Evolution of the resonance vibrations of an oscillator with dry friction

We study the problem of free and forced vibrations of a mechanical system modeling the functioning of seismic isolation systems of “kinematic foundation” type in the case when the sliding friction damper is not centrally located. We determine the damping characteristics in the case of free vibrations and the critical values of the damping parameter for the case of forced vibrations. For the case of resonance vibrations with one degree of freedom we establish the relation of the increase in the amplitude of the vibrations to time at subcritical values of the friction. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Effects of Damping Mechanisms in Free and Forced Vibrations of Some Structural Members

For continuous vibrating systems, such as bars and beams, end-mounted in the environment, knowledge about the mass, damping and stiffness properties of the resonating environment is important for analyzing free and forced vibrations of such structural members which are also damped themselves. To finally get an identification of the clamping parameters, examinations of both vibrating structural members for various restraint conditions and dynamic interaction with viscoelastic halfspaces, etc., are required. As a first step, longitudinal bar vibrations are studied in detail. The method of separation of variables combined with the reformatted orthogonality relation, and numerical integration is applied for calculating the free and forced oscillations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-stage procedure of estimation of reliability indexes from forced and service tests

Prolonged experiments are needed to control the reliability of modern technical products. One method of their shortening is the use of tests under forced regimes. To interpret the results of forced tests with regard to normal conditions the relationship between reliability indexes (RI) and test regime is used. This relationship is determined from preliminary tests. In this approach the above mentioned relationship between RI and test regime is assumed to be valid for all batches of products. It is noted in [1] that the technical characteristics of products are random variables whose distributions depend on production conditions. Therefore for different batches RI may vary rather widely under the same conditions. In the present paper the model of dependence of RI on some inner product parameters instead of those of the testing regime is used to find the common relationship for different products. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 52–58, Perm, 1991.     

Non-linear Proper Orthogonal Decomposition Reduced Order Modeling: Bifurcation Study of a Natural Convection Circuit

Boiling water reactors (BWRs) not only show growing power oscillations at high-power low-flow conditions but also amplitude limited oscillations with temporal flow reversal. Methodologies, applicable in the non-linear regime, allow insight into the physical mechanisms behind BWR dynamics. The present paper aims at a general non-linear methodology capable to achieve reliable and numerical stable reduced order models (ROMs). Model-specific options and prediction capabilities are adressed and illustrated by means of two strongly non-linear dynamical systems: Firstly, the tubular reactor (TR), treated as a generic “pathologic” test case for perturbation investigations violating the inlet conditions, and secondly, the natural convection in a closed circuit (NCC), considered due to dynamical similarities to BWRs. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multiple times scale solution for non-linear vibration of mass grounded system

In this paper, the vibration of a mass grounded system which includes two linear and non-linear springs in series has been considered. Since this system, depending on its parameters can oscillate symmetrically and asymmetrically, both cases have been solved using multiple times scales (MTS) method and some analytical relations have been obtained for natural frequency of oscillations. The results have been compared with previous work and good agreement has been obtained. Also forced vibrations of system in primary and secondary resonances have been studied for the first time and the effects of different parameters on the frequency-response have been investigated.     

Über nichtlineare Schwingungen sphärisch schwingender Gasblasen in Flüssigkeiten unter Berücksichtigung der Kompressibilität des Fluides

Zusammenfassung Es werden nichtlineare erzwungene Schwingungen einer Gasblase in einer kompressiblen Flüssigkeit untersucht. Die von Herring hergeleitete Differentialgleichung wird für die erzwungene Schwingung ergänzt und numerisch gelöst. Die Ergebnisse werden mit dem RPNNP-Modell, welches die Flüssigkeit als inkompressibel voraussetzt, verglichen. Es werden für die erzwungene Schwingung Resonanzkurven berechnet. Die Ergebnisse zeigen, daß die Kompressibilität mit berücksichtigt werden muß.

