Showing papers in "Journal of Sound and Vibration in 1995"

The calculation of thermoacoustic oscillations

Abstract: Thermoacoustic oscillations occur in a wide variety of practical applications in which heat is supplied to an acoustic resonator. A simple geometry is investigated systematically to determine the importance of various flow parameters on the frequency of the oscillations. Detailed consideration of elementary examples shows that the form of the coupling between the heat input and the unsteady flow has a crucial effect on the frequency of oscillation. The same elementary examples are used to compare how well (if at all) different calculation methods in the literature account for this influence. A mean flow and a distributed region of heat input significantly complicate the analysis of thermoacoustic oscillations and are often neglected. Model problems are used to illustrate that mean flow effects can become significant even at modest inlet Mach numbers, and to indicate circumstances under which a distributed heat input may be treated as concentrated.

Computation of propagative waves in free rail using a finite element technique

Abstract: A method is presented for the numerical computation of the wavenumbers and associated modes of the cross-section of solid waveguides. The method, based on the finite element technique, is well suited to the computation of both the propagative and the evanescent waves in a straight waveguide with an arbitrary cross-section. The solution is obtained by factorization, with the cross-section of the waveguide being modelled by numerical discretization. The dispersion curves of propagative waves in a free rail are computed by using triangular and quadrilateral “finite elements of the cross-section”. The evolution of cross-section modes as the frequency increases is evaluated and discussed.

Vertical dynamic interaction between train and track influence of wheel and track imperfections

Abstract: The vertical dynamic behaviour is investigated for a railway bogie moving on a rail which is discretely supported, via railpads, by sleepers resting on an elastic foundation. The transient interaction problem is numerically solved by use of an extended state-spacer vector approach in conjunction with a complex modal superposition for the track. Application examples are given in which the influences of three types of practically important imperfections in the compound vehicle/track system are investigated. The first is a sinusoidal corrugation of the railhead and the second a skid flat on the wheel tread (a wheelflat). The third imperfection is a case where a single sleeper has lost its support due to erosion of the ballast. Physical explanations of the calculated interaction behaviour are given.

Model reduction using dynamic and iterated IRS techniques

Abstract: Static or Guyan reduction is widely used to reduce the number of degrees of freedom in a finite element model but it is exact only at zero frequency. The Improved Reduced System (IRS) method makes some allowance for the inertia terms and produces a reduced model which more accurately estimates the modal model of the full system. In this paper the IRS method is extended by obtaining the equivalent transformation based on dynamic rather than static reduction. An iterative algorithm, based on the IRS method, is also described. On convergence this algorithm provides a reduced model which reproduces a subset of the modal model of the full system. The iterative version of the IRS method based on dynamic reduction is also described.

Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity

Abstract: A mechanical system exhibiting combined parametric excitation and clearance type non-linearity is examined analytically and experimentally in an effort to explain complex behavior that is commonly observed in the steady state forced response of rotating machines. The specific case of a preloaded mechanical oscillator having a periodically time-varying stiffness function and subject to a symmetric backlash condition is considered. A generalized solution methodology is proposed based on the harmonic balance method. The resulting non-linear algebraic equations are solved by using a direct Newton-Raphson technique, in which a closed form Jacobian matrix is computed using frequency domain methods. Analytical solutions are validated by comparison with numerical integration results and experimental measurements obtained from a gear dynamics test rig.

Active damping of self-excited torsional vibrations in oil well drillstrings

Abstract: A drillstring used for the drilling of oil or gas wells behaves as a rotating torsional pendulum. The drillstring is rotated at a constant angular velocity by an electric motor, but exhibits superimposed torsional vibrations caused by a non-linear relationship between torque and angular velocity at the rock-crushing tool. The vibrations are self-excited, and disappear when the mean angular velocity of the pendulum is raised above a threshold value. An active damping system is described that strongly reduces the threshold value by using feedback control, thus extending the working range for vibration-free rotation. It operates at the current and the voltage of the electric motor, and can be implemented with only electrical components. The active damping system is interpreted as an extension of the passive tuned vibration absorber for quenching of self-excited vibrations in the form of a resilient foundation, as described by Tondl (1975Journal of Sound and Vibration42(2), 251–260). The concept of quenching self-excited vibrations by modifying the drive system as described in this paper is directly applicable to other engineering systems which are driven by a separately excited DC motor. Furthermore, the concept can be applied to systems driven by a hydraulic motor with a continuously variable flow rate.

