Frequency analysis of multi-span shifts

Some basic properties of receptance functions are employed to determine the stationary values of the critical frequencies of a multi-span shaft in which the position of one of the bearings can be varied. Details are given of the determination of the maximum and minimum values of the first two critical frequencies. The theory is then extended to obtain the spacing of the intermediate supports for the first critical to be a maximum. The simple result thus obtained generalizes Darnley's solution for the special case of a uniform shaft with simply-supported ends. The method is also employed to determine the maximum fundamental frequency of a beam or shaft having a freely articulating internal joint of variable position.     

NON-LINEAR MODELLING OF ROTOR DYNAMIC SYSTEMS WITH SQUEEZE FILM DAMPERS—AN EFFICIENT INTEGRATED APPROACH

Squeeze film dampers used in rotor assemblies such as aero-engines introduce non-linear damping forces into an otherwise linear rotor dynamic system. The steady state periodic response of such rotor dynamic systems to rotating out-of-balance excitation can be efficiently determined by using periodic solution techniques. Such techniques are essentially faster than time marching techniques. However, the computed periodic solutions need to be tested for stability and recourse to time marching is necessary if no periodic attractor exists. Hence, an efficient integrated approach, as presented in this paper, is necessary. Various techniques have been put forward in order to determine the periodic solutions, each with its own advantages and disadvantages. In this paper, a receptance harmonic balance method is proposed for such a purpose. In this method, the receptance functions of the rotating linear part of the system are used in the non-linear analysis of the complete system. The advantages of this method over current periodic solution techniques are two-fold: it results in a compact model, and the receptance formulation gives the designer the widest possible choice of modelling techniques for the linear part. Stability of these periodic solutions is efficiently tested by applying Floquet Theory to the modal equations of the system and time marching carried out on these equations, when necessary. The application of this integrated approach is illustrated with simulations and an experiment on a test rig. Excellent correlation was achieved between the periodic solution approach and time marching. Good correlation was also achieved with the experiment.     

Numerical evaluation of the acoustic radiation from planar structures with general baffle conditions using wavelets

A method is presented for computing the acoustic radiation from baffled, unbaffled, or partially baffled planar structures. The surface displacement and the surface pressure are expressed in terms of wavelets, and the acoustic dynamic stiffness (baffled case) or the acoustic receptance (unbaffled case) between any two wavelets is derived in closed form. The wavelets are employed with translation only (i.e., no dilation), and the jinc function is used; the Hankel transform of this function is the Heavyside step function, and this feature greatly simplifies the analysis. There is a trivial mapping between the wavelet amplitudes and the physical motion of the structure, and hence the dynamic stiffness and receptance results can readily be used to derive the acoustic dynamic stiffness matrix (by inverting the receptance matrix in the unbaffled case) in any set of generalized coordinates. Partially baffled systems can then be studied by substructuring the dynamic stiffness matrix. A set of example problems is considered in which the method is used to compute the resistive and reactive radiation efficiency of a range of benchmark systems.     

The receptance method applied to ring-stiffened cylindrical shells: Analysis of modal characteristics

The receptance method is applied to determine the natural frequencies and mode shapes of circular cylindrical shells stiffened by rings. The receptances of cylindrical shell and of a ring to forces in the radial and circumferential directions are derived in terms of the modal characteristics of each. A matrix equation of free vibration, which must be solved by an iterative technique, results by eliminating the angular variable. An iterative solution is practical, since the size of the matrices remains at two times the number of stiffening rings, regardless of the number of modes of the unstiffened cylinder and rings included in the solution. The validity of the method is demonstrated by comparing results for specific cases with the results obtained theoretically and experimentally by others. When various stiffener configurations are being considered for a given cylindrical shell, the modal characteristics of the shell without stiffeners may be calculated once and used repeatedly to calculate the frequencies of the stiffened shell configurations. The form of the results offers potential for simplifications which are presented in a companion paper.     

Receptances of non-proportionally and continuously damped plates-equivalent dampers method

A method for the dynamic analysis of continuously and non-proportionally damped plates is discussed. The method is quite general and suitable for various damping treatments, such as in multilayer plates with damping layers. The transverse vibrations of partially coated plates under harmonic excitation are analyzed by the proposed method. The results of the undamped modal analysis made by classical finite element methods are used in the suggested lumped parameter analysis. The receptance matrices of coated plates have been computed at undamped natural frequencies. The computational results have been verified by comparison with experimental values for partially and fully coated rectangular plates.     