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Summary The nonlinear forced vibrations of a gas-bubble in a compressible liquid are investigated. The differential equation given by Herring for the case of the forced vibrations is completed and solved numerically. The results are compared with the known incompressible RPNNP-model. The resonance curves under forced vibrations are calculated. The results have shown that the fluid compressibility must be taken into consideration.

     

Effect of memory dependent derivative on forced transverse vibrations in transversely isotropic thermoelastic cantilever nano-Beam with two temperature

The present research deals with the study of forced vibrations in transversely isotropic thermoelastic (TIT) nanoscale beam with two temperature (2T). Memory dependent derivative theory of thermoelasticity for clamped-free/cantilever nano-beam has been considered. The mathematical model is prepared for the nanoscale beam in a closed form with the application of Euler Bernoulli beam theory. Laplace transform method is employed to solve the problem. Forced vibrations due to exponential decaying time varying load acting vertically downward along the thickness direction of the nano-beam, Uniform load, Time harmonic load have been considered. Dynamic analysis for these forced vibrations and Static analysis has been carried out in this research. The dimensionless expressions for lateral deflection, thermal moment, temperature change, and axial stress are solved for these three forced vibrations. Response ratio has also been calculated. The analytical results have been numerically analysed using programming in MATLAB. The effect of kernel function has been depicted graphically on the lateral deflection, thermal moment, temperature change, axial stress and response ratio for all the three types of forced vibrations. Some particular cases have also been discussed.     

New tools in non-linear modelling and prediction

In this paper we give an account of a new change of perspective in non-linear modelling and prediction as applied to smooth systems. The core element of these developments is the Gamma test a non-linear modelling and analysis tool which allows us to examine the nature of a hypothetical input/output relationship in a numerical data-set. In essence, the Gamma test allows us to efficiently calculate that part of the variance of the output which cannot be accounted for by the existence of any smooth model based on the inputs, even though this model be unknown. A key aspect of this tool is its speed: the Gamma test has time complexity O( ), where M is the number of data-points. For data-sets consisting of a few thousand points and a reasonable number of attributes, a single run of the Gamma test typically takes a few seconds. Around this essentially simple procedure a new set of analytical tools has evolved which allow us to model smooth non-linear systems directly from the data with a precision and confidence that hitherto was inaccessible. In this paper we briefly describe the Gamma test, its benefits in model identification and model building, and then in more detail explain and motivate the procedures which facilitate a Gamma analysis. We briefly report on a case study applying these ideas to the practical problem of predicting level and flow rates in the Thames valley river basin. Finally we speculate on the future development and enhancement of these techniques into areas such as datamining and the production of complex non-linear models directly from data via graphical representations of process charts and automated Gamma analysis of each input-output node.     

弹性扁壳由移动质量引起的强迫振动

本文用变分法讨论在弹性扁壳上有移动的质量所引起的壳体的强迫振动.文中讨论了关于强迫振动、共振条件及临界速度等一系列问题.     

Einfache Resonanzversuche am symmetrischen Kreisel

Summary Investigations are carried out on forced oscillations of a heavy symmetrical gyroscope-rotating with constant angular velocity about its axis of symmetry-as a function of an external linear or elliptical sinusoidal periodic force acting in horizontal direction on this gyroscope. The purpose of this work was to find a mechanic model for the phenomenon of nuclear induction. Experiments with the arrangement shown in figure 1 prove that the forced vibrations left over, after the natural vibrations have ceased, consist in a rotation of the body's axis on an elliptic cone. It is demonstrated how sense of rotation, orientation of the principal axis of the ellipse and phase difference between forced vibrations and acting external torque vary with frequency. Two frequencies of resonance are found, one at the natural frequency of precession and one at that of nutation, the first case corresponding to nuclear induction. An elementary calculation of the two frequencies of resonance is given.     