Research on thick plate vibration: a literature survey

Abstract: This paper is a review of existing literature on the vibration analysis of thick plates: 132 publications are cited, mostly recent and nearly all in the English language. Attention is mainly devoted to studies based on the Mindlin theory and the modified Mindlin plate theories for laminated plates, while some papers using high order shear deformation plate theories are also included. The aim of this paper is to provide a contemporarily relevant survey of studies on the vibration of thick plates. Studies are categorized by different plate shapes. This review paper should be useful for scientists and researchers in assisting them to locate relevant existing literature quickly.

Vibration of non-uniform rods and beams

Abstract: We show that for some non-uniform rods and beams the equation of motion can be transformed into the equation of motion for a uniform rod or beam. Then, when the ends are completely fixed, the eigenvalues of the non-uniform continuum are the same as those of uniform rods or beams. For other end support conditions, exact solutions are obtained. An efficient procedure is used to analyze the free vibration of non-uniform beams with general shape and arbitrary boundary conditions. Simple formulas are presented for predicting the fundamental natural frequency of non-uniform beams with various end support conditions.

On the direct solution of wave propagation for repetitive structures

Abstract: The direct method of solving the wave constants for a repetitive structure with given frequency ω is developed in this paper. The analogy between structural mechanics and optimal control theory is applied. The dynamic stiffness matrix approach is used to solve for the eigensolutions of wave propagation. The weighted adjoint symplectic orthonormality relationship and the eigenvector expansion theorem are also established for this approach. The symplectic eigenproblem is derived for skew-symmetric matrices, a cell triangular decomposition for skew-symmetric matrices is introduced to reduce the generalized eigenproblem, and the symplectic Householder transformation is introduced to cell tri-diagonalize the skew-symmetric matrix to solve the eigenequation. The problem of a wave impinging on the boundary is considered by the eigenvector expansion method, which could also be applied to a wave scattering when passing over an abnormal part. Numerical examples show that resonance will occur for the localized vibration case.

Effects of coupling misalignment on vibrations of rotating machinery

Abstract: Improper aligning of shafts through couplings often leads to severe vibration problems in many rotating machines. The rotor-bearing system is modelled using higher order finite elements by considering deflection, slope, shear force, bending moment with eight degrees of freedom per node. The reaction forces, moments developed due to flexible coupling misalignment are derived and introduced in the model. The imbalance response in two harmonics is evaluated. The increase in harmonics with misalignment can easily be modelled by using FEM analysis. The location of the coupling with respect to the bending mode shape has a strong influence on the vibrations.

Dynamics of flexible beams undergoing overall motions

Abstract: A modelling method for straight beams undergoing large overall motions as well as small elastic deformations is presented in this paper. Different from the classical linear Cartesian modelling method which employs three Cartesian deformation variables, the present modelling method uses a non-Cartesian variable along with two Cartesian variables to describe the elastic deformation. A quadratic form of the strain energy expressed with the hybrid set of deformation variables is used to obtain the generalized active forces and a geometric constraint equation relating the non-Cartesian variable and Cartesian variables is used to obtain the generalized inertia forces in the equations of motion. The present modelling method not only provides accurate simulation results but also clarifies the limit of validity of the classical linear Cartesian modelling method.

A general method of modelling sound propagation in layered media

Abstract: A general method of modelling acoustic fields in stratified media which include elastic solid, fluid and porous layers is presented. The simplicity and the versatility of the method is illustrated with several examples.

