Free vibration of joined conical-cylindrical shells

An analysis is presented for the free vibration of joined conical-cylindrical shells. The governing equations of vibration of a conical shell, including a cylindrical shell as a special case, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices of the shells and the point matrix at the joint, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method has been applied to a joined truncated conical-cylindrical shell and an annular plate-cylindrical shell system, and the natural frequencies and the mode shapes of vibration calculated numerically. The results are presented.     

The steady state out-of-plane response of a Timoshenko curved beam with internal damping

The steady state out-of-plane response of a Timoshenko curved beam with internal damping to a sinusoidally varying point force or moment is determined by use of the transfer matrix approach. For this purpose, the equations of out-of-plane vibration of a curved beam are written as a coupled set of the 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 steady state response of the beam is obtained. The method is applied to free-clamped non-uniform beams with circular, elliptical, catenary and parabolical neutral axes driven at the free end; the driving point impedance and force or moment transmissibility are calculated numerically and the effects of the slenderness ratio, varying cross-section and the function expressing the neutral axis on them are studied.     

Free vibration of non-circular cylindrical shells with longitudinal interior partitions

An analysis is presented for the free vibration of a simply supported non-circular cylindrical shell with longitudinal interior partitions. For this purpose, the governing equations of vibration of a non-circular cylindrical shell including a plate as special case are written in a matrix differential equation by using the transfer matrix in the circumferential direction. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrix of the shell and the point matrix at the joint of the structure matrix. The method is applied to approximately elliptical cylindrical shells with an interior plate, and the natural frequencies and the mode shapes of vibration are calculated numerically, giving the results.     

Free vibration of non-circular cylindrical shells with variable circumferential profile

An analysis is presented of the free vibration of non-circular cylindrical shells with a variable circumferential profile expressed as an arbitrary function. The applicability of thin-shell theory is assumed and the governing equations of vibration of a non-circular cylindrical shell are written in a matrix differential equation by using the transfer matrix of the shell. Once the transfer matrix has been determined by numerical integration of the matrix equation, the natural frequencies and mode shapes of vibration are calculated numerically in terms of the matrix elements. The method is applied to cylindrical shells of three or four-lobed cross-section, and the effects of the length of the shell and the radius at the lobed corners on the vibration are studied.     

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.     

Free vibration of a circular plate elastically restrained along some radial segments

An analysis is presented for the free vibration of a circular plate restrained against deflection along radial segments. With the reaction forces acting on the segments regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force distributions along the segments are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the supports. The natural frequencies and the mode shapes of the plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to circular plates supported along several radial segments located at equal angular intervals, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the supports is discussed.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

DYNAMIC RESPONSE OF TWO ROTORS CONNECTED BY RIGID MECHANICAL COUPLING WITH PARALLEL MISALIGNMENT

The effect of parallel misalignment on the lateral and torsional responses of two rotating shafts (Jeffcott rotors) is examined with theoretical and numerical analysis. The general equations of motion are derived and given in dimensionless form to represent the general case. The equations of motion revealed that parallel misalignment couples the translation and angular deflections through the stiffness matrix and the force vector. The non-linear equations are solved numerically using a combination of Newmark and Newton-Raphson methods to determine the dimensionless frequency and transient responses in terms of misalignment magnitude. The numerical results show that the system natural frequencies are excited at transient condition due to the presence of pure parallel misalignment. At steady state condition, the 1×-rotational speed excitation is present in the translation and angular directions, which indicates that parallel misalignment can be a source of both torsional and lateral excitations.     

The steady state out-of-plane response of an internally damped ring supported by springs in some bays

The steady state out-of-plane response of an internally damped ring supported by springs in some bays to a sinusoidally varying point force or moment is determined by use of the transfer matrix technique. For this purpose, the equations of out-of-plane vibration of a uniform circular ring based upon the Timoshenko beam theory are written as a coupled set of first order differential equations by using the transfer matrix of the ring. The matrix is obtained analytically and the steady state response of the ring is determined by the product of the matrices in free bays and those in supported bays. In this case, the elastic moduli of the ring and springs with internal damping are assumed to be complex quantities. The method is applied to rings supported against deflection and torsion in some bays of the same length located at equal angular intervals; the driving point impedance, transfer impedance and the distributions of the deflection, angular rotation, force and moment are calculated numerically, and the effects of the number, the stiffness and the length of supporting springs on them are studied.     

Free vibration of a conical shell with variable thickness

An analysis is presented for the free vibration of a truncated conical shell with variable thickness by use of the transfer matrix approach. The applicability of the classical thin shell theory is assumed and the governing equations of vibration of a conical shell are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined by quadrature of the equations, the natural frequencies and the mode shapes of vibration are calculated numerically in terms of the elements of the matrix under any combination of boundary conditions at the edges. The method is applied to truncated conical shells with linearly, parabolically or exponentially varying thickness, and the effects of the semi-vertex angle, truncated length and varying thickness on the vibration are studied.     

Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements

This paper employs the numerical assembly method (NAM) to determine the “exact” frequency–response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force. First, the coefficient matrices for the intermediate concentrated elements, pinned support, applied force, left-end support and right-end support of a beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix. The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency–response curve for any point of a multiple-span beam carrying a number of various concentrated elements. For the case of zero harmonic force, the above-mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained.     

Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate

The nonlinear natural frequency of a rectangular box, which consists of one flexible plate and five rigid plates, is studied in this paper. The flexible plate is assumed to vibrate like a simple piston. The behavior of the structural-acoustic coupling between the flexible plate and the air cavity is analyzed by using the proposed finite element modal method. The system finite element equation is reduced and expressed in terms of the modal coordinates with small degrees of freedom by using the proposed reduction method. The system nonlinear stiffness matrix representing the large amplitude vibration can be transformed to be a constant modal matrix. The natural frequencies are determined by using the harmonic balance method to solve the eigenvalue equations of the structural-acoustic system. The effect of the cavity depth on the natural frequencies and convergence studies are discussed in detail.     

Analytical description of the electrodynamic properties of metallic photonic crystals

A method of approximate calculation of the interaction inverse matrix in the method of discrete dipoles is proposed. The knowledge of this matrix makes it possible to determine the optical response of a system to the action of an electromagnetic wave with an arbitrary shape, which can be represented as a combination of vector spherical wave functions. The number of calculation operations of the matrix in the proposed method is considerably smaller than in the case of its direct calculation. In the case of a change in the refractive index of scattering particles, two methods of approximate calculation of the interaction inverse matrix are also proposed. This makes it possible to calculate the optical response of systems with new characteristics without direct solving equations of a system with a large dimension. The accuracy of the methods is numerically determined for particles with spherical and cubic shapes. It is shown that the methods are computationally efficient and can be used to calculate the values of polarization vectors inside particles and extinction and absorption cross sections of systems.     

An integral equation analysis of the harmonic response of three-layer beams

The integral equations of harmonic motion have been derived and solved for three-layer sandwich beams with a constrained linear viscoelastic core. The method of solution required first the construction of the Green's vector for a beam in analytical form. Following this, the integral equations were derived and readily approximated by matrix equations which were finally solved numerically. In addition to this analysis, the corresponding eigenvalue problem has been solved so that the modal frequencies and the beam loss factor could be calculated directly. The integral equation analysis offers a fast and efficient alternative to the traditional methods based on the solution of the differential equations of motion. The method has been verified by comparison with experimental results for three-layer cantilevers and simply supported beams.     

Non-linear dynamics of a driven two-level tunnelling defect

With increasing external-field amplitude the dynamics of a driven two-level system shows a cross-over from the weak-coupling or linear-response behaviour to a non-linear regime which is characterised by the appearance of Rabi oscillations, a field-dependent dynamical susceptibility, saturation of energy dissipation and additional frequencies in the correlation function. The system any initial distribution will relax towards a stationary state with a time-dependent polarisation vector. These features are derived from the propagator of the two-level system which is calculated using a perturbative scheme with a well-defined small parameter; the roatating-wave approximation of the Bloch equations is avoided. the dynamical behaviour is discussed by means of correlation and response functions.     

Steady state Behaviour of a Three level Ladder Atomic System with Incoherent Population Pumping in a Squeezed Vacuum

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Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

Iteration methods and algorithms for dielectric-resonator integral equations of the field

Quasi-eigenmodes of open cylindrical and rectangular dielectric resonators (DRs) are determined by the method of iterative solution of the volume integral and integro-differential equations with corresponding functionals. New forms of equations and iteration algorithms for the nonlinear input of the desired complex parameter are proposed. Frequencies and Q-factors of the H_01δ and H₀₁₁ modes of a cylindrical DR and the H mode of a rectangular DR for the uniform and nonuniform cases are obtained numerically. The influence of a thin semiconductor layer located at the ends of the DR and irradiated by high-power laser pulses on the frequencies and Q-factors of the DR modes is examined. It is shown that an up to ten or more percent tuning of resonant frequencies can be reached by transformation of a low conducting state to a high conducting state.     

Free vibration of a rotating non-uniform beam with arbitrary pretwist, an elastically restrained root and a tip mass

The governing differential equations for the coupled bending-bending vibration of a rotating beam with a tip mass, arbitrary pretwist, an elastically restrained root, and rotating at a constant angular velocity, are derived by using Hamilton's principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. The influence of the tip mass, the rotary inertia of the tip mass, the rotating speed, the geometric parameter of the cross-section of the beam, the setting angle, and the pretwist parameters on the natural frequencies are investigated. The difference between the effects of the setting angle on the natural frequencies of pretwisted and unpretwisted beams is revealed.     

Natural frequencies and modes of open cylindrical shells with a circumferential thickness taper

An analytical procedure to estimate not only the natural frequencies but also modes of open cylindrical shells with a circumferential thickness taper by the transfer matrix method is presented. The transfer matrix is derived from the non-linear differential equations for the cylindrical shells by numerical integration. The accuracy and convergence characteristics of this method are investigated, and the natural frequencies and modes of open cylindrical shells with a circumferential thickness taper are presented for various curvatures, aspect ratios, boundary conditions and thickness ratios. Furthermore, the influences of thickness variation of the cross-section on the natural frequencies and modes are examined.