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.     

Dynamics of finite-length tubular beams conveying fluid

The dynamics and stability of short tubes conveying fluid is re-examined by means of Timoshenko beam theory for the tube and a three-dimensional fluid-mechanical model for the fluid flow, rather than the plug-flow model utilized heretofore. The tubes considered are either clamped at both ends or cantilevered; in the latter case, special “outflow models” were introduced to describe the boundary conditions on the fluid exiting from the free end. By comparison with experiments, it is shown that this refined theory is necessary for describing adequately the dynamical behaviour of extremely short tubes, although Timoshenko beam theory together with a plug-flow model are quite satisfactory for relatively longer short tubes; for long tubes, Euler-Bernoulli beam theory and a plug-flow model are perfectly adequate.     

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.     

The stability of simply supported tubes conveying a compressible fluid

This paper is concerned with establishing the conditions of static stability of a simply supported tube conveying a compressible fluid by application of Euler's method of equilibrium. Timoshenko beam theory is used to describe the tube motion whiler Euler's equations of motion govern the compressible flow through the tube. The eigenvalue problem associated with the linearized equations of motion first derived by Niordson is solved by using Muller's method. The effects on critical velocity of fluid sound speed, tube shear, and tube aspect ratio are parametrically studied. When the flow is subsonic, the aspect ratio increases the critical velocity predicted by the theory while increased aspect ratio decreases the critical velocity when the flow is supersonic. Reduced sound speed and tube shear modulus always reflect a reduced critical velocity for the onset of tube buckling or divergence instability.     

Theoretical Treatment of Nonstationary Scattering by Phonons and Polaritons Part I. Derivation of the Basic Equations

The equations of motion of a two-level system and a harmonic oscillator coupled to a dissipative system are discussed; the dissipative system is assumed to consist of a large number of radiation oscillators. Special equations for the determination of the correlation functions of the fluctuation forces are derived under the condition of large time values, for which the atomic system has “forgotten” its initial state. The expectation values of the correlation functions are connected with the damping constant and the population operator of the excited state of the atomic system is in thermal equilibrium. Taking into account the influence of the coherent radiation field on the atomic system, the basic equations for the treatment of the nonstationary Raman scattering by polaritons are derived; the temporal range of validity is discussed. Using a time-dependent “variable” Fourier transformation, the nonstationary time- and spacedependent spectral densities are related to the correlation functions of the fields; here the Wiener-Khintchine theorem is applied in a nonstationary form. The limiting cases of the stationary scattering process as well as the usually introduced correlators of the slowly varying amplitudes are discussed.     

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.     

Stability analysis of partially loaded Leipholz column carrying a lumped mass and resting on elastic foundation

In the present study, the stability of a cantilever column resting on an elastic foundation under the action of a uniformly distributed tangential load is discussed. A Winkler type elastic foundation is considered. Moreover, the effect of a lumped mass located in an arbitrary position on the stability of the system when the column is subjected to a partially distributed follower force is investigated. The equations of motion are obtained using the extended Hamilton's principle and the influences of the lumped mass and applied load are included in the equations using the generalized functions theories. Applying the Ritz technique, the resulting equations are transformed into a general eigenvalue problem. The effects of several design parameters such as foundation elastic modulus, ratio of the lumped mass to the column's mass, position of the lumped mass and the distribution model of the follower force are examined. The validity of the present analysis is confirmed by comparing the results with those obtained in literature and excellent agreement is observed. The numerical results reveal that the load distribution length and model have significant effects on the flutter boundaries of the system.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

Stability of a deploying/extruding beam in dense fluid

The equations of motion of a flexible slender cantilevered beam with uniform circular cross-section, extending axially in a horizontal plane at a known rate while immersed in an incompressible fluid are derived. An “axial added mass coefficient” is implemented in these equations in order to better approximate the mass of fluid which stays attached to the oscillating beam while moving in the axial direction. Realistic initial conditions are given to the system and numerical solutions are obtained. The dynamical behaviour of the system is observed for cases of constant extension rate and for a trapezoidal deployment rate profile.In the case of low constant extension rates, the system displays a phase of oscillation with increasing amplitude and decreasing frequency until the motion is strongly damped and later becomes statically unstable. For faster deployment rates, the beam has a short flutter phase at the beginning of the deployment, followed by a brief phase of damped oscillation until it exhibits static divergence. For fast enough deployment rates, the system is unstable from the beginning and never stabilizes. The effect the axial added mass coefficient has on the system is studied and it is found that it plays two roles in the stability of the system. The trapezoidal deployment rate profile is studied because it is deemed more representative of real-life applications. For long deployment times, the system behaves in a very similar manner to one with low constant extension rate, except that it does not become statically unstable. For shorter deployment times, the maximum amplitude of the tip displacement is usually attained after the beam has stopped extruding.     

Dynamics of an array formed by three neutrally buoyant cylindrical cantilevers subjected to tensile follower forces

The flexural-rotational coupled motion of three identical flexible cylindrical cantilevers joined symmetrically to a central head is investigated. Effects of the tensile follower forces and inertia parameters on the natural frequencies of the system are studied. The analysis suggests two types of inplane motion: one corresponding to the oscillation of the cantilevers without any rotation of the central body, while the other involves coupled motion of the array. The former corresponds to the repeated eigenvalues which are identical to those of a single cantilever having the same axial tension parameter, P. Three sets of eigenvalues govern the out-of-plane motion: (a) the central head remaining stationary with no rolling motion of the array; (b) vertical motion of the central body without any rolling motion of the array; and (c) rigid body rolling motion without any vertical motion of the central head. There is a possibility of dynamic instability for small inertia parameters and large axial tension.     

