Elastic Oscillator Research Articles

Measurement of the damping coefficient of an elastic rubber band oscillator by a smartphone

The use of smartphones as laboratory tools for school physics experiments has recently received attention for the possibility of carrying out a wide variety of didactic experiments with low-cost equipments. This article presents a study on a damped oscillator consisting of an elastic rubber loop and a mass. The investigation of the oscillations was conducted by using a smartphone. The experimental data was interpreted by a simple model, obtaining information on the viscoelastic properties of the rubber material.

Influence of two kinds of clearance joints on the dynamics of planar mechanical system based on a modified contact force model

This study takes the slider-crank mechanism with revolute joint and translational joint as the research object and studies the contact force model of the clearance joint and the influence of the hybrid clearance joints on the nonlinear dynamic behavior of the mechanism. A modified contact force model is established based on the simplified elastic oscillator model, which can be used as a normal force in clearance joint. In the new contact force model, the component n of the indentation depth can be arbitrarily selected and it can support the calculation of contact force for both fully elastic recovery, non-elastic recovery and fully inelastic recovery. Based on the LuGre friction model, the tangential friction model of the clearance joint is given. Thus, the normal force and tangential force during the dynamic contact of the clearance joint are formed. Combining Lagrange’s equations of the first kind with the modified normal force and tangential friction force, the dynamic equations of the multi-body system with clearance joints are established. The Baumgarte stabilization method is used to improve the numerical stability. The correctness of the dynamic prediction model in the mechanism with clearance joint is verified by experiment. The dynamic analysis of the slider-crank mechanism with mixed clearance joints shows that the revolute clearance joint has a greater influence on the mechanism than the translational clearance, and the revolute clearance joint plays a leading role in the dynamic response.

Directionality and polarization of response spectral ordinates in the 2023 Kahramanmaras, Türkiye earthquake doublet

Until recently, the orientation of maximum horizontal spectral response was generally believed to not have a predominant orientation at rupture distances greater than 5 km. However, a recent study found that the orientation of maximum spectral response for strike-slip earthquakes in the NGA-West2 database tends to occur close to the epicentral transverse orientation, that is, an orientation perpendicular to a line connecting the epicenter to the station. This article investigates directionality in the 6 February 2023 Türkiye doublet earthquakes ( Mw 7.8 and 7.5) with strike-slip faulting. The orientation of the maximum response of 5%-damped linear elastic oscillators was studied. The spatial distribution of the level of polarization, which in this article refers to the amount of directionality, and intensities at specific orientations were also studied. The maximum spectral response was found to occur systematically close to the epicentral transverse orientation, consistent with previous observations for other strike-slip earthquakes. For the Mw 7.8 event where the location of maximum slip was relatively far from the epicenter, it was found that the orientation of maximum spectral response is, on average, closer to the maximum slip transverse orientation (i.e. perpendicular to a line connecting the station to the surface projection of the point of maximum slip) when compared to the epicentral transverse orientation over most period ranges. This suggests that the maximum slip transverse orientation may be a better estimator for determining the orientation of maximum spectral response in large-magnitude strike-slip earthquakes, although further study using more events is warranted. Polarized motions were observed over large geographical areas, and the orientation of maximum spectral response was found to be close to the epicentral or maximum slip transverse for Joyner–Boore distances up to the farthest studied (400 km). These findings further support the case for the development of orientation-dependent ground motion models for strike-slip earthquakes.

Haar wavelet for computing periodic responses of impact oscillators

Haar wavelets have an attractive property for non-smooth dynamical systems as they are capable of modelling sudden changes because of their local multi-resolution characteristics. In this paper, we applied the Haar wavelet collocation method embedding the segment technique to compute and detect periodic responses of an elastic impact oscillator. Comparisons between dynamical responses computed by a direct numerical simulation using a high accuracy Runge–Kutta algorithm and the proposed method are encouraging. Some key parameters of the impact oscillator were changed to demonstrate the effectiveness of the method. Moreover, the time–frequency index feature for the Haar wavelet coefficients describing impacts was proven to be an additional advantage of this method.