Partial pole placement in structures by the method of receptances: Theory and experiments

The theory and practical application of the receptance method for vibration suppression in structures by multi-input partial pole placement is described. Numerous advantages of the receptance method over conventional matrix methods such as state-space control based on finite elements have been demonstrated, in particular there is no need to know or to evaluate the structural matrices M, C, K and in practical experimentation the measurement of ‘receptance’ may be generalised so that explicit modelling of actuator dynamics becomes unnecessary. Active vibration control is demonstrated experimentally using two test rigs. In the first set of experiments partial pole placement is applied to a lightweight glass-fibre beam using macro fibre composite (MFC) actuators and sensors. In the second set of experiments active vibration control is implemented on a heavy modular test structure representative of systems of differing dynamic complexity using electromagnetic actuators and piezoelectric (ICP) accelerometers. It is demonstrated that chosen poles may be assigned to predetermined values without affecting the position of other poles of interest.     

Implication of conservative and gyroscopic forces on vibration and stability of an elastically tailored rotating shaft modeled as a composite thin-walled beam

Problems related with the implications of conservative and gyroscopic forces on vibration and the stability of a circular cylindrical shaft modeled as a thin-walled composite beam and spinning with constant angular speed about its longitudinal axis are addressed. Taking into account the directionality property of fiber reinforced composite materials, it is shown that for a shaft featuring flapwise-chordwise-bending coupling, a dramatic enhancement of both the vibrational and stability behavior can be reached. In addition, the effects played in the same context by transverse shear, rotatory inertias as well as by the various boundary conditions are discussed and pertinent conclusions are outlined.     

Active health monitoring in a rotating cracked shaft using active magnetic bearings as force actuators

We consider the active health monitoring of rotordynamic systems in the presence of breathing shaft cracks. The shaft is assumed to be supported by conventional bearings and an active magnetic bearing (AMB) is used in a mid-shaft or outboard location as an actuator to apply specified, time-dependent forcing on the system. These forces, if properly chosen, induce a combination resonance that can be used to identify the magnitude of the time-dependent stiffness arising from the breathing mode of the shaft crack.     

On the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies

A recent paper has shown that under certain conditions the cross-spectral matrix of the forces exerted by a vibrational or acoustic wave field on its surrounding boundaries can be expressed in terms of (i) the energy of the wave field, and (ii) the direct field dynamic stiffness matrix of the boundary. This "diffuse field reciprocity relation" was derived using wave mechanics, and it is not immediately clear how the required wave field properties translate to conditions on the vibrational modes of the system or the applied forcing. This issue is addressed here by deriving an extended version of the reciprocity relation using modal methods, and the conditions required for the extended version to reduce to the existing relation are delineated. It is shown that the existing diffuse field reciprocity relation leads to an anomalous result when used to predict the energy variance of a subsystem, and that this anomaly is resolved by using the present extended version of the relation. A supplementary result arising from the analysis is that for systems with a sufficient degree of randomness the ensemble average of the dynamic stiffness matrix of a random subsystem is equal to the inverse of the ensemble average of the receptance matrix.     

Classical and quantum kinetic equations with exact conservation laws

Stationary and dynamic properties of reduced density matrices can be determined from formal or approximate closures of an infinite hierarchy of equations. The local macroscopic conservation laws place weak but important constraints on the reduced density matrices which should be respected by any closure. For pairwise additive forces conditions on the closure of the one- and two-particle equations are obtained that preserve the exact functional dependence of the conserved densities and their fluxes on the reduced density matrices. To illustrate the nature of these conditions, a closure approximation suitable for a quantum gas is given, yielding an extension of the time-dependent Hartree-Fock equations for the dynamics of a nuclear fluid to include collisions.     

Correlation functions near instabilities in systems driven by parametric noise

For a quantum system weakly coupled to heat reservoirs, a statistical mechanical theory is developed in a formalism that dovetails perfectly with phenomenological thermodynamics. The present model is a modification of the many-reservoir model by Bergmann and Lebowitz. The method to be used here is based on the explicit introduction of external forces which bring about a deviation from equilibrium. These forces are assumed to arise from mechanical interaction with its surrounding which can be characterized thermodynamically. By an appropriate choice of reservoirs the Liouville-von Neumann equation is found to describe a heat conducting system. The nonequilibrium density matrices which describe such a system are found explicitly for some interesting cases. With these density matrices we obtain an expression for entropy production.     

