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.     

A new method for harmonic response of non-proportionally damped structures using undamped modal data

A method of calculating the receptances of a non-proportionally damped structure from the undamped modal data and the damping matrix of the system is presented. The method developed is an exact method. It gives exact results when exact undamped receptances are employed in the computation. Inaccuracies are due to the truncations made in the calculation of undamped receptances. Numerical examples, demonstrating the accuracy and speed of the method when truncated receptance series are used are also presented. Advantages of the method over classical methods are discussed, and it is concluded that the method is most advantageous when used for a structure with frequency and/or temperature dependent damping properties, or when the non-proportional part of the damping is local. The technique suggested can easily be applied to structural modification problems if there is no additional degree-of-freedom due to the modifying structure.     

Stability of damped membranes and plates with distributed inputs

This paper proves the stability of boundary and distributed damped membranes and Kirchhoff plates under distributed inputs. Distributed viscous or Kelvin-Voigt damping ensures a weakly bounded response to a bounded transverse loading for pinned membranes and clamped plates. Damping on part of the boundary can also weakly stabilize the forced response, provided the damped and undamped boundary normals satisfy certain conditions. For example, damping on half and one side of the boundary is sufficient for circular and rectangular domains, respectively.     

Effect of geometrical and material imperfections on damping flexural vibrations in plates with attached wedges of power law profile

In the present paper, an efficient method of damping structural vibrations using the acoustic black hole effect is further investigated experimentally. This method is based on some specific properties of flexural wave propagation in tapered plates (wedges) of power-law profile that have to be partially covered by narrow thin strips of absorbing layers. Ideally, if the power-law exponent of the profile is equal or larger than two, the flexural wave never reaches the sharp edge and therefore never reflects back, which constitutes the acoustic black hole effect. It has been previously established theoretically and confirmed experimentally that this method of damping structural vibrations is very efficient even in the presence of edge truncations. The present work describes the results of the experimental studies of the effects of manufacturing intolerances on damping flexural vibrations in wedge-like structures of power-law profile. In particular, the effect of mechanical damage resulting from the use of cutting tools to wedge tips is investigated, including tip curling and early truncation, as well as the placement of absorbing layers on different wedge surfaces. Also, the effects of welded and glued bonding of wedge attachments to basic rectangular plates (strips) are investigated. The results show that, although the above-mentioned geometrical and material imperfections reduce the damping efficiency by varying degrees, the method of damping structural vibrations using the acoustic black hole effect is robust enough and can be used widely without the need of high precision manufacturing.     

Effects of hysteretic and aerodynamic damping on supersonic panel flutter of composite plates

The governing equation for the finite element analysis of the panel flutter of composite plates including structural damping is derived from Hamilton's principle. The first order shear deformable plate theory has been applied to structural modelling so as to obtain the finite element eigenvalue equation. The unsteady aerodynamic load in a supersonic flow is computed by using the linear piston theory. The critical dynamic pressures for composite plates have been calculated to investigate the effects of structural damping on flutter boundaries. The effects are dependent on fiber orientation because flutter mode can be weak or strong in the fiber orientation of composite plates. Structural damping plays an important role in flutter stability with low aerodynamic damping but would not affect the flutter boundary with high aerodynamic damping.     

Time-domain simulation of damped impacted plates. II. Numerical model and results

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper [Part I, J. Acoust. Soc. Am. 109, 1422-1432 (2001)]. In this paper (Part II), the damped-plate model is extended to impact excitation, using Hertz's law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials (aluminum, glass, carbon fiber and wood). The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity.     

Refined vibration and damping analysis of multilayered rectangular plates

Refined vibration and damping analysis of a general multilayered rectangular plate consisting of an arbitrary number of layers of orthotropic materials has been developed by considering extension, bending, in-plane shear and transverse shear deformations in all the layers and taking into account the rotary and longitudinal translatory inertias along with the transverse inertia of the plate. The solution for a multilayered plate with simply supported edges has been taken in series summation form and resonating frequencies and associated loss factors for plates with alternate elastic and viscoelastic layers have been evaluated by application of the correspondence principle of linear viscoelasticity. Results for three-, five- and seven-layered plates obtained by the present refined analysis are compared with the results obtained by conventional analysis of multilayered plates.     

