Nicole Kessissoglou

Free vibration analysis of a cracked beam by finite element method

Abstract In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.

An integrated approach to fault diagnosis of machinery using wear debris and vibration analysis

Vibration and wear debris analyses are the two main condition monitoring techniques for machinery maintenance and fault diagnosis. In practice, these two techniques are usually conducted independently, and can only diagnose about 30–40% of faults when used separately. However, recent evidence shows that combining these two techniques provides greater and more reliable information, thereby resulting in a more effective maintenance program with large cost benefits to industry. In this paper, the correlation of vibration analysis and wear debris analysis was investigated. An experimental test rig consisting of a worm gearbox driven by an electric motor was set up to examine the correlation of the two techniques under various wear conditions. Three tests were conducted under the following conditions: (a) lack of proper lubrication, (b) normal operation, and (c) with the presence of contaminant particles added to the lubricating oil. Oil samples and vibration data were collected regularly. Wear debris analysis included the study of particle number and size distribution, the examination of particle morphology and types to determine possible wear mechanisms, and the analysis of chemical compositions to assess wear sources. Fault detection in the vibration signature was compared with the particle analysis. The results from this paper have given more understanding on the dependent and independent roles of vibration and wear debris analyses in machine condition monitoring and fault diagnosis.

Power transmission in L-shaped plates including flexural and in-plane vibration

In this paper, power flow propagation in plates connected in an L-joint is investigated in both the low and high frequency ranges. An exact solution is derived to describe the flexural, in-plane longitudinal and in-plane shear wave motion in the plates. The coupled plates are simply supported along two parallel sides, and free at the other two ends. A point force is used to generate flexural wave motion only. The flexural wave coefficients are determined from the boundary conditions, continuity equations at the driving force locations, and continuity equations at the corner junction of the plates. Structural intensity expressions are used to examine the structural noise transmission in the low and high frequency ranges. The contributions from the individual wave types are also examined.

Modal decomposition of exterior acoustic-structure interaction.

A modal decomposition technique to analyze individual modal contributions to the sound power radiated from an externally excited structure submerged in a heavy fluid is presented. The fluid-loaded structural modes are calculated by means of a polynomial approximation and symmetric linearization of the underlying nonlinear eigenvalue problem. The eigenvalues and eigenfunctions of a fluid loaded sphere with and without internal structures are presented. The modal sound power contributions using both fluid-loaded structural modes and acoustic radiation modes are presented. The results for the resistive and reactive sound power obtained from the superposition of the individual modal sound power contributions are compared to the harmonic solution of the forced problem.

Modal decomposition of exterior acoustic-structure interaction problems with model order reduction

A numerical technique for modal decomposition of the acoustic responses of structures submerged in a heavy fluid medium using fluid-loaded structural modes is presented. A Krylov subspace model order reduction approach to reduce the computational effort required for a fully coupled finite element/boundary element model is described. By applying the Krylov subspace to only the structural part of the global system of equations for the fully coupled problem, only the frequency independent finite element matrices are reduced. A fluid-loaded cylindrical shell closed at each end by hemispherical end caps is examined. The cylinder is excited by a ring of axial or transverse forces acting at one end. The individual contributions of the cylinder circumferential modes to the sound power and directivity of the radiated sound pressure are observed. The technique presented here provides a tool for greater physical insight into exterior acoustic-structure interaction problems using fully coupled numerical models, with significantly reduced computational effort.

