Random Vibration

Impulse response functions (IRFs) and frequency response functions (FRFs) are bases for modal parameter identification of single-input, single-output (SISO) and multiple-input, multiple-out (MIMO) systems, and the two functions can be transformed from each other using the fast Fourier transform and the inverse fast Fourier transform. An efficient iterative algorithm is developed in this work to directly and accurately calculate the IRFs of SISO and MIMO systems in the time domain using relatively short input and output data series. The iterative algorithm can avoid the time-consuming inversion of a large matrix in the conventional least-square method for calculating an IRF, greatly reducing the computation time. In addition, a fitting index and an error energy decreasing coefficient are introduced to evaluate the accuracy in calculating an IRF and to provide the termination criterion for the iterative algorithm. A new coherence function is also introduced to evaluate the accuracy of calculated IRFs and FRFs at different spectral lines. Two examples are given to illustrate the effectiveness and efficiency of the methodology.

The problem of digital modeling of random processes with given either correlation function or spectral density of the process is considered. These functions of a random process are interconnected by the Wiener–Khinchin theorem. The solution of one function can be used to solve another. The development of a mathematical representation of a stationary random process with a given correlation function based on the Karhunen-Loeve transformation, which is most often used to decorrelate the original process in order to describe it more concisely (data compression problem), has been completed. It is proposed to use the Karhunen–Loev transformation to impart the required correlation properties to the original uncorrelated random process by inverting (converting) this transformation. The form of the required transformation for a discrete (in time) representation of input and output processes of various lengths and methods for ensuring the required modeling accuracy are substantiated. A procedure for obtaining a correlation function from a given spectral density of a simulated random process is presented. An experimental study of the proposed method was carried out in the course of computer simulation in the Mathcad package which simplified the solution of the required computational problems. The initial random process was obtained as a sequence of independent (and, therefore, uncorrelated) random numbers, and the output process, as a result of the transformation, was obtained in the work. The calculated approximate correlation function is compared with the given one and the error variance is determined. The results of modeling random processes with given correlation functions and a homogeneous Markov process with a given transition probability are given as well as an example of the transition from a given spectral density of a random process to its correlation function. The results obtained confirm the effectiveness and feasibility of the developed modeling methods which will allow them to be used in computer research and design of various systems.

Chapter 12 presents a detailed analysis of continuous-time signals and systems in the frequency domain, including the theory of Fourier series and Fourier transforms, and key examples relevant for the analysis and synthesis of signals processed in the digital transceiver blocks of a communication system. The amplitude, magnitude, phase, and power spectra are defined and calculated for typical signals. In particular, the Fourier transform of periodic signals is presented, due to its importance in communication systems theory and practice. Using a unique notation that distinguishes energy and power signals, the correlation, power, and energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of a linear-time-invariant system, using the concepts of impulse response, system correlation function, and power spectral density, both for power signals and energy signals, is presented. In addition, Parseval’s theorem and the Rayleigh theorem are proven.

Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation

Errors involved in computing impulse response functions via frequency response functions

The power spectral density (PSD) is a fundamental technique of random vibration fatigue providing an effective statistical characterization that can be processed by linear systems theory and load spectrum estimators. This lays the basis for a statistical-based fatigue assessment. While the PSD assembles a full stochastic characterization of stationary Gaussian loading, for loading subjected to changing operational, environmental, and excitational conditions, it provides no means of a fluctuating spectral density. Therefore, the PSD neither qualifies to characterize varying loads, nor reproduces comparable stress amplitudes to a referencing non-stationary excitation following a statistical-based stress analysis. Consequently, this paper employs the non-stationarity matrix to characterize the varying evolution of realistic loading and proposes a system of equations that decomposes this characterization into stationary Gaussian portions. The fundamental idea is to approximate realistic loading by abstracting a series of stationary segments, whose assembly in return embodies a full statistical characterization. The resulting quasi-stationary load definition better reflects the fatigue damage potential of realistic, non-stationary loading and allows to implement load spectrum estimators, ensuing computationally efficient and statistically robust structural lifetime predictions. Further, quasi-stationary load definitions are utilized to advance the concept of damage-equivalent statistical load definitions to be independent of a specific Miner exponent.

Energy harvesting from ambient environment has received increasing attention over the last decade due to the need for minimizing the dependence on conventional batteries in wireless applications. Among the methods of vibration-to-electricity conversion, piezoelectric transduction has been investigated by numerous research groups due to the ease of application and high power density offered by piezoelectric materials. Electromechanical modeling efforts of piezoelectric energy harvesters have been mostly focused on deterministic forms of excitation input, as in the typical case of harmonic excitation. In most practical applications, however, ambient vibrational energy is often stochastic with broad frequency content. This paper presents analytical and numerical modeling, simulations, and experimental validations of piezoelectric energy harvesting from broadband random vibrations. The models employed herein are based on distributed-parameter electroelastic solution to ensure that the effects of higher vibration modes are included. The goal is to predict the expected value of the power output in terms of the given power spectral density (PSD) or time history of the random vibration input. The analytical estimations are based on the PSD of broadband random base excitation and distributed-parameter frequency response functions (FRFs) of the coupled voltage and vibration response. The numerical simulations use the Fourier series representation of base acceleration history in an ordinary differential equation solver that employs first-order electroelastic equations. The simulations are compared against the experiments for a brass-reinforced PZT-5H bimorph under different random excitation levels. The analytical and numerical simulations exhibit very good agreement with the experimental measurements. Soft and hard ceramic and single crystal bimorphs (made of PZT-5H, PZT-8, PMN-PZT, and PMN-PZT-Mn) are compared for broadband random excitation through a theoretical case study.