DYNAMICS BEHAVIOR OF AXISYMMETRIC AND BEAM-LIKE ANISOTROPIC CYLINDRICAL SHELLS CONVEYING FLUID

The flow-induced vibration characteristics of anisotropic laminated cylindrical shells partially or completely filled with liquid or subjected to a flowing fluid are studied in this work for two cases of circumferential wave number, the axisymmetric, where n=0 and the beam-like, where n=1. The shear deformation effects are taken into account in this theory; therefore, the equations of motion are determined with displacements and transverse shear as independent variables. The present method is a combination of finite element analysis and refined shell theory in which the displacement functions are derived from the exact solution of refined shell equations based on orthogonal curvilinear co-ordinates. Mass and stiffness matrices are determined by precise analytical integration. A finite element is defined for the liquid in cases of potential flow that yields three forces (inertial, centrifugal and Coriolis) of moving fluid. The mass, stiffness and damping matrices due to the fluid effect are obtained by an analytical integration of the fluid pressure over the liquid element. The available solution based on Sanders' theory can also be obtained from the present theory in the limiting case of infinite stiffness in transverse shear. The natural frequencies of isotropic and anisotropic cylindrical shells that are empty, partially or completely filled with liquid as well as subjected to a flowing fluid, are given. When these results are compared with corresponding results obtained using existing theories, very good agreement is obtained.     

A threshold for the use of Tikhonov regularization in inverse force determination

In the analysis of structure-borne sound from installed machinery, it is important to be able to estimate the operational forces. Assuming that their location is known, indirect approaches based on matrix inversion can be used to reconstruct the operational forces from a set of measured operational responses and corresponding matrix of frequency response functions. In common with many such inverse problems, matrix ill-conditioning can affect the reliability of the results. Methods such as pseudo-inversion of over-determined matrices, singular value rejection, and Tikhonov regularization have been used previously to overcome this and it has been found that Tikhonov regularization generally performs well in reducing the errors in the reconstructed forces. However, full-rank pseudo-inversion (unregularized solution) gives better results than Tikhonov regularization in some cases, particularly with low condition numbers. Since the need for regularization is greatest when the matrix is ill-conditioned, this suggests the introduction of a threshold above which Tikhonov regularization is used and below which pseudo-inversion is used. In this study, the extent to which response errors are amplified in the force estimates is considered and plotted against the matrix condition number. This allows a threshold condition number to be identified above which Tikhonov regularization gives improved results. It is found that the threshold is related not only to the condition number but also to the matrix dimensions including the extent of over-determination. A simple empirical formula is obtained for this threshold that is usable for matrices in a wide range of matrix dimensions.     

Receptances of non-proportionally and continuously damped plates—Reduced dampers method

This paper presents a method for the dynamic analysis of continuously and non-proportionally damped plates in bending modes. The damping can be in the form of constrained or unconstrained layers. The method is an extension of the equivalent dampers method discussed in a previous paper, in which the damping matrix of a discretized plate is replaced by a diagonal equivalent damping matrix. Each diagonal element represents an equivalent damper inserted between the structure and ground. In this method the number of equivalent dampers is reduced so that the receptance matrix of the damped structure can be obtained economically by a direct matrix method. The receptances of two different partially coated plates in transverse directions are computed by the method suggested. The verification of the results is demonstrated by comparison with the experimental values and also with the theoretical values obtained by the equivalent dampers method. The method presented can also be applied to the transverse vibration analysis of plates with discrete damping inserts.     

Application of the polar decomposition to light scattering particle systems

We have applied the polar decomposition theorem (PDT) to the scattering matrix of simple systems such as an isolated sphere, either metallic or dielectric. The discrete dipole approximation (DDA) method has been used as an intermediate tool to calculate these matrices. We show how the parameters introduced by the PDT for describing the scattering by a single sphere can be easily interpreted. It is also shown that the PDT provides us with an alternative frame to conventional Mueller matrix analysis, by characterizing the systems by means of a group of independent parameters representing magnitudes of simple (virtual) elements. This step is necessary in order to translate other complex system matrices into manageable information.     

Nonlinear analysis of cylindrical and conical hysteretic whirl motions in rotor-dynamics

The internal friction of a rotor–shaft-support system is mainly due to the shaft structural hysteresis and to some possible shrink-fit release of the assembly. The experimentation points out the destabilizing effect of the internal friction in the over-critical rotor running. Nevertheless, this detrimental influence may be efficiently counterbalanced by other external dissipative sources located in the supports or by a proper anisotropic configuration of the support stiffness. The present analysis considers a rotor–shaft system which is symmetric with respect to the mid-span and is constrained by viscous-flexible supports with different stiffness on two orthogonal planes. The cylindrical and conical whirling modes are easily uncoupled and separately analysed. The internal dissipation is modelled by nonlinear Coulombian forces and moments, which counteract the translational and rotational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are solved by averaging approaches of the Krylov–Bogoliubov type. In both the over-critical whirling motions, cylindrical and conical, stable limit cycles may be attained whose amplitude is as large as the external dissipation applied by the supports is low. The stiffness anisotropy of the supports may be recognised as quite beneficial for the cylindrical whirl.     