DEVELOPMENT OF COUPLED FE/BE MODELS TO INVESTIGATE THE STRUCTURAL AND ACOUSTIC RESPONSES OF A SUBMERGED VESSEL

An analytical model and a fully coupled finite element/boundary element model are developed for a simplified physical model of a submarine. The submerged body is modeled as a ring-stiffened cylindrical shell with finite rigid end closures, separated by bulkheads into a number of compartments and under axial excitation from the propeller-shafting system. Lumped masses are located at each end to maintain a condition of neutral buoyancy. Excitation of the hull axial modes from the propeller-shafting system causes both axial motion of the end closures and radial motion of the shell, resulting in a high level of radiated noise. In the low frequency range, only the axial modes in breathing motion are examined, which gives rise to an axisymmetric case, since these modes are efficient radiators. An expression for the structurally radiated sound pressure contributed by axial movement of the end plates and radial motion of the shell was obtained using the Helmholtz integral equation. In the computational model, the effects of the various influencing factors (ring stiffeners, bulkheads, realistic end closures, and fluid loading) on the free vibrational characteristics of the thin walled cylinder are examined. For both the analytical and computational models, the frequency responses, axial and radial responses of the cylinder, and the radiated sound pressure are compared.

Vibration of fluid loaded conical shells

An analytical model is presented to describe the vibration of a truncated conical shell with fluid loading in the low frequency range. The solution for the dynamic response of the shell is presented in the form of a power series. Fluid loading is taken into account by dividing the shell into narrow strips which are considered to be locally cylindrical. Analytical results are presented for different boundary conditions and have been compared with the computational results from a boundary element model. Limitations of the model to the low frequency range are discussed.

Surface contributions to radiated sound power

This paper presents a method to identify the surface areas of a vibrating structure that contribute to the radiated sound power. The surface contributions of the structure are based on the acoustic radiation modes and are computed for all boundaries of the acoustic domain. The surface contributions are compared to the acoustic intensity, which is a common measure for near-field acoustic energy. Sound intensity usually has positive and negative values that correspond to energy sources and sinks on the surface of the radiating structure. Sound from source and sink areas partially cancel each other and only a fraction of the near-field acoustic energy reaches the far-field. In contrast to the sound intensity, the surface contributions are always positive and no cancelation effects exist. The technique presented here provides a method to localize the relevant radiating surface areas on a vibrating structure. To illustrate the method, the radiated sound power from a baffled square plate is presented.

ENFORCING RECIPROCITY IN NUMERICAL ANALYSIS OF ACOUSTIC RADIATION MODES AND SOUND POWER EVALUATION

By identifying the efficiently radiating acoustic radiation modes of a fluid loaded vibrating structure, the storage requirements of the acoustic impedance matrix for calculation of the sound power using the boundary element method can be greatly reduced. In order to compute the acoustic radiation modes, the impedance matrix needs to be symmetric. However, when using the boundary element method, it is often found that the impedance matrix is not symmetric. This paper describes the origin of the asymmetry of the impedance matrix and presents a simple way to generate symmetry. The introduction of additional errors when symmetrizing the impedance matrix must be avoided. An example is used to demonstrate the behavior of the asymmetry and the effect of symmetrization of the impedance matrix on the sound power. The application of the technique presented in this work to compute the radiated sound power of a submerged marine vessel is discussed.

Active Control of Sound Radiated by a Submarine Hull in Axisymmetric Vibration Using Inertial Actuators

This paper investigates the use of inertial actuators to reduce the sound radiated by a submarine hull under excitation from the propeller. The axial forces from the propeller are tonal at the blade passing frequency. The hull is modeled as a fluid-loaded cylindrical shell with ring stiffeners and equally spaced bulkheads. The cylinder is closed at each end by circular plates and conical end caps. The forces from the propeller are transmitted to the hull by a rigid foundation connected to the propeller shaft. Inertial actuators are used as the structural control inputs. The actuators are arranged in circumferential arrays and attached to the internal end plates of the hull. Two active control techniques corresponding to active vibration control and discrete structural acoustic sensing are implemented to attenuate the structural and acoustic responses of the submarine. In the latter technique, error information on the radiated sound fields is provided by a discrete structural acoustic sensor. An acoustic transfer function is defined to estimate the far field sound pressure from a single point measurement on the hull. The inertial actuators are shown to provide control forces with a magnitude large enough to reduce the sound due to hull vibration.