Continuous Mechanical Systems Research Articles

Vibration confinement and localization patterns in weakly coupled oscillators with parameter gradients

Localization of vibration due to energy localized redistribution over a structure is defined by the presence of unusually large displacement at some specific locations of the structure. This article proposes a new understanding of the localization phenomena that occur in many different fields. These fields range from structural dynamics in lumped or continuous mechanical systems to the biomechanics of hearing. In this study, two kinds of turning point phenomena, are used as the fundamental blocs to explain different types of localization that were observed in various mechanical systems. A single system composed of coupled oscillators with parameter gradient is used to distinguish the influence of mass, stiffness and coupling on localization. It is shown that, depending on the parameters on which the gradient is applied as well as each parameter and gradient values, two kinds of turning points may occur. The turning points may combine to produce different patterns of localization in the system. These patterns are scrutinized in relation to the different localization phenomena with the literature.

Principal component analysis in the linear theory of vibrations: Continuous mechanical systems driven by different kinds of external noise

In this study, an analysis of mechanical vibrations influenced by external additive white Gaussian noise and colored noise is conducted using the principal component analysis. The principal component analysis is widely employed for encoding images in image processing, biology, economics, sociology, and political science. However, it is hereby applied to analyze nonlinear dynamics of continuous mechanical systems for the first time. A rich class of objects, including straight beams, beams on Winkler foundations and spherical shells, is investigated in the present paper. The basic differential equations are obtained based on the Bernoulli–Euler hypothesis, and solutions of the linear PDEs are analyzed by means of the principal component analysis. Results obtained with the principal component analysis are compared with those for the method of empirical modal decomposition and the wavelet-packet decomposition.

Hamilton's Principle for Material and Nonmaterial Control Volumes Using Lagrangian and Eulerian Description of Motion

The standard form of Hamilton's principle is only applicable to material control volumes. There exist specialized formulations of Hamilton's principle that are tailored to nonmaterial (open) control volumes. In case of continuous mechanical systems, these formulations contain extra terms for the virtual shift of kinetic energy and the net transport of a product of the virtual displacement and the momentum across the system boundaries. This raises the theoretically and practically relevant question whether there is also a virtual shift of potential energy across the boundary of open systems. To answer this question from a theoretical perspective, we derive various formulations of Hamilton's principle applicable to material and nonmaterial control volumes. We explore the roots and consequences of (virtual) transport terms if nonmaterial control volumes are considered and show that these transport terms can be derived by Reynolds transport theorem. The equations are deduced for both the Lagrangian and the Eulerian description of the particle motion. This reveals that the (virtual) transport terms have a different form depending on the respective description of the particle motion. To demonstrate the practical relevance of these results, we solve an example problem where the obtained formulations of Hamilton's principle are used to deduce the equations of motion of an axially moving elastic tension bar.

Generalized variational calculus for continuous and discrete mechanical systems

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory... This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems (see also [20]). Hence, this framework automatically includes reduced mechanical systems subjected or not to constraints. Finally, we show that our formalism can be used to tackle the case of discrete mechanics, including reduced systems, systems subjected to constraints and discrete optimal control theory.

Principal Component Analysis in the Nonlinear Dynamics of Beams: Purification of the Signal from Noise Induced by the Nonlinearity of Beam Vibrations

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA).

Generalized Noether’s theorem for continuous mechanical systems

In this paper, the generalized Noether’s theorem for continuous mechanical systems is formulated by analogy with Vujanovic and Djukic’s (Acta Mech 23:17–27, 1975) and Vujanovic’s (Int J Nonlin Mech 13:185–197, 1978) generalized Noether’s theorem in analytical mechanics of particles. This generalization, unlike by the mentioned authors, is directly performed, i.e., starting from the total variation of the action for continuous systems and the corresponding general Lagrangian equations. The thus formulated generalized Noether’s theorem allows for finding such transformations of the field and time functions that yield integrals (or constants) of motion, and it is also applicable to nonconservative systems, when energy integrals are obtained in a broader sense. In addition, the corresponding pseudoconservative continuous systems are introduced, analogous to those in analytical mechanics (Musicki in Acta Mech 223:2117–2133, 2012), whereby a second, complementary approach to this problem is made. The analysis of the obtained results particularly focuses on the nature of energy integrals (in both ordinary and broader sense), which are here closely associated with impulses and differ essentially from energy integrals in the mechanics of particles.

