Ivana Kovacic

University of Novi Sad · Faculty of Technical Sciences

Nonlinear metamaterials (NLMs) have recently garnered significant interest for their abilities to achieve wide bandwidth in vibration attenuation and wave manipulation. However, NLMs proposed have been predominantly theoretical and conceptual due to nonlinearity being introduced through discrete spring models rather than the practically achievable...

In this paper, a novel type of piezoelectric quasi-zero stiffness metastructure is proposed and the improved vibration isolation performance is investigated. A piezoelectric quasi-zero stiffness metastructure is composed of curved beams covered with piezoelectric macro-fiber composite patches, to which digital circuits are connected. Based on the p...

Three lecturers have been invited to present their results as being among the best experts in the world in this field: M. Kadic, NON-LOCAL MECHANICAL METAMATERIALS; G.K. Hu, DESIGN OF ELASTIC EXTREMAL METAMATERIALS AND THEIR APPLICATIONS; G.L. Huang, ACTIVE MECHANICAL METAMATERIALS WITH DIGITAL CIRCUITS: ACTIVE MECHANICS AND THEIR ENGINEERING APPLI...

The focus of this research is on designing a longitudinally excited lightweight metastructure that consists of external units distributed periodically, each enhanced with internal oscillators to serve as vibration absorbers. The metastructure initially exhibits uniformity, with all absorbers being identical to each other, being comprised of a canti...

A nondestructive testing method based on nonlinear vibration is usually implemented by measuring the intensity of the second harmonic signal. However, such intensity is much lower than that of the fundamental frequency signal. Particularly, without the help of sufficiently sensitive damage indication effect, the second harmonic signal from the smal...

A longitudinally excited metastructure with periodically distributed internal oscillators is considered with a view to determining how the tuning condition between their frequency and the frequency of the metastructure as a whole affects vibration attenuation. First, the mechanical model is created, and the tuning condition related to the first res...

Metamaterials are artificial microstructured media that exhibit unique effective material properties and can be tailored to achieve negative properties that inhibit the propagation of acoustic or elastic waves. However, the effectiveness of linear metamaterials (LMs) based on resonance mechanisms is limited to a narrow frequency band. To overcome t...

This research focuses on the analysis of the model and performance of lightweight metastructures encompassing a distributed array of internal homogenous oscillators, integrated into the host structure to create a single-piece element. This metastructure performs longitudinal vibrations, whose axis is colinear with the direction of the transverse vi...

Oscillators with a Duffing-type restoring force and quadratic damping are dealt with in this paper. Four characteristic cases of this restoring force are analyzed: hardening, softening, bistable and a pure cubic one. Their energy-displacement relationships are considered, and the corresponding closed-form exact solutions are obtained in terms of th...

This work is concerned with a mechanical model of a sympodial tree with first-level branches, which has been shown to exhibit certain properties potentially suitable for biomimetic applications. To investigate these potential benefits further from the viewpoint of the system nonlinear behaviour under external periodic excitation, modern numerical t...

This paper is concerned with an oscillator attached to a crank-slider mechanism, where the driving torque is not assumed as constant, but as dependent on its angular velocity. Moreover, it is affected by the motion of the oscillator, i.e. it represents the so-called non-ideal excitation. The displacement of the oscillator is limited by a barrier, a...

This study is concerned with bio-inspired hierarchically organised oscillatory chains from the viewpoint of a localisation phenomenon, when only certain parts of the chain oscillate. The chain consists of block masses attached mutually via tension-extension Hookean springs. The first- and second-order hierarchy are considered to determine their mod...

This study is concerned with modelling and analyses of a vibro-impact system consisting of a crank-slider mechanism and one oscillator attached to it, where the system is exposed to a non-ideal excitation. The impact occurs during the motion of the oscillator when it fits a base, and the excitation of the driving source is affected by this behaviou...

This study is concerned with modelling and analyses of a vibro-impact system consisting of a crank-slider mechanism and one oscillator attached to it, where the system can be exposed to ideal or non-ideal excitation. The impact occurs during the motion of the oscillator when it hits a base, and the excitation of the driving source is affected by th...

This work is concerned with generalized van der Pol oscillators, the damping-like force of which depends nonlinearly on the displacement and velocity with the powers that can be any positive real numbers, while the restoring force is either linear or purely nonlinear. The cases of small and large values of the damping parameter are considered. In t...

This chapter presents different techniques for designing a differential equation of motion of nonlinear oscillators that are isochronous, i.e. their frequency of vibration is amplitude-independent, or, in other words, their backbone curve is a straight line, like for the linear oscillator. These techniques include perturbations methods and certain...

This chapter is concerned with the techniques for obtaining both the exact solutions and the approximate solutions for the response of externally excited nonlinear oscillators. The exact solutions are achieved via tuning approaches, in which the external excitation or the nonlinearity in a restoring force are specially designed to achieve a desired...

This chapter provides an overview of the archetypical nonlinear conservative oscillators whose nonlinearity stems from their restoring terms. The overview starts with linear oscillators, which are presented as a reference point for the proceeding nonlinear oscillators as well as for their mutual comparisons. What is common for all the nonlinear osc...

