Frequency Domain Analysis for Detecting Pipeline Leaks

This paper introduces leak detection methods that involve the injection of a fluid transient into the pipeline, with the resultant transient trace analyzed in the frequency domain. Two methods of leak detection using the frequency response of the pipeline are proposed. The inverse resonance method involves matching the modeled frequency responses to those observed to determine the leak parameters. The peak-sequencing method determines the region in which the leak is located by comparing the relative sizes between peaks in the frequency response diagram. It was found that a unique pattern was induced on the peaks of the frequency response for each specific location of the leak within the pipeline. The leak location can be determined by matching the observed pattern to patterns generated numerically within a lookup table. The procedure for extracting the linear frequency response diagram, including the optimum measurement position, the effect of unsteady friction, and the way in which the technique can be extended into pipeline networks, are also discussed within the paper.     

Range of Validity of the Transient Damping Leakage Detection Method

In a recent paper, an elegant, efficient, and easy to apply transient-based leakage detection method was proposed. The method exploits the fact that friction and leakage damp the modes of transient waves in a different manner. The method involves six major assumptions. These are: (1) the periodic motion in time of each mode is linearly independent of all other modes; (2) the amplitude of the induced transient is small; (3) the magnitude of the leak is small in comparison with the flow rate; (4) the wall friction can be represented by the Darcy–Weisbach equation; (5) the transient is initiated by an instantaneous small amplitude disturbance; and (6) the pipe system is a simple reservoir–pipe–valve type system or reservoir–pipe–reservoir type system. These six assumptions are relaxed and the validity of the transient damping method is assessed. The analysis shows that the first four assumptions do not pose any serious restriction to the applicability of the damping rate method provided that the mathematical model, used to generate the transient head trace in the leak-free pipe, accurately represents the frictional damping in the system. On the other hand, Assumptions (5) and (6) restrict the applicability of the method to systems that do not involve internal boundary conditions, such as junctions and pumps, and to transients triggered by impulses whose duration is smaller than the wave travel time. Extension of this method to complex pipe systems requires that the linearized waterhammer equations are solved under more general initial and boundary conditions. In addition, more investigation in relation to the frequency content of the input signal and its importance in leakage detection is warranted. The general framework used to derive the damping rate method has led to an efficient and direct algorithm for identifying leaks and future research should seek ways to adapt this framework to more complex pipe systems.     

Extensive Development of Leak Detection Algorithm by Impulse Response Method

The oscillatory flows in pipeline systems due to excitation by valve operation are efficiently analyzed by the impulse response method. The impact of leakage is incorporated into the transfer functions of the complex head and discharge. Frequency-dependent friction is used to consider the impact of unsteady friction for laminar condition. Extensive development of the impulse response method was made by considering the sources of friction associated with the local and convective acceleration of velocity for turbulent flow. The genetic algorithm was integrated into the impulse response method to calibrate the location and the quantity of leakage. The calibration function for leakage detection can be made using the pressure-head response at the valve, or the pressure-head and flow response at the section upstream from the valve. The proposed leak detection algorithm shows the potentials for being applied to a simple pipeline system with a single leak or multiple leaks.     

减振复合板阻尼测试样品模态分析及尺寸确定

针对减振复合板样品在阻尼测试中样品尺寸标准不一,样品尺寸在确定时缺乏理论依据,无法确定测试样品尺寸的问题,利用有限元方法对减振复合板进行模态分析,用悬臂梁法进行阻尼实测。有限元模拟结果表明,在一定的宽度和厚度下,随着试样长度的增加,其1阶到6阶共振频率均出现降低的规律;在一定的长度和厚度下,随着样品宽度的增加,前4阶共振频率有略微增加的趋势;在一定的长度和宽度下,随着样品厚度的增加,1阶到3阶的共振频率基本保持稳定。实测结果表明,选取自由端长度为216 mm、宽度为12.7 mm的样品,利用2阶和3阶共振频率下的损耗因子进行阻尼表征较为合理。     