A fuzzy finite element procedure for the calculation of uncertain frequency-response functions of damped structures: Part 1—Procedure

This work introduces a numerical algorithm to calculate frequency-response functions (FRFs) of damped finite element (FE) models with fuzzy uncertain parameters. Part one of this paper describes the numerical algorithm for the solution of the underlying interval finite element (IFE) problem. First, the IFE procedure for the calculation of undamped envelope FRFs is discussed. Starting from the undamped procedure, a strategy is developed to analyse damped structures based on the principle of Rayleigh damping. This is achieved by analysing the effect of the proportional damping coefficients on the subsequent steps of the undamped procedure. This finally results in a procedure for the calculation of fuzzy damped FRFs based on an analytical extension of the undamped algorithm. Part one of this paper introduces the numerical procedure. Part two of this paper illustrates the application of the methodology on four numerical case studies.     

Inspection on SiC coated carbon–carbon composite with subsurface defects using pulsed thermography

An investigation on SiC coated carbon–carbon (C/C) composite plates has been undertaken by pulsed thermography. The heat transfer model has been built and the finite element method (FEM) is applied to solve the thermal model. The simulation results show that defects with DA/DP smaller than one can hardly be detected by an infrared camera with the sensitivity of 0.02 °C. Certificated experiments were performed on the built pulsed thermography system. The thermal wave signals have been processed by subtracting background image method (SBIM), pulsed phase thermography (PPT), and temperature–time logarithm fitting method (TtLFM). The limit DA/DP of defects in SiC coated C/C composite plates with the thickness of 6 mm that can be detected by pulsed thermography with the presented signal analysis algorithms has been obtained.     

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.     

Resonance frequency of microbubbles: effect of viscosity

The transmitted frequency at which a gas bubble of millimeter or submillimeter size oscillates resonantly in a low-viscosity liquid is approximately equal to the undamped natural frequency (referred to as the Minnaert frequency if surface tension effects are disregarded). Based on a theoretical analysis of bubble oscillation, this paper shows that such an approximation cannot be validated for microbubbles used in contrast-enhanced ultrasound imaging. The contrast-agent microbubbles represent either encapsulated bubbles of size less than 10 microm or free (nonencapsulated) bubbles of submicron size. The resonance frequency of the microbubbles deviates significantly from the undamped natural frequency over the whole range of microbubble sizes due to the increased viscous damping coefficient. The difference between these two frequencies is shown to have a tremendous impact on the resonant backscatter by the microbubbles. In particular, the first and second harmonics of the backscattered signal from the microbubbles are characterized by their own resonance frequencies, equal to neither the microbubble resonance frequency nor the undamped natural frequency.     

Inverse iteration for damped natural vibration

Inverse iteration is extended to internally and/or externally damped natural vibration. Each iteration involves one matrix multiplication and one linear equation solution of order n. The symmetric band form of the original undamped eigenvalue problem is preserved. If the undamped mode is taken as the first approximation, the inverse iteration will converse to the corresponding damped mode in about four iterations. However, the one step method is divergent for heavy damping. Therefore, it is advisible to subdivide the damping into successive steps if inverse iteration does not converge in say five iterations. The method is successful for both discrete systems and distributed systems. The implementation is very simple by means of complex arithmetic which is readily available in many FORTRAN compilers.     

Damped vibration analysis of an elastically connected complex double-string system

The main purpose of the present paper is to consider theoretically damped transverse vibrations of an elastically connected double-string system. This system is treated as two viscoelastic strings with a Kelvin-Voigt viscoelastic layer between them. A theoretical analysis has been made for a simplified model of the system, in which assumed physical parameters make it possible to decouple the governing equations of motion by introducing the principal co-ordinates. Applying the method of separation of variables and the modal expansion method, exact analytical solutions for damped free and forced responses of the system subjected to arbitrarily distributed transverse continuous loads are determined in the case of arbitrary magnitude of linear viscous damping. It is important to note that the solutions obtained are explicitly expressed in terms of parameters characterizing the physical properties of the system under discussion. For the sake of completeness of the analysis, solutions for undamped free and forced vibrations are also formulated.     

Damping lateral vibrations in rotary machinery using motor speed modulation

This paper outlines a new principle for damping lateral vibrations of rotary systems. According to this principle, no changes in the visco-elastic properties of the system to be damped are required. The method is based on the generation of a harmonic additive to the constant speed of rotation that provides significant damping of lateral vibrations at critical speeds of rotation. This concept is validated analytically using the method of averaging and additionally with the help of direct numerical integration. The solution is shown to represent Fourier series containing Bessel functions. Consequently, proper choice of the parameters of the additional harmonic component ensuring that the Bessel functions have minimum values results from a minimization of the solution itself. Thus, the analytical solution and numerical results prove this concept by showing an essential decrease of the amplitudes of lateral vibrations of the damped system compared with those of the undamped system. The physical explanation of this effect is presented.     

