Linearly Coupled,Slowly Varying Oscillators: The Interaction of a Positive-Energy Mode With a Negative-Energy Mode

A linear dynamical system is considered whose normal frequencies and normal modes come into close coincidence. The case when both modes have positive energy has been discussed by Grimshaw and Allen (1979). Here the case when one mode has positive energy and the other mode has negative energy is discussed. Coupled equations are derived and solved exactly using parabolic cylinder functions. It is found that the action in both modes grows during the coupling.     

The equations of the perturbed motion of a rocket as a thin-walled structure with a liquid

The perturbed motion of a rocket as an elastic thin-walled structure with compartments partially filled with liquid propellant is considered. It is assumed that the normal modes of the hydroelastic oscillations of the rocket are determined under the condition that the velocity potential on the free surface of the liquid is equal to zero and with standard remaining conditions. Certain features of these modes with zero fundamental frequencies are pointed out and the “loss” of mass effect associated with this is explained. Equations are derived for the perturbed motion of a rocket taking account of the hydroelastic oscillations of its structure and the oscillations of the liquid with deviations of the free surface from the equilibrium position under the action of mass forces. The coefficients of these equations, characterizing the relation between the different type of oscillations, are expressed in terms of known hydrodynamic parameters and the values of the oscillation modes at certain points.     

The coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features

The purpose of this paper is to investigate the coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features. There are some new phenomena due to the coupling effects of shaft-bending, shaft-torsion, disk-transverse and blade-bending. In this investigation, this paper mainly focuses on the influence of mistuned features of the blade's length and the stagger angle. It is found that there are four types of coupling modes: the coupling mode of shaft bending, disk transverse and blade bending (SDB), the coupling mode of shaft torsion disk transverse-blade bending (TDB), the coupling mode of disk transverse and blade bending (DB), the repeated mode of blade bending-blade bending (BB). With the effect of mistuned features, the natural frequencies and the coupling mode type will change correspondingly. With the mistuning value of blade length employed in this study, the TDB mode in the tuned system will disappear and shift into TSDB mode instead, and one of the repeated SDB modes will be replaced by STDB modes. Due to this mistuned features, the blades and disk experience a certain degree of vibration localization phenomenon. Different from the length feature, the influence of mistuning values of blade's stagger angle mainly take effect on the coupling modes. At last, by inspection on the Campbell diagrams, the influence of rotational speed on the transformation of natural frequencies is illustrated on the tuned/mistuned flexible shaft-disk-blades coupling structure.     

Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness

Based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, this paper presents the vibration behavior of an isotropic and orthotropic oval cylindrical shell with parabolically varying thickness along its circumference. The governing equations of motion of the shell, which have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling, which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to eight first‐order differential equations in the circumferential coordinate and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different type modes of vibration is investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

Multimodal admittance method in waveguides and singularity behavior at high frequencies

This work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi-trapped modes. The coupled mode equations are obtained by projecting the Helmholtz equation on the local transverse modes. To solve this problem we integrate the Riccati equation governing the admittance matrix (Dirichlet-to-Neumann operator). For many propagating modes, i.e. at high frequencies, the numerical integration of the Riccati equation shows that the rule is that this matrix has quasi-singularities associated to quasi-trapped modes.     

Non-stationary nonlinear analysis of a composite rotating shaft passing through critical speed

In this paper, nonlinear non-stationary dynamics of a nonlinear composite shaft passing through critical speed is studied. The nonlinearity is due to the large amplitude of shaft vibration. The equations of motion are obtained by three-dimensional constitutive relationships of composite materials. The gyroscopic effect, rotary inertia and coupling caused by material anisotropy are considered but shear deformation is neglected. Without any simplification, axial-flexural-flexural-torsional equations of motion (EOM) for the elastic composite shaft with variable rotational speed are obtained. The approximate analytical method namely asymptotic method is applied to analyze the nonstationary behavior of the composite shaft with constant acceleration. First, the EOMs are discretized using one and two-term Galerkin method. Then, the resulted equations are transformed to normal coordinates. Finally, the asymptotic method is applied to equations described in normal coordinates. Analytical expressions governing the amplitude and phase of motion during passage through critical speeds are obtained. By comparing the results obtained from analytical solutions, it is shown that discretization by one mode is not enough due to the existence of coupling in the equations and at least two modes are necessary for this purpose. Effects of damping, eccentricity, initial angular velocity and fiber angle on response amplitude are investigated. For verification, the results of perturbation theory are compared with numerical simulations and it is shown that there is good agreement between both methods.     