Simple models of the energetics of transversely vibrating plates

Abstract: Approximate energy models for infinite and finite plates are derived using three relationships; an energy balance, a loss factor damping model and an approximate energy transmission model. The energy transmission relationship for the infinite plate relates the far field radial intensity, the group speed and the local energy density. The resulting energy equation for infinite plates is an excellent approximation in the far field. For finite plates, the far field energy density and intensity expressions for plane wave approximations are smoothed in order to derive an energy transmission relationship. The resulting relationship is analogous to Fourier’s law of heat conduction. The energy model is a second order equation which models the smoothed far field energy distribution. The equations model the general behavior of finite plates well and explain the dependence of plate energetics on frequency and damping.

A non-linear dynamic model for cables and its application to a cable-structure system

Abstract: A set of governing equations for dynamic transverse motions of a cable with small sag is firstly obtained where effects of finite motions of the cable and small support motions are included. Cable motions are separated into two parts; quasi-static motions and modal motions. The quasi-static motions are the displacements of the cable which moves as an elastic tendon due to the support movements. The modal motions are expressed as a combination of the linear undamped modes of a cable with fixed ends. By Lagrange's equations of motion, the governing equations of the non-linear cable motions are obtained, where quadratic as well as cubic non-linear couplings appear. The cable model developed is next applied to a cable–structure system. A global/local mode approach is employed; the total motions are expressed in terms of global and local motions. The local motions are the modal motions of the cable, while the global motions are 3-D motions of the structure which include quasi-static motions of the cables only. The global are expressed as a combination of the eigenmodes computed by 3-D FEM in which cables are treated as tendons. By using Lagrange's formulation, algebraic governing equations are finally obtained in which global–local interaction appears as linear and quadratic couplings. The model for the system is modified to include the actuator motions at the cable supports; the actuator motions are in the cable axis direction. The study shows many possibilities for the control of global and/or local modes.

Experimental verification of flexibility difference method for locating damage in structures

Abstract: A flexibility difference method for locating damage in structures is presented. This method is based on the estimation of changes in the flexibility matrix. First, with the help of an analytical model of beams, it is shown that a clear pattern exists for the changes in the flexibility matrix produced due to the presence of cracks. Next, the method is applied to several damage cases in three wide-flange steel beams. Damage in the beams is created by cutting through the beam using a saw. Experimental modal analysis techniques are used to estimate the natural frequencies and mode shapes for both the intact and the damaged beams. The measured natural frequencies and mode shapes are then used to estimate the flexibility matrix. Using only the lowest three modes, the flexibility difference method is successful in locating a 13 mm long saw cut in a 102 mm wide flange. The flexibility difference method is also successful in locating multiple damage locations in the beam.

Scattering of flexural waves on thin plates

Abstract: Some general results are presented concerning the scattering of flexural waves from regions of inhomogeneity on flat plates. We derive a flux conservation relation for arbitrary motion, and show that it simplifies for periodic motion. An optical theorem is obtained relating the total scattered flux for a straight-crested incident wave to the scattering amplitude in the forward direction. Scattering by circular inclusions with different plate properties is discussed, and numerical results are presented. The response simplifies for the limiting cases of a hole and a rigid obstacle, but with quite different behavior for each. A rigid obstacle can produce an unbounded scattering cross-section as the frequency tends to zero, whereas the cross-section of a hole vanishes in the same limit.

Analysis of rotating laminated cylindrical shells by different thin shell theories

Abstract: The natural frequencies of the forward and backward modes of thin rotating laminated cylindrical shells are determined by using four common thin shell theories, namely Donnell’s, Flugge’s, Love’s and Sanders’. The objective is to present a unified analysis, formulated with the use of tracers so that it can be reduced to any of the four shell theories by giving appropriate values to the tracers. For simplicity, results are presented only for the case of simply supported-simply supported boundary conditions, which can be satisfied by expressing the displacement fields in terms of the products of sine and cosine functions. Numerical results presented are the non-dimensional frequency parameters of the forward and backward travelling modes for rotating cylindrical shells and the non-dimensional frequency parameters for non-rotating cylindrical shells. The present analysis is verified by comparing the numerical results with those in the literature and very good agreement is obtained.