Нестационарные колебания в системе из двух связанных осцилляторов в условиях кубической нелинейности и кубической связи. Часть 2. Вариация параметров системы

The task about the excitation of magnetoelastic vibrations in geometry of magnetostriction transducer which contains normal magnetized ferrite plate excited by alternating magnetic field is investigated. The dynamics of transducer is investigated on the basis of equations system for nonlinear vibrations of two connected oscillators – magnetic and elastic. The equation for magnetic oscillator is nonlinear and for elastic oscillator – linear. The nonlinearity and connections between oscillators have cubic character. In connection with excitation level there are five basal regimes: synchronize, three-multiplication of frequency, chaos, gigantic oscillations and delayed stabilization. The first two regimes are stable, without-threshold and completely determined. The remaining three regimes are non-stable and separated form stable regimes by sharp threshold by the excitation level. It is shown that by the increasing excitation level the vibrations amplitude approach to saturation experiences in value jumps about one and half or two times. The dependence of vibrations threshold of non-stable vibrations from excitation frequency is investigated. It is shown that by increasing the excitation frequency the threshold on non-stable vibrations in very high precision increases by power low of third ore fourth order. The character of vibrations by variation of oscillator dissipation is investigated. It is found that the primary influence on the character first oscillator vibrations exercise the dissipation not first but second oscillator. The variation of own oscillators frequencies is investigated. It is shown that the increasing of own oscillator frequencies leads the system to chaos and further to regularization of vibrations. The decreasing of own oscillator frequencies leads to compressed sine with centre displaced regime and leaving of system to infinity. The variation of cubic nonlinearity of first oscillator is investigated. The large nonsymmetry of variation of vibration character by nonlinearity parameter to both sides is established. The decreasing of nonlinearity parameter leads to compressed sine with displaced centre and further leaving of system to infinity. The increasing of nonlinearity parameter leads to chaos and regularization. The variation of connection between oscillators is investigated. It is found that the variation of connection parameters any of oscillators leads to the same order of regime replacement – from regularization to chaos, gigantic oscillations, delayed stabilization, compressed sine and leaving to infinity. On the plane in coordinates first and second connection parameters the regions of existence different regimes of vibrations is constructed. For the interpretation of described phenomena the model of dynamic potential is proposed. It is shown that the potential of first oscillator has minimum the place of which is determined by value of displacement of second oscillator. The course of chaotic vibrations in non-stable regimes is the difference of phase between vibrations of oscillator and jumps of potential in which these vibrations take place. The ring character of vibrations excitation in system as a whole is considered. The analogy with autovibrations in radioengineering system with positive opposite connection is proposed. The recommendations for further development of investigation are proposed.

Design and Analysis of a Sloshing Liquid Damping Device with Elastic Baffle Vibrator

Sloshing liquid damping technology is a typical passive damping technology, which has many advantages in application. To improve the sloshing intensity of liquid, it is necessary to install internal components to increase damping of liquid. Elastic materials, especially rubber materials, are widely used as internal components of the deep-water damper, which would interact well with damping liquid. In this paper, a damping device combining elastic material with sloshing liquid damper is adopted, and a new-type elastic baffle oscillator sloshing liquid damper is proposed by using the lag function of the spring device to transmit the force. The damping mechanism of the structure and the core factors affecting the damping performance are studied. The numerical simulation results show that by adjusting the stiffness of the internal spring, the phase difference between the structural displacement and the acting force of the internal spring on the structure can be changed in a large range. Under the working condition of this paper, when the elastic coefficient is 2000 N·m−1, the phase difference is close to π/2, and the amplitude of the last wave trough of the structural displacement is only 0.5% of the initial amplitude, which indicates that the damping performance is excellent.

The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM).

Complex response analysis of a non-smooth oscillator under harmonic and random excitations

It is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo (MC) simulation. Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.

VIBRATIONS OF AN INSTANTLY LOADED DISSIPATIVE OSCILLATOR

The work investigates non-stationary oscillations of dissipative oscillators. The joint influence of resistance forces of different nature in the composition of the dissipative force on the oscillations of an elastic linear oscillator caused by its instantaneous loading by a constant external force is investigated. The work was limited to the case of small displacements, when the elastic characteristic of the system can be considered linear. The problem is nonlinear due to the account of the action of the dry friction force, but it allows the construction of exact analytical solutions in elementary functions. In this work, by the method of adding solutions, formulas are derived for calculating the amplitude of oscillations and the duration of half cycles. First, a variant is considered when the resistance force consists of linear viscous and dry friction forces. Then a generalization is made to the case of three resistance forces. The third force is the force of positional friction, which arises in elastic elements such as a leaf spring. It is shown that as a result of the action of the total resistance force, the oscillatory process of a loaded oscillator has a finite number of cycles and is limited in time, which is usually observed in practice. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. To check the adequacy of the derived calculation formulas, numerical computer integration of the nonlinear differential equations of the oscillator motion was additionally carried out. The full agreement of the numerical results obtained using various methods is shown. In addition to direct problems of dynamics, the inverse problems of determining the characteristics of the load and resistance from the results of measuring the amplitude of oscillations are also considered. The derived calculation formulas are also suitable for identifying the characteristics of the load and resistance based on the results of experimental measurements of the oscillation ranges. Examples of identifying these characteristics are given. The study showed that the nonlinear problem of motion of an instantly loaded oscillator with the total resistance of several forces of different nature has an analytical solution in elementary functions. The presence of such resistance significantly affects the motion of the oscillator after loading. The constructed analytical solutions give results such as the numerical integration of the original nonlinear differential equation on a computer, which confirms the adequacy of the formulas obtained.