Non-symmetric and chaotic vibrations of Euler-Bernoulli beams under harmonic and noisy excitations

In this paper we study non-linear dynamics of flexible Euler-Bernoulli beams subjected to harmonic load and white noise. We report that in the case of continuous mechanical systems like the studied Euler-Bernoulli beams the action of white noise yieldes novel and unexpected phenomena, i.e. symmetric transversal loads and symmetric boundary conditions imply non-symmetric beam vibration. In addition, regions of non-symmetric beam vibrations for two values of white noise intensity are given and the charts of regular and chaotic vibrations of the studied beams are presented.

Continuous Mechanical Ventilation in Housing ⬜ Understanding the Gap between Intended and Actual Performance and Use

Improved air tightness and mechanical ventilation systems are regarded as vital elements of low energy strategy in housing. Mechanical ventilation (MV) has become part of the heating energy load optimization model due to its capacity to secure recommended air change levels without depending on daily active control by inhabitants or uncontrolled air leaks. Numerous in- use issues related to continuous mechanical ventilation systems have been identified through field studies. They relate to underperformance of the as built system compared to design targets as well as unintended operation modes. The gap between design intention and actual performance and use of continuous MV in housing context is generally regarded as a threat to expected energy savings or inhabitant's health and as such it should be narrowed as far as possible. Reducing the gap partly depends on the improvement of the continuous MV model to allow for better match with inhabitants needs. This paper proposes a dynamic framework linking factors that influence the emergence of this gap. The framework is based upon findings of previous studies as well as results of a one year-long in-depth Building Performance Evaluation of 40 households in two UK developments. Organizing the pitfalls of embedding MV design into practice in a sequence as well as indicating important links within the process can help to make the complexity more comprehensive and possible to tackle efficiently. Importance of natural ventilation design to backup for MV to increase redundancy and resilience is also highlighted.

Overview of the UK Residential Ventilation Market and Initiatives to Improve the Quality of the Installed Systems

New homes currently being built within the UK all incorporate some type of ventilation system, the majority of which are of the fixed mechanical fan type. These generally come in three generic designs known as single room background ventilators, continuous mechanical systems and continuous mechanical systems with heat recovery. Installation, inspection and commissioning of these systems is covered by Building Regulations, and there are training schemes in place which allow individuals to become Competent Persons to undertake these tasks. Regulation 42 of the Building Regulations (2010) requires builders to provide evidence appertaining to the installation, inspection and commissioning process to be lodged with the Building Control Body within 5 days of the tests being carried out. The method of testing and reporting is described in a Domestic Ventilation Compliance Guide, but since 2010 there has been a growing body of evidence cataloguing problems with the quality of work being undertaken. This has led to organisations such as BSRIA developing initiatives to assist all the parties involved in the design, installation, inspection, and commissioning processes to improve their outputs.

Natural Ventilation Potential in Portuguese Residential Buildings under Winter Conditions – An Opportunity for Hybrid Ventilation

This paper presents a study of the potential for the use of natural ventilation systems in Portuguese multi-family residential buildings under winter climatic conditions. The behaviour of various natural ventilation systems is tested in a standard residential dwelling, using the TRNSYS 15 and COMIS 3.1 software programs. This study leads to the conclusion that the use of hybrid ventilation systems can save a considerable amount of the energy normally spent on continuously operating mechanical ventilation systems. Hybrid ventilation systems will not be adequate if there are no ventilation inlets in the façade and air intake is only by means of infiltration, especially in the case of certain modern windows with better seals; in such situations, the use of continuous mechanical ventilation systems is recommended. Therefore, it is essential to equip buildings with air inlets in the façades, preferably self-adjustable grilles, to ensure the best use of hybrid systems.

Quantification and control of non-Markovian evolution in finite quantum systems via feedback

We consider the unitary time evolution of continuous quantum mechanical systems confined to a cavity in contact with a finite bath of variable size. Measures for Markovianity for such finite system-bath configurations are developed in terms of Hilbert-Schmidt distances of time evolving wave packets. The relevant time scales are identified, which characterize pseudo-Markovian transient behavior, boundary scattering induced non-Markovian oscillations at intermediate times, and non-Markovian rephasing events at long time scales. It is shown how these time scales can be controlled by tunable parameters such as the bath size and the strength of the system-bath coupling.