Vibration and damping analysis of a three-layered composite plate with a viscoelastic mid-layer

Abstract: The natural frequencies and loss factors of a rectangular three-layered plate with a viscoelastic core layer and laminated face layers are considered. The equations of free vibration of the plate together with the corresponding boundary conditions are derived for a non-symmetric plate with general anisotropy of the face layers. A first order shear deformation theory is used to describe the deformation of the faces. Simplified forms of these equations, for a symmetric plate or for specially orthotropic face layers, are then discussed. Equations are also given for a model with no shear deformation in the face layers and all but transverse inertia terms neglected (simplified model). Results of numerical examples are presented for simply supported plates with specially orthotropic face layers. Complex eigenvalues are found numerically, and from these, both frequencies and loss factors are extracted. Comparison is made between the shear deformation and the simplified models, in the case of high modulus composite face layers.

Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions

Abstract: A four-degree-of-freedom model able to capture the main phenomena which are likely to occur in the non-planar finite dynamics of an elastic suspended cable subjected to external forcings and support motions is developed from the continuum equations. It contains two in-plane and two out-of-plane components, one symmetric and one antisymmetric. An order two multiple time scales solution is obtained for the case of primary external resonance and three simultaneous internal resonances (cable at crossover frequency). Possible classes of steady state (planar/non-planar, unimodal/multimodal) regular motions of the system are identified, and their linearized stability analysis is performed. Some results are presented to highlight the role played by key perturbations on the onset of various classes, the rich behaviour of the system and the influence of some control parameters on the response.

High amplitude vortex-induced pulsations in a gas transport system

Abstract: High Reynolds number, low Mach number gas flows in pipe systems with closed side branches exhibit spectacular low frequency self-sustained pulsations driven by periodic vortex shedding at specific values of the Strouhal number. A detailed study is presented of the behaviour of the flow in a system with two opposite closed side branches of equal length in a cross configuration. For junctions with both sharp and rounded edges the acoustic flow velocity amplitude is comparable to the main flow velocity. A two-dimensional potential flow model based on the vortex blob method, used to simulate the flow in the junction, describes accurately the flow visualization and laser Doppler data obtained in pipes with square cross-sections and with sharp edged junctions. The numerical simulation is used to calculate the acoustical power generated by the vortical flow at a given amplitude of the acoustic velocity field and Strouhal number. In reality, for a pulsation with constant amplitude, this power is balanced by the viscothermal losses and acoustic radiation, which is the basis for the indirect measurement of the source power. It is shown that, for the acoustic amplitude observed, radiation losses due to the generation of non-resonating harmonics by wavesteepening has to be taken into account in the energy balance. This finding is confirmed by the appearance of shock waves in the geometry with rounded edges.

A study of passive and adaptive hydraulic engine mount systems with emphasis on non-linear characteristics

Abstract: Passive hydraulic mounts exhibit excitation frequency variant and deflection amplitude sensitive stiffness and damping properties. Such non-linear dynamic characteristics are examined by using analytical and experimental methods, both at the device level and within the context of a simplified vehicle model. A new lumped parameter non-linear mathematical model of the hydraulic mount is developed by simulating its decoupler switching mechanism and inertia track dynamics. The low frequency performance features and limitations of several passive mounts are made clear through the non-linear vehicle model simulation and comparable laboratory vibration tests. The high frequency performance problems of the passive hydraulic mount are identified by applying the quasi-linear analysis method. Based on these results, a new adaptive mount system is developed which exhibits broad bandwidth performance features up to 250 Hz. It implements an on-off damping control mode by using engine intake manifold vacuum and a microprocessor based solenoid valve controller. A laboratory bench set-up has already demonstrated its operational feasibility. Through analytical methods, it is observed that our adaptive mount provides superior dynamic performance to passive engine mounts and comparable performance to a small scale active mount over a wide frequency range, given the engine mounting resonance control, shock absorption and vibration isolation performance requirements. Although technical prospects of the proposed adaptive system appear promising, thein situperformance needs to be evaluated.