Analytical investigation of the seismic response of elastic oscillators placed on the top of rocking storey

In the present study the seismic response of elastic single degree of freedom oscillators placed on the top of a rocking storey is examined. A storey comprised of columns that may uplift under ground motion excitations can be regarded as an innovative and cost effective base isolation technique. Regarding the podium structural systems, apart from studying the rocking response in order to avoid the rocking overturn, it is important to examine the seismic demand on the superstructure so as to assess the rocking storey as base isolation system. From this point of view, structural as well as ground motion parameters that affect the seismic demands of the superstructure are examined. For this purpose, the superstructure’s response spectra, of various structural configurations, under free rocking response, analytical pulses and natural ground motions are presented and discussed. Finally, the effect of different assumptions regarding energy dissipation during impact on the response of the superstructure is investigated. The study of the podium structures results in useful remarks concerning the performance of the rocking storey as a base isolation system.

Simplified analysis of bilinear elastic systems exhibiting negative stiffness behavior

SummaryThis work studies the dynamics of the Negative Stiffness Bilinear Elastic (NSBE) oscillator. Such a mathematical idealization can be used to describe deformable rocking systems equipped with restraining tendons or with curved extensions of their bases. First, this paper establishes the characteristic quantities of the bilinear system to make it equivalent to the actual rocking structures. Then, it proceeds by proposing a simpler “equivalent” system that can be used to study the behavior of the NSBE. The equivalent system is not some linear elastic oscillator but a bilinear elastic system with zero stiffness of the second branch: the Zero Stiffness Bilinear Elastic (ZSBE) system. ZSBE is useful because it needs one parameter less than NSBE to be defined. Next, “Equal Displacement” and “Equal Energy” rules that provide estimates of the maximum displacement of the NSBE based on the response of the ZSBE are defined. The concept is similar to the RμΤ relations that provide estimates of the response of bilinear yielding systems based on the response of an equivalent linear elastic system, with one major difference: it does not resort to a linear elastic system but to the ZSBE. The proposed methodology is applied on the FEMA P695 ground motions scaled at three different levels. The results show that ZSBE is a good proxy of NSBE and, hence, indicate that an exhaustive study of the ZSBE is useful for the design of rocking structures.

FREE OSCILLATOR OSCILLATIONS IN THE PRESENCE OF QUADRATIC VISCOUS RESISTANCE AND DRY FRICTION

The article is devoted to the derivation of formulas for calculating the ranges of free damped oscillations of a double nonlinear oscillator. Using the Lambert function and the first integral of the nonlinear differential equation of motion, formulas are derived for calculating the ranges of free damped oscillations of a linearly elastic oscillator under the combined action of the forces of quadratic viscous resistance and Coulomb dry friction. The calculations involve a table of the specified special function of the negative argument. It is shown that the presence of viscous resistance reduces the duration of free oscillations to a complete stop of the oscillator. The set dynamics problem is also approximately solved by the energy balance method, and a numerical integration of the nonlinear differential equation of motion on a computer is carried out. The satisfactory convergence of the numerical results obtained in various ways confirmed the suitability of the derived closed formulas for engineering calculations. In addition to calculating the magnitude of the oscillations, the energy balance method is also used for an approximate solution of the inverse problem of dynamics, by identifying the values of the coefficient of quadratic resistance and dry friction force in the presence of an experimental vibrogram of free damped oscillations. An example of identification is given. This information on friction is needed to calculate forced oscillations, especially under resonance conditions. It is noted that from the obtained results, in some cases, well-known formulas follow, where the quadratic viscous resistance is not associated with dry friction.

Stochastic analysis of strongly non-linear elastic impact system with Coulomb friction excited by white noise

The stochastic response of frictionally damped strongly non-linear elastic impact oscillator subjected to white noise excitation and its stochastic bifurcation are considered. By the stochastic averaging method based on generalized harmonic function, one can obtain the stationary probability density function of this system. The effects of system parameters on the responses are investigated and the analytical results were verified by comparing with numerical results from Monte Carlo simulations. Stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution, which indicates that the coefficient of friction, damping constant of the elastic impact force respectively, can be treated as bifurcation parameters.