Partial Blockage Detection in Pipelines by Frequency Response Method

A new technique is presented utilizing the frequency response for the detection of partial blockages in a pipeline. In the system frequency response, a partial blockage increases the amplitude of the pressure oscillations at even harmonics. Such an increase in amplitude has an oscillatory pattern, the frequency and amplitude of which may be used to predict the location and size of a partial blockage. In this technique, the pressure transient history at only one location is sufficient, and the history of the transient in the pipe prior to blockage is not needed, which is an advantage over a number of other available techniques, in addition to being simpler to use. It is shown that the technique successfully detects the location of a blockage in a number of simple systems with blockage size as small as 10%. The technique is verified by comparing the computed results with those computed by the method of characteristics and with measurements from simple laboratory setups. A number of practical issues and limitations for field implementations are discussed.     

A Note on Nonproportional Damping

This note deals with three aspects of nonproportional damping in linear damped vibrating systems in which the stiffness and damping matrices are not restricted to being symmetric and positive definite. First, we give results on approximating a general damping matrix by one that commutes with the stiffness matrix when the stiffness matrix is a general diagonalizable matrix, and the damping and stiffness matrices do not commute. The criterion we use for carrying out this approximation is closeness in Euclidean norm between the actual damping matrix and its approximant. When the eigenvalues of the stiffness matrix are all distinct, the best approximant provides justification for the usual practice in structural analysis of disregarding the off-diagonal terms in the transformed damping matrix. However, when the eigenvalues of the stiffness matrix are not distinct, the best approximant to a general damping matrix turns out to be related to a block diagonal matrix, and the aforementioned approximation cannot be justified on the basis of the criterion used here. In this case, even when the damping and stiffness matrices commute, decoupling of the modes is not guaranteed. We show that for general matrices, even for symmetric ones, the response of the approximate system and the actual system can be widely different, in fact qualitatively so. Examples illustrating our results are provided. Second, we present some results related to the difficulty in handling general, nonproportionally damped systems, in which the damping matrix may be indefinite, by considering a simple example of a two degrees-of-freedom system. Last, we use this example to point out the nonintuitive response behavior of general nonproportionally damped systems when the damping matrix is indefinite. Our results point to the need for great caution in approximating nonproportionally damped systems by damping matrices that commute with the stiffness matrix, especially when considering general damping matrices. Such approximations could lead to qualitatively differing responses between the actual system and its proportionally damped approximation.     

Attenuation and Damping of Multireflected Transient Elastic Waves in a Pile

Early time transient waves multireflected in a finite pile, governed by a damped wave equation, are analyzed by a reverberation-ray matrix. The pile is surrounded by compacted soil, and the composite is modeled by elastic springs and viscous dampers distributed along the length and at the tip of an elastic rod. Steady-state waves with complex frequencies and wave numbers that are generated by a source of harmonic time function at the top and reverberated between the top and bottom surface of the pile are sorted in matrix form into ray-groups, arriving at a receiver in successive orders of reflections from the bottom. The steady-state ray groups are synthesized into a series of nonsingular Fourier integrals that can be evaluated accurately with a fast Fourier-transform algorithm. The first integral (zeroth-order) has also been reduced by complex contour integration to the well-known closed-form solution in Bessel functions for a semi-infinite pile. Detailed time records of velocity response received at the top after three reflections are calculated to illustrate attenuation and damping; arrival times and amplitude-phase change on each reflection for various lateral and base supports. The calculated records resemble ultrasonic nondestructive testing data.     