Modelling and vibration analysis of a cantilever composite beam with a transverse open crack

Abstract: Eigenfrequencies of a cantilever beam, made from graphite-fiber reinforced polyimide, with a transverse on-edge non-propagating open crack are investigated. Two models of the beam are presented. In the first model the crack is modelled by a massless substitute spring. The flexibility of the spring is calculated on the basis of fracture mechanics and the Castigliano theorem. The second model is based on the finite element method (FEM). The undamaged parts of the beam are modelled by beam finite elements with three nodes and three degrees of freedom at the node. The damaged part of the beam is replaced by the cracked beam finite element with degrees of freedom identical to those of the non-cracked one. The effects of various parameters (the crack location, the crack depth, the volume fraction of fibers and the fibers’ orientation) upon the changes of the natural frequencies of the beam are studied. Computation results indicate that the decrease of the natural frequencies not only depends on the position of the crack and its depth, as in the case of isotropic material, but also that these changes strongly depend on the volume fraction of the fibers and the angle of the fibers of the composite material.

Vibration analysis of beams traversed by uniform partially distributed moving masses

Abstract: An investigation into the dynamic behavior of beams with simply supported boundary conditions, carrying either uniform partially distributed moving masses or forces, has been carried out. The present analysis in its general form may well be applied to beams with various boundary conditions. However, the results from the computer simulation model given in this paper are for beams with simply supported end conditions. Results from the numerical solutions of the differential equations of motion are shown graphically and their close agreement, in some extreme cases, with those published previously by the authors is demonstrated. It is shown that the inertial effect of the moving mass is of importance in the dynamic behavior of such structures. Moreover, when considering the maximum deflection for the mid-span of the beam, the critical speeds of the moving load have been evaluated. It is also verified that the length of the distributed moving mass affects the dynamic response considerably. These effects are shown to be of significant practical importance when designing beam-type structures such as long suspension and railway bridges.

Effects of boundary conditions on frequencies of a multi-layered cylindrical shell

Abstract: A formulation based on Love's first approximation theory and with beam functions used as axial modal functions in the Ritz procedure is used to study the effects of boundary conditions on the free vibration characteristics for a multi-layered cylindrical shell. Altogether, nine different boundary conditions are considered. Four of the boundary conditions are with ends having the same end conditions: clamped–clamped, free–free, simply supported–simple supported and sliding–sliding. The other five boundary conditions are with ends having different end conditions: clamped–free, clamped–simply supported, clamped–sliding, free–simply supported and free-sliding. Mode shapes for the fundamental frequencies of nine boundary conditions are also presented. To validate the analysis, comparisons of frequency parameters with those from exact three-dimensional linear elasticity analysis, Flugge theory, higher order displacement analysis and shear deformation theory in the literature are made; very good agreement is achieved by using the present method with results available in the literature.

A measurement method for determining the source data of acoustic two-port sources

Abstract: Acoustic source data for fluid machines are of importance for calculating the acoustic field generated in a duct system and for analysis of source-load interaction effects. In this paper the problem of measuring source data for acoustic two-port sources in general is first described and critically discussed. A measurement method to determine the source data of an acoustic two-port source in the form of a source strength vector and a scattering matrix is then presented. This method consists of two steps. First the scattering matrix is determined by using external noise sources to create two independent acoustic test cases. As the second step, the source strength is determined by using a number of different acoustical loads to obtain an overdetermined problem. By solving the overdetermined matrix problem one can improve the prediction of the source data and it is also possible to verify whether or not the machine under test behaves as a linear source. For this purpose a special linearity measure has been defined. The measurement method has been verified and also successfully tested on an axial flow fan.