Forced oscillations with a sliding regime range of a two-mass system interacting with a fixed stop : PMM vol. 37, n6, 1973, pp. 999–1006

We investigate, by the method developed in [1]. the forced oscillations with a sliding regime range of a two-mass system with elastic connection between the elements, impacting a fixed stop. The system being considered is a dynamic model for a number of vibrational mechanisms. Forced oscillations with a sliding regime range of a system with shock interactions are periodic motions accompanied by a period of an infinite succession of instantaneous collisions of two fixed elements of the model [2]. Within the framework of conditions of roughness of the parameter space [3], in this paper we study by the method of [1] periodic motions with a sliding regime range of a two-mass system with a stop. This problem was posed because in real systems the velocity recovery factor R changes from shock to shock, mainly taking small values (0, 0.2). At the same time, the regions of realizability of one-impact oscillations, in practice the most essential ones among motions with a finite number of interactions over a period, narrow down sharply as R decreases and becomes very small even for R < 0.6 [4]. Thus, the stability of the given operation can be ensured by a law of motion which is independent or weakly dependent on R (^*) (see footnote on the next page). By virtue of what has been said above, finite-impact periodic modes are little suitable for this purpose. Regions, delineated in the parameter space of the model being considered, of existence of stable periodic motions with a sliding regime range have proved to be sufficiently broad. By virtue of the adopted approximation of the sliding regime, the dynamic characteristics of these motions do not depend upon R. The circumstances mentioned confirm the practical value of motions with a sliding regime range in dynamic systems with impact interactions.     

Frequency-dependent vibration analysis of symmetric cross-ply laminated plate of Levy-type by spectral element and finite strip procedures

This research describes spectral finite element formulation for vibration analysis of rectangular symmetric cross-ply laminated composite plates of Levy-type based on classical lamination plate theory (CLPT). Formulation based on SFEM includes partial differential equations of motion, spectral displacement field, dynamic shape functions, and spectral element stiffness matrix (SESM). In this paper, vibration analysis of composite plate is investigated in two sections: free vibrations and forced vibrations. In free vibrations, natural frequencies are calculated for different Young’s moduli ratios and boundary conditions. In forced vibrations, plate vibrations are investigated under high-frequency concentrated impulsive loads. The resulting responses due to spectral element formulation are compared with those of (time-domain) finite element and analytical formulations, whenever available. The results demonstrate the superiority of SFEM with respect to FEM, in reducing computational burden, simultaneously increasing numerical accuracy, specifically for excitations of high-frequency content. By reducing the time duration of impulsive loads, and consequently increasing the modal contribution of higher modes in the transient response of plate, the accuracy of FEM responses decreases substantially accompanied with a high volume of computations, while the accuracy of the SFEM response results is very high and simultaneously, with a low computational burden. Practically, SFEM follows very closely exact analytical solutions.     

A Model for Determining Tactical Parameters for Materials Requirements Planning Systems

This paper develops an approximate model for simultaneously determining reorder intervals and planned lead times for MRP systems. Unlike previous research, this model explicitly considers commonality, non-instantaneous production, multiple work centres (with multiple machines and with limited capacity), and queue inventory carrying cost. Four heuristic solution procedures for the non-linear integer optimization problem are proposed and compared with an exact branch-and-bound algorithm on a set of forty test problems. The results suggest that two of the heuristics are both effective and efficient.     

Sliding mode control of the forced generalized Burgers equation

^** Email: smaoui{at}mcs.sci.kuniv.edu.kw This paper deals with the sliding mode control (SMC) of theforced generalized Burgers equation via the Karhunen-Loève(K-L) Galerkin method. The decomposition procedure of the K-Lmethod is presented to illustrate the use of this method inanalysing the numerical simulations data which represent thesolutions of the forced generalized Burgers equation for viscosityranging from 1 to 100. The K-L Galerkin projection is used asa model reduction technique for non-linear systems to derivea system of ordinary differential equations (ODEs) that mimicsthe dynamics of the forced generalized Burgers equation. Thedata coefficients derived from the ODE system are then usedto approximate the solutions of the forced Burgers equation.Finally, static and dynamic SMC schemes with the objective ofenhancing the stability of the forced generalized Burgers equationare proposed. Simulations of the controlled system are givento illustrate the developed theory.     