Valve Design for Extracting Response Functions from Hydraulic Systems Using Pseudorandom Binary Signals

The analysis of the dynamic response of a pressurized water pipeline system is important for the design and also the integrity monitoring of these systems. An efficient method for summarizing the behavior of a pipeline system is through the determination of their system response functions. These functions can be extracted by injecting a pressure signal with a wide bandwidth that persists over the length of a pipeline system. Unlike electrical and mechanical systems, generating such signals in pressurized water systems is difficult. Valves capable of generating a signal against the system back-pressure often lack the necessary maneuverability to ensure the signal is sharp (and hence with high-frequency content) and the generated transient is often large in amplitude, risking damage to the system. A method for generating a small amplitude transient signal with a wide band of frequencies is desirable. This paper presents the design for a side discharge valve for generating a pseudorandom binary sequence of pressure changes that are of a small magnitude in relation to the steady state head of the pipeline. The pseudorandom pressure sequence is used to provide an estimate of the system response function. The continuous form of the signal allows the amplitude of each individual pulse within the signal to be small while maintaining the same signal bandwidth. The valve has been tested experimentally and was found to provide a good match with the theoretical response of the pipeline. The method provides a practical alternative to frequency sweeping using sinusoidal signals or sharp valve closures for the extraction of the response functions. Once determined, the system response function can be utilized to detect system faults such as leaks and blockages.     

Pipeline Network Features and Leak Detection by Cross-Correlation Analysis of Reflected Waves

This paper describes progress on a new technique to detect pipeline features and leaks using signal processing of a pressure wave measurement. Previous work (by the present authors) has shown that the analysis of pressure wave reflections in fluid pipe networks can be used to identify specific pipeline features such as open ends, closed ends, valves, junctions, and certain types of bends. It was demonstrated that by using an extension of cross-correlation analysis, the identification of features can be achieved using fewer sensors than are traditionally employed. The key to the effectiveness of the technique lies in the artificial generation of pressure waves using a solenoid valve, rather than relying upon natural sources of fluid excitation. This paper uses an enhanced signal processing technique to improve the detection of leaks. It is shown experimentally that features and leaks can be detected around a sharp bend and up to seven reflections from features/leaks can be detected, by which time the wave has traveled over 95?m. The testing determined the position of a leak to within an accuracy of 5%, even when the location of the reflection from a leak is itself dispersed over a certain distance and, therefore, does not cause an exact reflection of the wave.     

Numerical Sensitivity Study of Unsteady Friction in Simple Systems with External Flows

This paper investigates the importance of unsteady friction effects when performing water hammer analyses for pipe systems with external fluxes due to demands, leaks, and other system elements. The transient energy equation for a system containing an orifice-type external flow is derived from the two-dimensional, axial momentum equation. A quasi-two-dimensional flow model is used to evaluate the relative energy contribution of total friction, unsteady friction, and the external flow, in a 1,500?m pipeline, with orifice flows ranging from steady-state flows of 2–70% of the mean pipe flow, and a Reynolds number of 600,000. It is found that for initial lateral flows larger than around 30% of the mean flow, unsteady friction effects can probably be neglected, whereas for external flows smaller than this, unsteady friction should generally be considered. Overall, the relative role of unsteady friction is found to diminish as the external flux increases, implying that unsteady friction is not critical for systems with large external flows. These results imply that unsteady friction may have a significant impact on the validity of transient leak detection techniques that have been derived assuming quasi-steady friction. To demonstrate this point, an existing transient leak detection method, originally derived under quasi-steady conditions, is tested with unsteady friction included.     

Interaction of Nonconservative 1D Continuum and Moving MDOF Oscillator

This paper presents a method for computing the response of a 1D elastic continuum induced by a multi-degree-of-freedom (MDOF) oscillator traveling over it. The continuum and the oscillator are nonconservative systems with proportional damping. Unlike most studies in the field, the solution method does not address a particular type of continuous structure and oscillator. Instead, a rigorous mathematical formulation is presented that can be applied to a broad class of proportionally damped 1D continua and MDOF oscillators, regardless of boundary conditions. The problem is reduced to the integration of a system of linear differential equations with time-dependent coefficients. These coefficients are found to depend on natural frequencies, damping ratios, and eigenfunctions and eigenvectors of the continuum and the oscillator. The method is tested on numerical examples and results are compared to those available in the literature. As a practical application, the method can be used to analyze vehicle-bridge interaction problems.     

