability Analysis of Circular Plate Subjected to Follower Force and Thermal Load

Transverse vibration and stability analysis of circular plate subjected to follower force and thermal load are analyzed . B ased on the thin plate theory in involving the variable temperature, the differential equation of transverse vibration for the axisymmetric circular plate subjected to follower force and thermal load is established. Then, the differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Meanwhile, the generalized eigenvalue under three different boundary conditions are calculated. In this case, the change curve of the first order dimensionless complex frequency of the circular plate subjected to the follower force in the different conditions with the variable temperature coefficient and temperature load is analyzed. The stability and corresponding critical loads of the circular plate subjected to follower force and thermal load with simply supported edge, clamped edge and free edge are discussed. The results provide theoretical basis for improving the dynamic stability of the circular plate.     

Vertical accelerations of simple beams due to successive loads traveling at resonant speeds

In this paper, the vertical acceleration response of a simple beam traveled by a series of equally spaced moving loads at constant speeds is studied by the superposition method. From the closed-form solution derived, the key parameters dominating the resonant response of the beam are identified, along with the effect of higher modes of vibration on the acceleration response investigated. For the loads moving at resonant speeds, the higher modes can have significant influence on the acceleration amplitude. This is true especially for beams with light damping, for which the maximum acceleration on the beam depends on which vibration mode is excited. As such, the maximum acceleration of the beam need not occur at the mid-point. By considering the resonant speeds associated with the first and second modes, a simplified formula is proposed for checking whether the maximum acceleration may occur at the mid-point of the beam. For the case when the structural damping is taken into account, the contribution of higher modes to the acceleration response tends to be damped out. It is concluded that for a beam properly damped, the maximum acceleration response of the beam is dominated by the fundamental vibration mode.     

Bending–torsional flutter of a cantilevered pipe conveying fluid with an inclined terminal nozzle

Stability analysis of a horizontal cantilevered pipe conveying fluid with an inclined terminal nozzle is considered in this paper. The pipe is modelled as a cantilevered Euler–Bernoulli beam, and the flow-induced inertia, Coriolis and centrifugal forces along the pipe as well as the follower force induced by the jet-flow are taken into account. The governing equations of the coupled bending–torsional vibrations of the pipe are obtained using extended Hamilton's principle and are then discretized via the Galerkin method. The resulting eigenvalue problem is then solved, and several cases are examined to determine the effect of nozzle inclination angle, nozzle aspect ratio, mass ratio and bending-to-torsional rigidity ratio on flutter speed of the system.     

Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads

The vibration and stability of an elastically supported beam carrying an attached mass and subjected to axial and tangential compressive loads are investigated. The analysis is based on the Timoshenko beam theory and the effects of the attached mass are expressed with Dirac delta functions. The influences of the support stiffness, the direction of loading, and the slenderness ratio on the natural frequency and critical load of a beam are discussed.     

On the occurrence of flutter in the lateral-torsional instabilities of circular arches under follower loads

The lateral-torsional stability of circular arches subjected to radial and follower distributed loading is treated herein. Three loading cases are studied, including the radial load with constant direction, the radial load directed towards the arch centre, and the follower radial load (hydrostatic load), as treated by Nikolai in 1918. For the three cases, the buckling loads are first obtained from a static analysis. As the case of the follower radial load (hydrostatic load) is a non-conservative problem, the dynamic approach is also used to calculate the instability load. The governing equations for out-of-plane vibrations of circular arches under radial loading are then derived, both with and without Wagner's effect. Flutter instabilities may appear for sufficiently large values of opening angle, but flutter cannot occur before divergence for the parameters of interest (civil engineering applications). Therefore, it is concluded that the static approach necessarily leads to the same result as the dynamic approach, even in the non-conservative case.     

Vibration of a beam due to a random stream of moving forces with random velocity

The problem of dynamic response of a beam to the passage of a train of concentrated forces with random amplitudes and velocities is considered. Force arrivals at the beam are assumed to constitute the point stochastic process of events. Thus, the excitation process is an idealization of vehicular traffic loads on a bridge. An analytical technique is developed to determine the response of the beam. Explicit expressions for the expected value and the variance of the beam deflection are provided. As an example, the response of a beam to a stationary stream of forces is determined for some practical situations, and discussed.     

Suspended bridges subjected to moving loads and support motions due to earthquake

The paper deals with the vibration of suspended bridges subjected to the simultaneous action of moving loads and vertical support motions due to earthquake. The basic partial integro-differential equation is applied to the vertical vibration of a suspended beam. The dynamic actions of traffic loads are modelled as a row of equidistant moving forces, while the earthquake is considered by vertical motions of supports. The governing equation is solved first analytically to receive an ordinary differential equation and next numerically. Moreover, the designed world's largest suspended bridge—Messina Bridge—is investigated (central span of length 3.3 km). The paper studies the effect of various lags of the earthquake arrival because the earthquake may appear at any time when the train moves along a large-span bridge. The modified Kobe earthquake records have been applied to calculations. The results indicate that the interaction of both the moving and seismic forces may substantially amplify the response of long-span suspended bridges in the vicinity of the supports and increase with the rising speed of trains.     

Coupling effects on a cantilever subjected to a follower force

In this investigation a solution methodology is presented for studying the stability of a uniform cantilever having a translational and rotational spring at its support, carrying two concentrated masses, one at the support and the other at its tip, and subjected to a follower compressive force at its free end. The analysis is based on Timoshenko's beam theory by considering the cantilever as a continuous elastic system. The coupling effects on the flutter load are fully assessed for a variety of parameters such as translational and rotational springs at the support, translational and rotational inertia of the concentrated masses, and cross-sectional shape, as well as transverse shear deformation and rotatory inertia of the mass of the column.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Dynamic stability of functionally graded cantilever cylindrical shells under distributed axial follower forces

In this paper, flutter of functionally graded material (FGM) cylindrical shells under distributed axial follower forces is addressed. The first-order shear deformation theory is used to model the shell, and the material properties are assumed to be graded in the thickness direction according to a power law distribution using the properties of two base material phases. The solution is obtained by using the extended Galerkin's method, which accounts for the natural boundary conditions that are not satisfied by the assumed displacement functions. The effect of changing the concentrated (Beck's) follower force into the uniform (Leipholz's) and linear (Hauger's) distributed follower loads on the critical circumferential mode number and the minimum flutter load is investigated. As expected, the flutter load increases as the follower force changes from the so-called Beck's load into the so-called Leipholz's and Hauger's loadings. The increased flutter load was calculated for homogeneous shell with different mechanical properties, and it was found that the difference in elasticity moduli bears the most significant effect on the flutter load increase in short, thick shells. Also, for an FGM shell, the increase in the flutter load was calculated directly, and it was found that it can be derived from the simple power law when the corresponding increase for the two base phases are known.     

A simplified shear and normal deformations nonlocal theory for bending of nanobeams in thermal environment

This article presents a simplified three-unknown shear and normal deformations nonlocal beam theory for the bending analysis of nanobeams in thermal environment. Eringen's nonlocal constitutive equations are considered in the analysis. Governing equations are derived according to the present refined theory using Hamilton's principle. Central deflections of nanobeams under uniform and point loads are given and compared with the available ones in the literature. Additional results of displacement and stresses are presented for future comparison. The effects of nonlocality, temperature parameters, length of beam, length-to-depth ratio as well as shear and normal strains are all investigated.