There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

Dynamic Properties of Electrotonic Coupling between Cells of Early Xenopus Embryos

For the further development we use the state-variable description of linear time-invariant systems. This is the standard form used in linear system theory. Introducing this notation brings some advantages. The state-variable description is the most general form and is equally applicable for scalar and for multivariable systems. Matrix calculus is employed instead of working with polynomials. The whole theory of linear systems is available.

The isotropic turbulence assumption, though widely used, may not accurately represent real-world conditions in stochastic wind models for turbulence relative to moving vehicles. This study derives the statistical properties of non-isotropic turbulence relative to moving vehicles, focusing on correlation functions, power spectral densities (PSDs), and coherence functions of wind speed fluctuations. A quantitative parameter characterizing the impact of non-isotropic turbulence is introduced, enabling an in-depth analysis of its effects on turbulence statistics. Non-isotropic turbulence notably increases PSDs in frequency regions critical to vehicle stability under crosswinds, underscoring the need to account for these effects in quantitative assessments. Without considering non-isotropic influences, crosswind stability may be overestimated. Non-isotropic turbulence impacts the PSDs both indirectly, by altering the lateral-to-longitudinal turbulence length scale ratio, and directly, through additional influences on PSDs. Prior studies have largely focused on the indirect effect, but both effects should be considered, as omitting the direct effect results in up to a 28.9% underestimation of PSDs at low frequencies, potentially overestimating running safety. The analysis also reveals that the relative wind angle, defined as the difference between the vehicle's running direction wind angle and its yaw angle, is the primary factor influencing turbulence characteristics, consistent for both isotropic and non-isotropic conditions. Additionally, the spatial correlation of wind speed fluctuations depends largely on the separation distance relative to the longitudinal turbulence length scale. Therefore, in qualitative studies of wind fluctuation effects on crosswind stability, non-isotropic turbulence may be disregarded. This work enhances the accuracy of stochastic wind models across different scenarios, providing a foundation for future research on crosswind safety in railway operations under turbulent wind conditions.

The close connections between convolutional coding theory and linear system theory are well known. A convolutional encoder is simply a discrete-time linear time-invariant (LTI) system over a finite field, always finite-dimensional, in general multivariable. These connections were exploited in the 1970’s to develop an algebraic structure theory of convolutional codes [5], the key elements of which turned out later to be useful in linear system theory [6, 14].

Finite element model updating methodologies allow the identification of structures by improving the accuracy of an initial finite element model considering data acquired from a vibration test. Usually, numerical predictions and experimental results disagree thus requiring an updated model that best reproduces the dynamic behavior of the actual structure. Several updating methodologies have been developed in the last years such as the modal-based and response-based ones, which minimize the difference between the measured and predicted modal parameters and Frequency Response Functions, respectively. This thesis investigates a response-based method which iteratively minimizes a residual vector defined on the correlation functions between the Frequency Response Functions, which are directly available from experimental tests, to identify the mismodeled regions of a numerical model and improve the correlation with its experimental counterpart. The minimization problem is generally solved by means of the weighted least-squares approach and starting from a reference formulation available in literature, some enhancements are proposed, validated numerically, and extended to real experimental data using simple structures such as beams and plates. Finally, the proposed methodology is applied to identify the physical properties of a reduced-scale helicopter blade model that incorporates a semi-active device for vibration control, named Smart Spring. As a further study, the effectiveness of this device to reduce vibration is analytically verified through the development of its mathematical model in the frequency domain that makes use of an open-loop control law. Moreover, the capability of the Smart Spring to modulate the structural properties of a system and alter its dynamic behavior, thus acting as a semi-active device, is experimentally assessed through vibration tests performed at different operative working conditions of the device.

Finite element (FE) model updating is an important tool to improve the accuracy of an initial FE model so that the predicted dynamic behavior matches as closely as possible that tested during the experiment. Two sets of new frequency response function (FRF) correlation functions, i. e. the amplitude correlation coefficient and the shape correlation coefficient are introduced, and an updating algorithm has been obtained by using the sensitivity of FRF correlation functions with respect to the selected design parameters. The use of physical parameters in model updating is implemented from a large ship model. The results indicate that this technique is feasible for very incomplete experimental FRF data and any updating frequency range, even in the case of measurement noise.

Modal parameters, i.e., modal frequencies, damping ratios, and mode shapes of a two disk-shaft system are estimated in Multiple Input/Multiple Output (MIMO) scheme. The response at the output degrees of freedom (dof) of the considered structure due to the excitation at the input dofs is estimated from theoretical analysis. The generated theoretical data is used to estimate the Frequency Response Function (FRF) matrix, which relates the output (vibration) at response points with the input at the excitation points in the frequency domain. The corresponding Impulse Response Function (IRF) matrix, which relates response and excitation in the time domain, is obtained by Inverse Fast Fourier Transform (IFFT) of the FRF matrix. The unified Matrix Polynomial Approach (UMPA) is employed in the frequency and time domains with the estimated FRF and IRF matrices, respectively, to estimate the desired modal parameters. The obtained results are compared with results from Theoretical Modal Analysis (TMA).