Response of Damped Oscillators to Cycloidal Pulses

In this paper, the transient response of a damped oscillator subjected to cycloidal pulses is investigated. The response is computed analytically by considering viscous and friction damping, and response spectra for relative and absolute quantities are presented for the linear viscous and sliding oscillator. The study complements the list of numerous shock spectra mostly published for the undamped linear oscillator. Subsequently, a numerical procedure based on a state-space formulation is developed to compute the response of damped oscillators when subjected to ground motions recorded near the source of strong earthquakes. It is found that although in several occasions such motions resemble to cycloidal pulses, the response of structures with low to moderate periods is substantially affected by the high-frequency fluctuations that often override the long duration pulse.     

Wavelet-Galerkin Solution to the Water Hammer Equations

In this paper, a wavelet-Galerkin method is utilized to solve the hyperbolic partial differential equations describing transient flow in a simple pipeline. Two wavelets (Haar and Daubechies) are utilized as bases for the Galerkin scheme. The governing equations are solved for the expansion coefficients, which are then used to reconstruct the signal at the downstream end of the pipeline; the computed results are in an excellent agreement with those calculated by using the method of characteristics including laminar or linearized turbulent friction terms. Most importantly, the wavelet-Galerkin approach allows the transient flow equations to be solved directly for the expansion coefficients at a certain level of resolution. This can be used to form the wavelet multiresolution framework that can be utilized for further analysis, such as feature extraction and signal identification.     

Enhanced Realization∕Identification of Physical Modes

Physical structures are often sufficiently complicated to preclude constructing an accurate mathematical model of the system dynamics from simple analysis using the laws of physics. Consequently, determination of an accurate model requires utilization of (generally noisy) output measurements from dynamic tests. In this paper, we present a robust method for constructing accurate, structural‐dynamic models from discrete time‐domain measurements. The method processes the measurements in order to determine the number of modes present, the damping and frequency of each mode, and the mode shape. The structure may be highly damped. Although the mode‐shape identification is more sensitive to measurement noise than the order, frequency, and damping identification, the method is considerably less sensitive to noise than other leading methods. Accurate detection of the modal parameters and mode shapes is demonstrated for modes with damping ratios exceeding 15%.     

Processing and damping behaviour of porous copper

Abstract

The well distributed open cellular porous copper was fabricated by present powder metallurgy technique based on space holder method, Depending on the volume fraction and size of the space holding particle, the porosity can be varied in the wide range of 30–85% and pore size from micron to millimetre in magnitude of order respectively. The damping behaviour and related relative dynamic modulus of the porous copper were investigated by a multifunction internal friction apparatus as a function of temperature from room temperature to 600°C. The results of investigation disclose that the porous copper can obtain a higher damping capacity than that of bulk one. In addition to this, an internal friction peak was found in the spectra of internal friction against temperature for the porous copper, it was proposed that the viscous sliding of the grain boundaries should be responsible for the appearance of the peak, and the dependence of the peak on porosity can be understood in terms of the anelastic relaxation mode of grain boundary.     

Complex Frequencies in Elastodynamics, with Application to the Damping-Solvent Extraction Method

This paper addresses the use of complex frequencies in problems of wave propagation and structural vibrations. The most common form of application is as artificial damping that is extracted after the response in the time domain has been obtained. Then again a rather unorthodox application is in the simulation of systems of infinite spatial extent by means of finite systems modeled with discrete methods such as finite elements, a task that can be accomplished even when no transmitting or absorbing boundaries are used. This latter application of complex frequencies, which goes by the name damping-solvent extraction method or its acronym DSE, is assessed herein by means of exact solutions to canonical problems that are used to establish the conditions that must be met by the finite models to work as intended, especially the size of the models, the magnitude of the imaginary component of frequency, and the limitations of the method.     