The Lyapunov–Movchan method in problems of the stability of flows and deformation processes

The extension of Lyapunov's method to continuous mechanical systems are discussed. An annotated bibliography of papers is given in which, based on the Lyapunov–Movchan method, with the construction of corresponding functionals, a direct analysis is carried out of the stability of motion (deformation) of continuous mechanical systems. The material is divided into sections, devoted to the following: (a) the extension of the mathematical apparatus as a whole to continuous and dynamic systems, (b) the stability of elastic, elastoplastic and viscoelastic deformable solids, (c) stability in aeroelasticity and hydroelasticity theory, (d) the linearized theory of hydrodynamic stability, and (e) the stability with reference to perturbations of material functions in the theory of constitutive relations.

Solution of Inverse Task of Mechatronic Systems by Means the Classical and Unclassical Methods of Modelling and Analysis

AbstractIn this paper the modelling by means of the different category graphs and analysis of vibrating subsystem of mechatronic systems by means of the exact and approximate methods have been presented. That approach was to nominate the relevance or irrelevance between the characteristics obtained by means of the exact method (only for the mechanical subsystem) and the approximate method. Such formulation concerns mostly the relevance of the natural frequencies‐poles of the characteristics both mechanical subsystems and mechatronic systems. Approximate solutions requiring all the conditions for vibrating mechanical and/or mechatronic systems have been finding and it may be introduction to synthesis of these systems, modeled by different category graphs. Research limitation is following, that torsional vibrating continuous mechanical subsystem and mechatronic systems are linear type. Presentation the introduction to synthesis of considered class of mechatronic bar‐systems with a constant changeable cross‐section is practical implications of this work. Originality of such formulation is focused on the use of the different category graphs for modelling and synthesising by different methods represented by graphs of vibrating bars to the synthesis of considered discrete‐continuous mechatronic systems. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Indoor environmental quality benefits of apartment energy retrofits

Sixteen apartments serving low-income populations in three buildings were retrofit with the goal of simultaneously reducing energy consumption and improving indoor environmental quality (IEQ). Retrofit measures varied among apartments and included, among others, envelope sealing, installation of continuous mechanical ventilation systems, upgrading bathroom fans and range hoods, attic insulation, replacement of heating and cooling systems, and adding wall-mounted particle air cleaners. IEQ parameters were measured, generally for two one-week periods before and after the retrofits. The measurements indicate an overall improvement in IEQ conditions after the retrofits. Comfort conditions, bathroom humidity, and concentrations of carbon dioxide, acetaldehyde, volatile organic compounds, and particles generally improved. Formaldehyde and nitrogen dioxide levels decreased in the building with the highest concentrations, were unchanged in a second building, and increased in a third building. IEQ parameters other than particles improved more in apartments with continuous mechanical ventilation systems installed. In general, but not consistently, larger percent increases in air exchange rates were associated with larger percent decreases in indoor levels of the pollutants that primarily come from indoor sources.

Approximate Galerkin method applied to an analysis of vibration of continuous mechanical systems

AbstractPaper presents fundamental assumptions of the approximate Galerkin method application in order to vibration analysis of continuous mechanical systems with different form of vibration and different boundary conditions. Flexural vibration of beams, longitudinal vibration of rods and torsional vibration of shafts with all possible ways of fixing were considered. Analyzed mechanical systems were treated as subsystems of mechatronic systems with piezoelectric transducers. This work was done as an introduction to the analysis of mechatronic systems with piezoelectric transducers used as actuators or passive vibration dampers [1–3]. It is impossible to use an exact Fourier method in this case. This is the reason why the approximate Galerkin method was chosen and analysis of its exactness was done as a first step of this work. Dynamic flexibilities of considered mechanical systems were calculated twice, using exact and approximate methods. Obtained results were juxtaposed and it was proved that in some cases the approximate method should be corrected while in the other it is precise enough. A correction method was proposed and it was assumed that the approximate method can be used in mechatronic systems analysis. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Smooth decomposition of random fields

The Smooth Decomposition (SD) method was introduced to analyze discrete-time signals and generalized to continuous-time vector-valued random processes. The SD is obtained solving a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process which defines the decomposition basis. This paper presents a new extension of the SD to continuous-time and continuous-space vector-valued random processes, classically named random fields. This generalization is a major step since one now deals with operators in infinite-dimensional spaces and not matrices. It is shown that in this new context the main properties of the SD are preserved. Applied to the responses of randomly excited continuous mechanical systems, the SD can be considered as an output-only analysis tool. Moreover, two natural orderings are defined to classify the decomposition terms which permit to interpret the SD in terms of modal analysis or in terms of Karhunen–Loève analysis.

Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyper-hyper and spatial–temporal chaos

Third part of the paper is devoted to analysis of the hyper, hyper-hyper and spatial–temporal chaos of continuous mechanical systems using the Lyapunov exponents. The constructed algorithms for the Lyapunov exponents’ computation allowed detecting and analysing novel phase transitions from chaos through hyper chaos to hyper-hyper chaos. In addition, a novel characteristic “maximal deflection versus excitation amplitude” has been introduced to study stability properties of the investigated continuous systems. It should be emphasized that the latter characteristic yields results in full agreements with those obtained via the Lyapunov exponents’ spectrum estimation. The introduced methods and tools of analysis allowed detecting the Sharkovskii windows of periodicity in all continuous mechanical systems investigated in this paper. Finally, the approach to study the space-temporal chaos exhibited by shell structural-members is also proposed.

Routes to chaos in continuous mechanical systems: Part 2. Modelling transitions from regular to chaotic dynamics

In second part of the paper both classical and novel scenarios of transition from regular to chaotic dynamics of dissipative continuous mechanical systems are studied. A detailed analysis allowed us to detect the already known classical scenarios of transition from periodic to chaotic dynamics, and in particular the Feigenbaum scenario. The Feigenbaum constant was computed for all continuous mechanical objects studied in the first part of the paper. In addition, we illustrate and discuss different and novel scenarios of transition of the analysed systems from regular to chaotic dynamics, and we show that the type of scenario depends essentially on excitation parameters.

Routes to chaos in continuous mechanical systems. Part 1: Mathematical models and solution methods

In this work chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells is studied. Namely, a wide class of the mentioned objects is analyzed including flexible plates and cylinder-like panels of infinite length, rectangular spherical and cylindrical shells, closed cylindrical shells, axially symmetric plates, as well as spherical and conical shells. The considered problems are solved by the Bubnov–Galerkin and higher approximation Ritz methods. Convergence and validation of those methods are studied. The Cauchy problems are solved mainly by the fourth Runge-Kutta method, although all variants of the Runge-Kutta methods are considered. New scenarios of transition from regular to chaotic orbits are detected, analyzed and discussed.First part of the paper is devoted to the validation of results obtained. This is why the same infinite length problem is reduced to that of a finite dimension through the FDM (Finite Difference Method) with the approximation order of O(c2), BGM (Bubnov–Galerkin Method) or RM (Ritz Method) with higher approximations. We pay attention not only to convergence of the mentioned methods regarding the number of partitions of the interval [0, 1] in the FDM or regarding the number of terms in the series applied either in the BGM or RM methods, but we also compare the results obtained via the mentioned different approaches. Furthermore, a so called practical convergence of different Runge-Kutta type methods are tested starting from the second and ending with the eighth order.Second part of the work is devoted to a study of routes to chaos in the so far mentioned mechanical objects. For this purpose the so-called “dynamical charts” are constructed versus control parameters {q0, ωp}, where q0 denotes the loading amplitude, and ωp is the loading frequency. The charts are constructed through analyses of frequency power spectra and the largest Lyapunov exponent (LE). Analysis of the mentioned charts indicates clearly that different routes to chaos exist and allow us to control the objects being investigated. In some cases we detect the classical Feigenbaum scenario and we compute also the Feigenbaum constant. This scenario accompanied all problems which we studied. In addition, we detect and illustrate novel scenarios of transition from regularity into chaos including the Ruelle–Takens–Newhouse–Feigenbaum scenario, and the so called modified Pomeau–Manneville scenario.Third part of the paper is devoted to analysis of the Lyapunov exponents. Namely, while investigating evolutions of vibration regimes of a shell associated with an increase of excitation amplitude q0 phase transitions chaos–hyper chaos as well as chaos-hyper chaos–hyper–hyper chaos dynamics are illustrated and studied. Furthermore, for all investigated plates and shells the Sharkovskiy windows of periodicity are detected. In particular, a space-temporal chaos/turbulence is studied.

Nonlinear structural identification by the “Reverse Path” spectral method

When dealing with nonlinear dynamical systems, it is important to have efficient, accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data. An approach for nonlinear system identification widely studied in recent years is “Reverse Path”. This method is based on broad-band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system. Parameter estimation is performed in the frequency domain using conventional multiple-input–multiple-output or multiple-input–single-output techniques. This paper presents a generalized approach to apply the method of “Reverse Path” on continuous mechanical systems with multiple nonlinearities. The method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities. The proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures. The results show that the method is effective on both simulated as well as experimental data.