Identification of joint properties of a structure using frf data

Abstract: The connections between structural components, i.e., joints, are extremely difficult to model accurately by using a pure analytical approach. Alternatively, joint properties can be extracted from experimental data. A method for identifying (extracting) joint properties from the measured frequency response function (FRF) data is presented in this paper. One of the major difficulties in accurate identification of joint properties is that the identification results can be significantly affected by various errors in the measured data. Even if errors in the measured data are at an insignificantly low level (say, 5% in FRFs), their effects on the identification can be severe enough so that in some cases the predicted joint properties can be completely irrelevant to the true properties of joints. Apart from the accuracy of the data used for the identification, the accuracy of the identification also significantly depends upon the mathematical formulae used for the identification. Even with the same experimental data, the accuracies of the identification by different methods can be significantly different. Therefore, developing methods that are insensitive to measurement errors is of utmost importance for the identification of joint properties from experimental data. In this paper, techniques for improving the accuracy of joint identification are discussed. It is demonstrated that by using all available information effectively, the accuracy of the identification can be much improved. Both numerical and experimental examples are presented.

The effect of frequency and clearance variations on single-degree-of-freedom impact oscillators

Abstract: A simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro-impacting systems. The presence of impacts makes the system discontinuous and non-linear and to study it effectively the theory of dynamical systems is used. By a combination of analytical and numerical techniques, many phenomena are identified which are characteristic of non-linear dynamical systems as the system parameters are varied. In particular, these include chaotic regimes and multiply coexisting stable states. The domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states. In addition, phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems; chatter, when infinitely many impacts occur in a finite time, and grazing, when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters.

Dynamic behaviour of beams and rectangular plates under moving loads

Abstract: A theory concerning the dynamic response of finite elastic structures (Rayleigh beams and plates) having arbitrary end supports and under an arbitrary number of moving masses is developed. The versatile solution technique presented is based on modified generalized finite integral transforms and the modified Struble’s method. Various analytical results are presented. Numerical examples are given and the results compare favourably with existing ones. The efficiency of the solution technique is demonstrated and discussed.

On crack identification and characterization in a beam by non-linear vibration analysis

Abstract: The weakly non-linear character of a cracked vibrating beam is exploited for the purpose of determining crack location, depth and opening load. The approach is motivated by examining the response of a bilinear spring-mass system to excitation at two frequencies, such that the difference between the two frequencies is the resonant frequency of the system. The numerically generated steady state response of the system clearly betrays the presence of the bilinear spring, even if the difference between the compressive and tensile stiffness is very small. The same idea is applied to a cracked beam forced at two frequencies, with the crack providing a local bilinear stiffness in the beam. The numerically generated steady state response shows the effect of the opening and closing of the crack. The prominence of this non-linear effect is then correlated with crack position and depth. It is shown that the non-linear effect is maximized if a static load is also placed on the beam that would cause the crack to be on the verge of opening, thus determining the opening load.

A modal strain energy approach to the prediction of resistively shunted piezoceramic damping

Abstract: The use of piezoceramic materials with resistive shunting circuits has previously been shown to increase passive structural vibration damping The ability to tailor the frequency dependence of damping is especially attractive when active linear time-invariant control of uncertain structures is to be attempted A method for predicting the damping performance of resistively shunted piezoceramics based on a variation of the modal strain energy approach has recently been developed Using this approach, the damping for a structural mode of vibration may be found as the product of the effective fraction of modal strain energy stored in the piezoceramic material, an effective piezoceramic material loss factor and a frequency shaping factor A finite element model may be used to accurately determine the effective modal strain energy fraction; the effective material loss factor is closely related to the piezoceramic electromechanical coupling coefficient; and the frequency shaping factor results from the dynamics of the shunting circuit Design concerns include the effect of stiff piezoceramic material on mode shapes, the frequency dependence of piezoceramic elastic properties, and the effect of adhesive on load transfer from the structure to the piezoceramic Analytical and experimental results are presented for a uniform cantilevered beam with two pairs of resistively shunted piezoceramic plates The results show good agreement between predicted and measured added damping