Existence of frequency modes coupling seismic waves and vibrating tall buildings

We prove in this paper an existence result for frequency modes coupling seismic waves and vibrating tall buildings. The derivation from physical principles of a set of equations modeling this phenomenon was done in previous studies. In this model all vibrations are assumed to be anti-plane and time harmonic so the two dimensional Helmholtz equation can be used. A coupling frequency mode is obtained once we can determine a wavenumber such that the solution of the corresponding Helmholtz equation in the lower half plane with relevant Neumann and Dirichlet conditions at the interface satisfies a specific integral equation at the base of an idealized tall building. Although numerical simulations suggest that such wavenumbers should exist, as far as we know, to date, there is no theoretical proof of their existence. This is what this present study offers to provide.     

Effect of Torsion and Warping on the Free Vibration of Sandwich Beams

The problem on free vibrations of wide sandwich beams is tackled in this paper. Torsional and warping effects in addition to flexure are included in the formulation of the dynamic problem. In order to show the effects of bending-torsion coupling and warping on the natural frequencies and the corresponding vibration modes, three cases are considered. First, the warping and torsion effects are neglected, second, the effect of warping on the coupling terms is neglected, and third, the effect of warping on the coupling terms is included. The viscoelastic core is modeled by elastic translational and rotational springs. The finite-difference method is used to solve the partial differential equations of motion with different boundary conditions for the top and bottom layers. Results for different materials, fiber orientations, depth-to-width ratios, and boundary conditions are found. The natural frequencies and the corresponding vibration modes obtained are in a good agreement with those cited in the literature. If the bending-torsion coupling is pronounced, the inclusion of warping affects the natural frequency considerably.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 163–176, April–May, 2005.     

Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators

A system of two non-linear differential equations is considered that simulates the dynamics of two completely identical weakly coupled oscillators both in the case of dissipative and active coupling. The system of normal modes is investigated. All the self-similar periodic solutions, including the asymmetric solutions describing the natural ascillations of oscillators with dissimilar amplitude's, are found analytically. The stability is investigated as well as the local bifurcations of the self-similar cycles when there is a change in stability. In particular, the possibility of the creation of two-dimensional invariant tori is pointed out. In the case of active coupling, it is shown that the basic version of the natural oscillations is a stable antiphase cycle that was observed in Huygens experiments.     

The stability of the equilibrium of a pendulum of variable length

The frequencies and modes of parametric oscillations of a pendulum of variable length for values of the modulation index from the smallest to the limit admissible values are investigated. The limits of the resonance zones of the first four oscillation modes are constructed and investigated by analytical and numerical methods, and the fundamental qualitative properties of the higher modes are established. Complete degeneracy of the modes with even numbers, i.e., coincidence of the frequencies of the odd and even eigenmodes for admissible values of the modulation parameter, is proved. The global pattern of the limits of the regions of stability of the lower position of equilibrium is constructed and it is shown that it differs considerably from the Ince–Strutt diagrams. Specific properties of the eigenmodes are established.     

Effect of an elastic core on the dynamic stability of an orthotropic cylindrical shell

A method of determining the regions of dynamic instability of an orthotropic cylindrical shell "bonded" to an elastic cylinder is proposed. An expression for the core reaction is obtained from the coupling conditions for the forces normal to the lateral surface and the radial displacements of the shell and the core at the contact surface. When the reaction is substituted in the system of equations of motion of the shell, the part corresponding to the free vibrations of the cylinder is discarded. The system of equations of motion of the shell is reduced to an equation of Mathieu type, from which transcendental equations for determining the boundaries of the regions of dynamic instability are obtained. These regions are analyzed for various modes of loss of stability and different values of the core modulus of elasticity.     

Dynamic motion of whirling rods with Coriolis effect

Longitudinal vibrations coupled with transverse vibrations of whirling rods are investigated. It is known that longitudinal and transverse vibrations are governed by second and fourth order differential equations, respectively. Due to the Coriolis effect, a system of equations that governs the longitudinal and transverse displacements will be constructed by coupling these two equations together. Solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the whirling rod. Exact and approximate solutions are obtained from the proposed governing equations, where the approximate solutions on displacements and natural frequencies are acquired by neglecting the Coriolis effect. A proposed numerical scheme known as complete function collocation method is implemented to solve the governing equations coupled with longitudinal and transverse displacements. The approximate results on both longitudinal and transverse natural frequencies show that natural frequencies are decreasing while the angular velocity of the rod is increasing. Exact and numerical results on both longitudinal and transverse natural frequencies show that there are no predictable trends whether natural frequencies are increasing or decreasing while the angular velocity of the rod is increasing.     

Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory

Transverse dynamical behaviors of axially moving nanoplates which could be used to model the graphene nanosheets or other plate-like nanostructures with axial motion are examined based on the nonlocal elasticity theory. The Hamilton's principle is employed to derive the multivariable coupling partial differential equations governing the transverse motion of the axially moving nanoplates. Subsequently, the equations are transformed into a set of ordinary differential equations by the method of separation of variables. The effects of dimensionless small-scale parameter, axial speed and boundary conditions on the natural frequencies in sub-critical region are discussed by the method of complex mode. Then the Galerkin method is employed to analyze the effects of small-scale parameter on divergent instability and coupled-mode flutter in super-critical region. It is shown that the existence of small-scale parameter contributes to strengthen the stability in the super-critical region, but the stability of the sub-critical region is weakened. The regions of divergent instability and coupled-mode flutter decrease even disappear with an increase in the small-scale parameter. The natural frequencies in sub-critical region show different tendencies with different boundary effects, while the natural frequencies in super-critical region keep constants with the increase of axial speed.     

The dependence of the natural frequencies of a one-dimensional elastic system on its length

The dependence of the natural frequencies and modes of the oscillations of distributed elastic system with characteristics of the stiffness and density that are variable along a coordinate of the cross section for arbitrary boundary conditions is investigated. It is proved that the presence of an external elastic medium, described by the Winkler model, may lead to an increase in the natural frequencies of the lower oscillation modes when the length of a one-dimensional elastic system is increased. The fine properties of the change in the natural frequencies as a function of the length of the system and the number of the oscillation mode are also established. A numerical-analytical investigation of examples which illustrate the characteristic anomalous behaviour of the lowest natural frequencies is presented.     

An mode-matching approach for ridge-guided wave investigation

An analytical method based on partial wave decomposition, mode matching and transverse resonant analysis is proposed to solve the pure feature-guided modes in a rectangular elastic ridge embedded in a flat plate. Compared with the semi-analytical finite element (SAFE) models and experiment results, the effectiveness of the method on dispersion curve and some mode shape calculation are proven. Without the drawbacks of the SAFE, which induced by insufficient discretization at high frequencies and large additional absorption zones at low frequencies, the method can render more accurate results wherein. As a result, in a symmetrically embedded ridge, four more non-leaky ridge-guided modes, meaningful for nondestructive evaluation are founded, with the dispersion curves and most of the mode shapes precisely predicted.     

Molecular dynamics by the Backward-Euler method

This paper introduces a new computational method for molecular dynamics. The method combines the Backward-Euler scheme for the solution of stiff differential equations with a Langevin-equation approach to the establishment of thermal equilibrium. The method allows the user to choose a cutoff frequency ω_c. Vibrational modes with frequencies below ω_c will be fully excited (receive a mean energy of kT per mode), while modes with frequencies greater than ω_c will be effectively frozen by the method. By setting ω_c = kT/h, one can obtain reasonable agreement with the quantum-mechanical energy distribution among the various modes, despite the classical character of the computation.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.     

On the effect of damping on the stabilization of mechanical systems via parametric excitation

The effect of damping on the re-stabilization of statically unstable linear Hamiltonian systems, performed via parametric excitation, is studied. A general multi-degree-of-freedom mechanical system is considered, close to a divergence point, at which a mode is incipiently stable and n ? 1 modes are (marginally) stable. The asymptotic dynamics of system is studied via the Multiple Scale Method, which supplies amplitude modulation equations ruling the slow flow. Several resonances between the excitation and the natural frequencies, of direct 1:1, 1:2, 2:1, or sum and difference combination types, are studied. The algorithm calls for using integer or fractional asymptotic power expansions and performing nonstandard steps. It is found that a slight damping is able to increase the performances of the control system, but only far from resonance. Results relevant to a sample system are compared with numerical findings based on the Floquet theory.     

Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments

Interaction curves for vibration and buckling of thin-walled composite box beams with arbitrary lay-ups under constant axial loads and equal end moments are presented. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. The resulting coupling is referred to as triply flexural–torsional coupled vibration and buckling. A displacement-based one-dimensional finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, bending moment, fiber orientation on the buckling loads, buckling moments, natural frequencies and corresponding vibration mode shapes as well as axial-moment–frequency interaction curves.