Accurate identification of the frequency response functions for the rotor-bearing-foundation system using the modified pseudo mode shape method

In this paper, an identification technique in the dynamic analyses of rotor-bearing-foundation systems called the pseudo mode shape method (PMSM) was improved in order to enhance the accuracy of the identified dynamic characteristic matrices of its foundation models. Two procedures, namely, phase modification and numerical optimisation, were proposed in the algorithm of PMSM to effectively improve its accuracy. Generally, it is always necessary to build the whole foundation model in studying the dynamics of a rotor system through the finite element analysis method. This is either unfeasible or impractical when the foundation is too complicated. Instead, the PMSM uses the frequency response function (FRF) data of joint positions between the rotor and the foundation to establish the equivalent mass, damping, and stiffness matrices of the foundation without having to build the physical model. However, the accuracy of the obtained system's FRF is still unsatisfactory, especially at those higher modes. In order to demonstrate the effectiveness of the presented methods, a solid foundation was solved for its FRF by using both the original and modified PMSM, as well as the finite element (ANSYS) model for comparisons. The results showed that the accuracy of the obtained FRF was improved remarkably with the modified PMSM based on the results of the ANSYS. In addition, an induction motor resembling a rotor-bearing-foundation system, with its housing treated as the foundation, was taken as an example to verify the algorithm experimentally. The FRF curves at the bearing supports of the rotor (armature) were obtained through modal testing to estimate the above-mentioned equivalent matrices of the housing. The FRF of the housing, which was calculated from the equivalent matrices with the modified PMSM, showed satisfactory consistency with that from the modal testing.     

Organic fiber reinforced plastics based on complex hybrid matrices including polysulfone and carbon nanotubes as modifiers of epoxy resins

Organic Fiber Reinforced Plastics (OFRP) based on aramid fibers are as a rule used in constructions working under extremal conditions. In view of this, the possibility of increasing the resistance of OFRP to destruction by modifying matrices with thermoplastic polymers and carbon nanotubes (CNTs) offers much promise. In this work, we present the results obtained in a study of the properties of OFRP based on Rusar fibers and epoxy matrices containing either CNTs or a thermoplastic (PSK-1 polysulfone) or both these components simultaneously. The data obtained substantiate the possibility of using epoxypolysulfone matrices for the preparation of wound composites. This modification noticeably increases crack and impact resistance of OFRP based on aramid fibers without decreasing the glass transition temperature, as when matrices are plasticized by rubber and active diluents. The strongest effect of polysulfone introduced into an epoxy matrix is observed at a large (20 wt %) content of PSK-1. The modification of epoxypolysulfone matrices by CNTs also increases the shear strength of OFRP and almost does not change the fracture toughness and compression strength. The introduction of CNTs into epoxy matrices is less effective and can increase crack growth resistance of OFRP by approximately 30% only at a large (1%) content of CNTs. Small CNT admixtures (0.3–0.6%) do not influence the fracture toughness. Possible mechanisms of the changes observed are considered.     

Partial synchronization in stochastic dynamical networks with switching communication channels

In this paper, the partial synchronization problem of stochastic dynamical networks (SDNs) is investigated. Unlike the existing models, the SDN considered in this paper suffers from a class of communication constraint-only part of nodes' states can be transmitted. Thus, less nodes' states can be used to synchronize the SDN, which makes the analysis of the synchronization problem much harder. A set of channel matrices are introduced to reflect such kind of constraint. Furthermore, due to unpredictable environmental changes, the channel matrices can switch among some communication modes. The switching considered here is governed by a Markov process. To overcome the difficulty, a regrouping method is employed to derive our main results. The obtained conditions guarantee that partial synchronization can be achieved for SDNs under switching communication constraint. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results and how the communication constraint influences synchronization result.     

Random vibration of an initially stressed thick plate on an elastic foundation

The mean-square bending moment of a thick rectangular plate excited by a uniform distribution of stationary random forces that are uncorrelated in space is calculated. The plate has in-plane compressive or tensile stresses. In addition, the plate is mounted on an elastic foundation. Numerical results are given for plates with uniform initial stress when the temporal correlation function of the excitation possesses an exponential decay. In general it can be said that the position on the plate where the mean-square moment takes on a maximum value depends upon the relative values of the initial stress, the stiffness of the foundation and the aspect ratio of the plate. The mean-square response amplitude of the plate on a foundation never exceeds that of the plate without a foundation, regardless of the intensity of the initial stress or the geometrical configuration of the plate.