Active damping of forced resonance vibrations of an isotropic shallow viscoelastic cylindrical panel under the action of an unknown mechanical load

Using a new approach, we consider the problem of active damping of the forced resonance vibrations of a viscoelastic isotropic cylindrical panel with simply supported edge faces. The mechanical load is assumed to be unknown; it can be found from the experimental data of a sensor. The problem is solved by the Bubnov–Galerkin method. A relation was obtained for the potential difference that has to be supplied to an actuator for the damping of resonance vibrations of the panel. We also study the influence of sizes of sensors and actuators, the dissipative properties of materials, and temperature on the efficiency of active damping of the forced resonance vibrations of a cylindrical panel.     

Vibration elimination in cable-stayed footbridges and bridges

In this paper, the problems of eliminating vibration in cable stayed footbridges and bridges by dynamically steering the tension in cables during construction vibration are considered. A physical and a mathematical model of cable stayed footbridges and bridges has been formulated. A damping model has been selected on the basis of an analysis of vibrations due to periodic excitation. Standard computer software and the author's own program, both leaning on Finite Element Method (FEM), have been used to analyze the eigenproblem and the forced vibration of bridges. On the basis of the formulated theory an efficient algorithm of forced vibration reduction analysis was established. The reduction is achieved by dynamically steering the tension in cables. A sensitivity analysis has been used. An example has been provided. The theory presented takes into account the new method in the elimination of vibration in cable stayed bridges and footbridges. It can therefore be applied to dynamic analysis of modern cable stayed bridges with typical structure systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010).     

One-way and two-way couplings of CFD and structural models and application to the wake-body interaction

The current study focuses on the wake-body interaction of a circular cylinder, whose transverse free vibration is modeled by a mass-spring-damper system coupled to a computational fluid dynamics (CFD) model for the flow and wake. We first simulate the free vibration of the elastically-mounted cylinder and the wake, and analyze the transverse load it exerts on the cylinder and its phase with the vibration. We vary the damping by three orders of magnitude and examine the difference in the wake-body interaction for slightly-damped and highly-damped systems. We then use the spectral properties of the free vibration and use them to construct two different types of forced vibrations: one consists only of the fundamental component of the free vibration, and the other accounts for all spectral properties of it. We compare the wake load for each type to that corresponding to the free vibration. The forced vibrations correspond to a one-way coupling and the information is communicated from the CFD model to the structural model, whereas the free vibration corresponds to a two-way coupling of the models. By comparing the spectral properties of the wake load, including the phase relation of its components with the vibration, which we obtained for the free vibration and for the equivalent forced vibration, we identify the effects of the wake feedback. The findings show that a forced vibration does not reproduce exactly the wake load at small and intermediate levels of structural damping. As the damping increases, the vibration changes from being in-phase with the wake load to being 90° out-of-phase with it, corresponding to two different wake states, and the forced vibration gives wake load that is very close to the one occurring in the case of full wake-body interaction.     

Influence of physical nonlinearity on resonance vibrations and dissipative heating of inelastic rectangular plates

On the basis of a concept of complex characteristics, we present a statement of nonlinear conjugate problems of forced resonance vibrations and dissipative heating of inelastic plates of physically nonlinear materials, for which the real and imaginary parts of mechanical complex characteristics depend on the amplitude of deformations. Numerical-analytical methods for the solution of the indicated nonlinear problems are presented. An analysis of the influence of physical nonlinearity on the amplitude–frequency and temperature–frequency characteristics and on the coefficient of damping of vibrations of rectangular plates is performed.