Numerical Error in Weighting Function-Based Unsteady Friction Models for Pipe Transients

The accurate simulation of pressure transients in pipelines and pipe networks is becoming increasingly important in water engineering. Applications such as inverse transient analysis for condition assessment, leak detection, and pipe roughness calibration require accurate modeling of transients for longer simulation periods that, in many situations, requires improved modeling of unsteady frictional behavior. In addition, the numerical algorithm used for unsteady friction should be highly efficient, as inverse analysis requires the transient model to be run many times. A popular model of unsteady friction that is applicable to a short-duration transient event type is the weighting function-based type, as first derived by Zielke in 1968. Approximation of the weighting function with a sum of exponential terms allows for a considerable increase in computation speed using recursive algorithms. A neglected topic in the application of such models is evaluation of numerical error. This paper presents a discussion and quantification of the numerical errors that occur when using weighting function-based models for the simulation of unsteady friction in pipe transients. Comparisons of numerical error arising from approximations are made in the Fourier domain where exact solutions can be determined. Additionally, the relative importance of error in unsteady friction modeling and unsteady friction itself in the context of general simulation is discussed.     

Address-Oriented Impedance Matrix Method for Generic Calibration of Heterogeneous Pipe Network Systems

The generic evaluation of pipeline parameters is one of the most demanding technological tasks in the efficient management of a water distribution system. Information about current pipeline status is feasible by monitoring the pressure variation online. Conventional methods of transient computation and parameter calibration for a heterogeneous pipeline network suffer from cost issues both in time and storage as well as several other constraints associated with the numerical representation of a real-life system. As an alternative approach, an extension of the impulse response method, namely the address-oriented impedance matrix method (AOIMM), has been developed for a more robust calibration of a heterogeneous and multilooped pipe network system. The genetic algorithm was incorporated into the AOIMM for generic calibration of several parameters, such as the location and quantity of leakage, friction factor, and wave propagation speed. The potential of the proposed calibration algorithm over other conventional approaches was demonstrated when it was applied to a hypothetical heterogeneous pipe network system.     

Closely Spaced Roots and Defectiveness in Second-Order Systems

When two closely spaced eigenvalues merge the associated eigenvectors can either (1) form a subspace where every vector in the span is an eigenvector or (2) coalesce into a single eigenvector. In the second alternative the repeated eigenvalue is associated with a bifurcation point in the eigenvector space and the system is said to be defective. In defective systems a set of coordinates that uncouple the dynamics does not exist and the closest thing possible is the basis of eigenvectors and generalized eigenvectors (sometimes called power vectors) that lead to the Jordan form. Although true defectiveness does not occur in practice, because eigenvalues are never exactly repeated, one anticipates that the features associated with defective conditions will have a bearing on the behavior of systems that are perturbed versions of defective ones. In viscously damped second order systems with symmetric matrices the potential for defectiveness is determined by the structure of the damping. This paper focuses on identification of conditions connecting the damping matrix with defectiveness. A numerical example of a two degree-of-freedom system that varies from being classically damped, to nonclassical, to defective, depending on the position of a dashpot, is used to illustrate the features of the eigensolution as defectiveness is approached.     

Dynamic Response Solution in Transient State of Viscoelastic Road under Moving Load and Its Application

In order to study the dynamic response of an asphalt road, a dynamic model of the road under a moving load is proposed, in which the viscoelastic characteristics of the base and pavement are all considered, the pavement is regarded as an infinite beam on a Kelvin viscoelastic base. By using Green’s functions, Laplace transforms, and Fourier transforms, the analytical solution in transient is deduced. As the viscosity of the pavement is included in the model, the analytical solution can be used to investigate more of the factors that affect the dynamic response, such as vehicle speed, temperature, and road material properties. Using this analytical solution, some numerical calculations are given to illustrate the effects of vehicles’ speeds and different damping on the deflection with the displacement.