Вільні коливання пружно нелінійного осцилятора з сухим тертям

Методом енергетичного балансу виведено рекурентні співвідношення для розрахунку послідовності спадаючих амплітуд розмахів дисипативного осцилятора з сухим тертям Кулона. Розглянуто різні варіанти нелінійної пружності коливальної системи. У статті розглянуті випадки коливань осцилятора з наступними показниками нелінійності коефіцієнту пружності: степенево-нелінійної пружності з варіантами лінійного пружного осцилятора, осцилятора з м’якою характеристиками пружності при різних її значеннях, квадратичною та кубічною нелінійністю; коливання осцилятора за наявності в виразі сили пружності лінійної складової з різними показниками нелінійності. Виділено випадки, коли виведені рекурентні співвідношення мають замкнені аналітичні розв’язки і побудовано їх. Складені рекурентні співвідношення між амплітудами розмахів та запропоновано їх числове розв’язання методом ітерацій Ньютона. Проведено порівняння числових результатів, до яких призводять такі розв’язки та комп’ютерне інтегрування диференціального рівняння руху. Показано повну узгодженість результатів, одержаних різними способами. Запропоновано компактні формули для розрахунку ширини зони застою при різних нелінійностях. Викладений спосіб розрахунку простий в реалізації , бо не потребує розв’язування нелінійного диференціального рівняння вільних коливань осцилятора. При виведенні рекурентних співвідношень задіяно точний аналітичний розв’язок кубічного рівняння, запропонований Кардано. Виявлено варіанти нелінійностей, коли розрахунок зводиться до використання явних рекурентних співвідношень, пов’язаних з розв’язками квадратного та кубічного рівнянь. У випадку довільного показника нелінійності розрахунок амплітуд доводиться проводити методом ітерацій.

A New Method for Estimating the Input-Energy of MDOF Structures

Input-energy is the amount of energy that is imposed by an earthquake on a structure and its correlation with structural damage has been studied and demonstrated by many researchers. Since studies concerning seismic energy in multi-degree-of-freedom systems are relatively limited compared with single-degree-of-freedom, in this paper, firstly a theoretical exact method is discussed to calculate the input-energy of the multi-degree-of-freedom elastic oscillators. It is proved that unlike the general rule in mechanics, the superposition theorem is valid for input-energy in conventional modal analysis. To estimate the input-energy, an approach based on PHSA to predict the Fourier amplitude spectrum, is proposed. The results indicate that the modal mass ratio is not the only decisive parameter in input-energy. Modal input-energy decomposition also confirms the possibility of greater input-energy of higher modes in comparison with fundamental ones or the ones with the higher mass participation ratio, especially for tall buildings located in the near-field seismic zones.

Reliability of elastic impact system with Coulomb friction excited by Gaussian white noise

Abstract The reliability of nonlinear Elastic impact oscillator with Coulomb friction subjected to Gaussian white noise excitation is investigated. Firstly, the stochastic averaging method for the nonlinear Elastic impact system with Coulomb friction is proposed and the averaged Ito equation for the system energy is obtained. Secondly, the Backward Kolmogorov(BK) equation and Generalized Pontryagin(GP) equation are established to achieve the conditional reliability function and the mean first-passage time of the Elastic impact system. Finally, two examples are worked out in detail to illustrate the feasibility of the proposed procedures. The accuracy and efficiency of the proposed procedures are validated by Monte Carlo simulations. We can conclude that the increase of the Coulomb friction, noise intensity and damping constants of the elastic impact force can improve the reliability of the Elastic impact system.

Filtered incremental velocity: A novel approach in intensity measures for seismic collapse estimation

SummaryThis study evaluates the performance of a new intensity measure, referred to as filtered incremental velocity FIV3, which is computed using time‐domain features extracted from an acceleration time series and is aimed at the evaluation of structural collapse. This novel approach focuses on the area under a small number of acceleration pulses in the ground motion instead of focusing on the peak response of linear elastic oscillators as in many recently proposed measures of ground motion intensity. FIV3 is developed based on previous research that has highlighted the close relation between the incremental velocity of a ground motion and its potential to induce large inelastic incursions on structures. However, unlike the original definition of incremental velocity which provides a single level of intensity for a ground motion, this new intensity measure is period‐dependent and computed as the sum of the three largest incremental velocities obtained from a low‐pass filtered ground acceleration time series. Efficiency and sufficiency with respect to several ground motion parameters such as magnitude, source‐to‐site‐distance, spectral shape, scale factor, and duration are carefully evaluated and compared against those computed with some traditional and recently proposed intensity measures using collapse results from a four‐story reinforced concrete frame. Results indicate that FIV3 leads to lower variability of collapse estimates and therefore higher efficiency as well as high sufficiency compared with those of other ground motion intensity parameters indicating that this new intensity measure is a promising parameter for structural collapse risk assessment.