A distributed-mass approach for dynamic analysis of Timoshenko plane frames

The dynamic stiffness method is the exact method for the dynamic analysis of plane frames using the continuous-coordinate system to consider the effect of mass distribution in beam elements. The dynamic stiffness method may create some null modes where the joints of beam element have null deformation. Unlike the Bernoulli–Euler frames, adding an interior node at the middle of the beam elements cannot normalize all the null modes of flexural vibration in the Timoshenko frames. The floating interior-node scheme is proposed to eliminate the null modes of flexural vibration in the Timoshenko frames. Orthogonal properties of vibration modes in Timoshenko plane frames are theoretically derived, through which the equations of motion in beam elements can be transformed into the decoupled equations of motion in terms of mode amplitudes.     

Solvable and/or Integrable and/or Linearizable N-Body Problems in Ordinary (Three-Dimensional) Space. I

Abstract

Several N -body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (“acceleration equal force;” in most cases, the forces are velocity-dependent) and are amenable to exact treatment (“solvable” and/or “integrable” and/or “linearizable”). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider “few-body problems” (with, say, N =1,2,3,4,6,8,12,16,...) as well as “many-body problems” (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N -body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases.     

A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants

A N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of N pointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of the N “particles”) at any later time t can be obtained from its configuration at the initial time by algebraic operations. It features specific nonlinear velocity-dependent many-body forces depending on N² arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous—so that all its solutions are then periodic with a fixed period independent of the initial data. A variant with twice as many arbitrary coupling constants, or even more, is also identified.     

Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.     

Modal characteristics of a flexible beam with multiple distributed actuators

A flexible structure with surface-bonded piezoceramic patches is modelled using Timoshenko beam theory. Exact mode shapes and natural frequencies associated with the flexural motion are computed for various piezoceramic distributed actuator arrangements. The effects of patch placement and of shear on the modal characteristics are demonstrated using a cantilevered beam as an example. Perfect bonding of the piezoceramic to the beam substructure is assumed, and for the purposes of this paper only passive piezoceramic properties are considered. The modelling technique and results obtained in a closed form are intended to assist investigations into the modelling and control of active structures with surface-bonded piezoceramic actuators.     

Vibrations of double-nanotube systems with mislocation via a newly developed van der Waals model

This study deals with transverse vibrations of two adjacent-parallel-mislocated single-walled carbon nanotubes (SWCNTs) under various end conditions. These tubes interact with each other and their surrounding medium through the intertube van der Waals (vdW) forces, and existing bonds between their atoms and those of the elastic medium. The elastic energy of such forces due to the deflections of nanotubes is appropriately modeled by defining a vdW force density function. In the previous works, vdW forces between two identical tubes were idealized by a uniform form of this function. The newly introduced function enables us to investigate the influences of both intertube free distance and longitudinal mislocation on the natural transverse frequencies of the nanosystem which consists of two dissimilar tubes. Such crucial issues have not been addressed yet, even for simply supported tubes. Using nonlocal Timoshenko and higher-order beam theories as well as Hamilton's principle, the strong form of the equations of motion is established. Seeking for an explicit solution to these integro-partial differential equations is a very problematic task. Thereby, an energy-based method in conjunction with an efficient meshfree method is proposed and the nonlocal frequencies of the elastically embedded nanosystem are determined. For simply supported nanosystems, the predicted first five frequencies of the proposed model are checked with those of assumed mode method, and a reasonably good agreement is achieved. Through various studies, the roles of the tube's length ratio, intertube free space, mislocation, small-scale effect, slenderness ratio, radius of SWCNTs, and elastic constants of the elastic matrix on the natural frequencies of the nanosystem with various end conditions are explained. The limitations of the nonlocal Timoshenko beam theory are also addressed. This work can be considered as a vital step towards better realizing of a more complex system that consists of vertically aligned SWCNTs of various lengths.     

H-theorem for a mean field model describing coupled oscillator systems under external forces

This article studies the asymptotic behavior of solutions of Fokker–Planck equations describing mean field approximations of weakly coupled oscillator systems subjected to external forces. Using an H-theorem we show that transient probability densities converge to stationary ones. Furthermore, stability criteria are derived for the stationary solutions of these Fokker–Planck equations. The obtained results are applied to a model that combines the Haken–Kelso–Bunz model and the models of weakly coupled oscillators proposed by Winfree and Kuramoto. The stability criteria based on the H-theorem agree with those derived in our earlier analyses.     

Equations of motion,commutation relations and ambiguities in the Lagrangian formalism

The inverse problem of the calculus of variations is analysed in the case of Newtonian mechanics. Its connection with the question: “Does the equation of motion determine commutation relations?” raised some time ago by Wigner, is exhibited. All the Lagrangians that yield variational equations equivalent to a given set of second order differential equations are explicitly determined for some simple classes of forces.     

A note on the equations of motion for the transverse vibration of a Timoshenko beam subjected to an axial force

The equations of motion are derived for a Timoshenko beam of arbitrary section in transverse vibration with an applied axial loading. Some discrepancies are noted by comparison with earlier work and these are examined and accounted for. The application of these more general equations is then considered for certain special cases, including a rotating blade.