An approximate solution for the bending vibrations of a combination of rectangular thin plates

The appropriate method often used for calculating the bending vibration of a single rectangular plate is extended to calculate the bending vibrations of a global system of combinations of rectangular plates with elastically supported and damped non-coupled edges. Two examples, a series of T-combinations and an L-combination of rectangular thin isotropic plates, are considered and the input and transfer mobilities due to point excitation derived. Numerical results are presented for the case of combinations of concrete plates and the effects of varying the material damping of the plates and edge damping are investigated.The eigenfrequencies of an L-combination of plates with one plate of very high bending stiffness are calculated and results compare well with the eigenfrequencies of a single plate calculated by means of the classical Ritz-Rayleigh method.     

Influence of non-linear forces on beam behaviour in flutter conditions

Appropriate researches on non-linear panel flutter behaviour have been already performed by many authors. In most cases the intent of them focuses on the limit cycle determination, with particular interest towards its amplitude versus the flow dynamic pressure. This paper deals first with a study of all the solutions without damping of beam flutter versus the vibration frequency in non-linear post-critical conditions. A numerical model, which takes into account the influence of the non-linear contribution of the structural forces, due to the axial stretching of the beam, has been implemented. A complete analysis of all the possible non-linear solutions without damping leads to the possibility of characterizing the most appropriate conditions for the presence of the post-critical panel flutter limit cycles. Then the complete model, which also takes into account aerodynamic damping, has been utilized, according to the “Piston Theory”, to verify the state evolution of the fluttering damped beam towards the limit cycle, which is very near to the undamped vibrating beam state with minimum amplitude. This convergence test is an interesting aspect of the numerical results.     

Analysis of an anisotropic plate Faraday rotator coated with multilayer dielectric plates

The analysis of an anisotropic plate Faraday rotator coated with multilayer dielectric plates is presented by means of the matrix methods in conjunction with Fourier transformation technigues. It provides a closed-form representation of the electromagnetic field over the whole multilayer regions. The treatment does not limit the direction of the biased magnetic field and the numbers of the coated dielectric plates. Numerical results are presented and compared with that calculated with the method in ref.[1], and full-waveguide-band bandwidth for 47.5°±1.5° rotation angle is obtained for transmission type rotator in W band. Calculation for reflection type rotator is also presented and compared with the experiments.     

Vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment

The free vibration analysis of functionally graded annular plates with mixed boundary conditions in thermal environment is carried out by the 3D elasticity theory and the Chebyshev–Ritz method. The material properties are assumed to be temperature dependent and graded in the thickness direction. The mixed boundary conditions which include upper and lower surfaces partially fixed, inner side partially fixed and outer side partially fixed are considered, respectively. The accuracy of the present approach for solving the free vibration of the plates with different boundary conditions is validated by comparing the present numerical results with the results available. The effects of the different mixed boundary conditions, the temperature rise, the material graded index and the geometrical parameters on the eigen-frequencies are studied.     

Vibration and damping analysis of multilayered rectangular plates with constrained viscoelastic layers

Governing equations of motion for vibrations of a general multilayered plate consisting of an arbitrary number of alternate stiff and soft layers of orthotropic materials are derived by using variational principles. Extension, bending and in-plane shear deformations in stiff layers and only transverse shear deformations in soft layers are considered as in conventional sandwich structural analysis. In addition to transverse inertia, longitudinal translatory and rotary inertias are included, as such analysis gives higher order modes of vibration and leads to accurate results for relatively thick plates. Vibration and damping analysis of rectangular simply supported plates consisting of alternate elastic and viscoelastic layers is carried out by taking a series solution and applying the correspondence principle of linear viscoelasticity. The damping effectiveness, in term of the system loss factor, for all families of modes for three-, five- and seven-layered plates is evaluated and its variations with geometrical and material property parameters are investigated.     

Electrical capacitance of dielectric coated metallic parallelepiped and closed cylinder isolated in free space

This paper presents a method for the evaluation of the capacitance and the charge distribution of a dielectric coated metallic parallelepiped and a dielectric coated metallic hollow cylinder with the top and bottom cover plates using the method of moments (MoM) based on the pulse basis function and the point matching. Boundary conditions for the potential on the conductor surfaces and continuity of the normal component of the displacement density at the dielectric-free space interface is used to generate two integral equations. Two sets of simultaneous equations are formed from the two integral equations using the MoM. The total charge on the conductor surface is found from the solution for the set of simultaneous equations. The validity of the analysis has been justified by comparing the data on the capacitance available in the literature for metallic cube and hollow cylinder with top and bottom cover plates with the data on capacitance, computed by the present method for similar structures considering a very low dielectric constant as well as a very thin dielectric coating.