Seismic response assessment of rocking systems using single degree‐of‐freedom oscillators

SummaryA new modeling for the seismic response assessment of free‐standing, rigid or flexible, pure rocking systems is presented. The proposed modeling is based on equivalent single degree‐of‐freedom (SDOF) oscillators that can be implemented with common engineering software or user‐made structural analysis codes. The SDOF models adopted use beam elements that are connected to a nonlinear rotational spring with negative stiffness that describes the self‐centering capacity of the rocking member. The loss of energy at impact is treated with an “event‐based” approach consistent with Housner's theory. Different variations pertinent to rigid blocks are first presented, and then the concept is extended to the flexible case. The implementation of the method requires some minor programming skills, while thanks to the versatility of the finite element method, it is capable to handle a variety of rocking problems. This is demonstrated with two applications: (a) a vertically restrained block equipped with an elastic tendon and (b) a rigid block coupled with an elastic SDOF oscillator. The accuracy and the efficiency of the proposed modeling is demonstrated using simple wavelets and historical ground motion records.

Band gap of structure coupling Helmholtz resonator with elastic oscillator

<sec> In order to improve the low-frequency acoustical insulation performance of Helmholtz phononic crystals, a structure coupling Helmholtz resonator with elastic oscillator is designed. This structure combines the characteristics of Helmholtz resonators with those of the local resonant solid-solid phononic crystals. In this structure, the elastic oscillator is bonded to the inner wall of the conventional Helmholtz resonator by rubber. The structure has two bandgaps in the low-frequency range, i.e. 24.5−47.7 Hz and 237.6−308.6 Hz for a lattice constant of 6 cm. However, for the same lattice constant, the lower limit of the bandgap of the traditional Helmholtz resonator without the elastic oscillator structure is only 42.1 Hz. Our structure reduces the minimum lower limit of the bandgap by 40% compared with the traditional Helmholtz structure and has better low-frequency acoustical insulation characteristics. </sec><sec> In this study, the generation mechanism of the bandgap is analyzed with the sound pressure field and vibration mode. It is found that the elastic oscillator and the air in the air passage of the resonator vibrate in the same direction at the frequency of upper and lower limit for the first bandgap while they vibrate in the reverse direction for the second bandgap. Outside the resonator, air sound pressure is zero at the lower limit of the bandgap. The spring-oscillator system is established as an equivalent model. In the model, the elastic oscillator and the air in the passage are regarded as oscillators, and the air separated by the elastic oscillator, the air outside the resonator, and the rubber connected with the elastic oscillator are all regarded as springs. Besides, it can be found that the air in the resonator shows different equivalent stiffness for different vibration mode. </sec><sec> In the discussion, the effects of structural parameters on the bandgap are studied by theoretical calculation and the finite element method. The results show that when the lattice constant decreases without changing the side length of the resonator, the bandgap width increases without affecting the lower limit of the bandgap. The increase of the length of the air passage can increase the width of the first bandgap while the second bandgap decreases. However, the increase of the mass effect of the elastic oscillator results in the first bandgap width decreasing and the second bandgap width increasing. The increase of the length of the air passage and the mass of the elastic oscillator both can reduce the bandgap frequency. It can be found that the volume of the right cavity only affects the frequency of the second bandgap, while the volume of the left cavity can influence the frequency of each bandgap. Therefore, the shorter distance between the elastic oscillator and the passage, the better low-frequency acoustical insulation performance of the structure can be reached. Finally, the increase of the length of the rubber produces new vibration modes, which leads to the generation of new small bandgaps and the change of the frequency of the original bandgaps. However, it is found that the influence of the mode of vibration on the bandgap is smaller than that of the mass of the elastic oscillator, and the regularity of its impact is not apparent.</sec>

Practical examples of ISS systems

In this work we collect several practical examples that illustrate ISS-like properties of nonlinear finite dimensional systems. Depending on the nonlinearities involved we show under which conditions ISS, strong iISS, iISS or LISS property holds in these examples. Strict Lyapunov functions for these systems are provided. Supply rates or gains are calculated explicitly in most cases. Water tank, nonlinear damped elastic oscillator and nonlinear mathematical pendulum are considered as illustrative examples.