This chapter is concerned with damped oscillators. Nonlinear oscillators are mainly dealt with, but linear oscillators are also considered or referred to for the sake of comparison or clear extension of the related methodology. First, Lagrangians and conservation laws of viscously damped linear oscillators as well as Duffing oscillators are present...

This chapter starts with several exemplary systems that illustrate a vast variety and richness of oscillatory systems, oscillatory responses and the related phenomena. Although they appear to be of complicated and diverse nature, some of them can be mechanically modelled in a simplified way that reflects their essential characteristics. The ones co...

This chapter is concerned with chains of nonlinear oscillators, which are analysed first from the viewpoint of obtaining similar normal modes and the related exact solutions for free and forced motion. The number of masses is increased gradually, starting from two masses in a chain, proceeding with three masses in a chain, and ending up with an inf...

This study presents how the motion, velocity and acceleration of conservative antisymmetric constant force oscillators can be expressed in terms of Jacobi elliptic functions. Two approaches have been developed. In the first approach, one starts from the known period of vibrations and the solution for motion expressed in terms of Jacobi elliptic fun...

The free vibration displacement of an undamped hardening Duffing oscillator is described in exact form by a Jacobi elliptic function. Unlike an undamped linear oscillator, whose displacement is described by a trigonometric function, a Jacobi elliptic function is difficult to interpret by a simple inspection of the function arguments. The displaceme...

This study is concerned with a double pendulum and its regular behaviour associated with low energy levels and the influence of the associated initial conditions on the frequency of normal modes. The case of nonlinear oscillations described by the exact equations of motion is examined. A global qualitative insight is provided via energy diagrams an...

This bio-inspired study deals with the dynamics of tree-like models whose main structural part mimics a trunk, while the first- and second-order branches are modelled as pendula coupled to it. Conditions for the existence of modes localized in branches are determined in terms of the system parameters. This is then compared with the behaviour of the...

This work presents an analytic technique aimed at designing the external excitation of linear and nonlinear oscillators so that a prescribed form of their steady-state response can be achieved. The technique exploits the exact analytic solutions of the oscillator response having quadratic and/or cubic nonlinearities. Both single-frequency and multi...

Analysis of global behaviour of low-dimensional, strongly nonlinear dynamical systems has been well explored in the past, and modern trends are directed toward the investigation of dynamics of systems whose dimension is large, i.e. higher or equal to six. To deal with the huge number of computations required for high-dimensional global analysis, we...

This study presents two different groups of models of bio-inspired hierarchically organised oscillatory mechanical models exhibiting a localization phenomenon, when only certain parts of the system oscillate. The first group of models corresponds to an idealized sympodial-tree-type branched structure consisting of branches represented by physical p...

This book presents exact, closed-form solutions for the response of a variety of nonlinear oscillators (free, damped, forced). The solutions presented are expressed in terms of special functions. To help the reader understand these `non-standard' functions, detailed explanations and rich illustrations of their meanings and contents are provided. In...

U ovom radu je dat teorijski prikaz fraktalnih struktura, kao i primer praktične primene istih u oblasti građevinarstva. Teorijsko-istraživački deo rada obuhvata analitičku i numeričku analizu fraktalnog simpodijalnog drveta. U analitičkoj analizi određeni su pravci glavnih krutosti fraktalnog drveta kao i njihove vrednosti. Takođe je dobijena i el...

This study is concerned with forced damped purely nonlinear oscillators and their behaviour at different excitation frequencies. First, their dynamics is considered numerically for the response determined in the vicinity of a backbone curve with the aim of detecting coexisting responses that have not been found analytically so far. Both the cases o...

This work first presents an analytical technique on how to exploit exact solutions for the response of certain oscillators to design specific external excitation and get a desired form of the exact steady-state response, both in the undamped and damped systems. Related benefits are then discussed, which include: (i) producing the response of free n...

A state-of-the-art infrared marker-tracking system that consists of eight cameras outfitted with infra-red optical filters and an array of infra-red light-emitting diodes as well as a set of reflective markers, is used to record the motion of a set of markers arranged along a trunk-dominated potted tree, which was pulled and released to perform fre...

This study presents quantitative and qualitative insights into the analysis of data obtained by tracking the motion of reflective markers arranged along the trunk of a pole-like potted tree, which was recorded by a state-of-the-art infrared motion-tracking system. The experimental results showed in-plane damped trajectories of the markers with late...

(In Serbian) O oscilatornim sistemima i oscilatornim fenomenima - prvi deo. Prof. dr Ivana Kovačić sa Fakulteta tehničkih nauka Univerziteta u Novom Sadu, član tima Mech-in-NS, govori o oscilatornim sistemima i različitim aspektima primene i pojave vibracija. Projekat Mech-in-NS „Multimedijalna i interaktivna nastava i učenje Inženjerske mehanike“...

This work presents a methodology on how to use exact closed-form solutions for the response of free undamped linear and nonlinear oscillators to design the external excitation of undamped or damped nonlinear oscillators to get such steady-state response. A variety of examples, including Duffing-type oscillators and purely nonlinear oscillators, are...

This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalisations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlin...

This work is concerned with the design of planar frames consisting of rigidly connected beams so that their ending point, which oscillates in the same plane, is characterized by equal stiffness in all directions. Consequently, when being under the action of a static force in the same plane, this point lies on a circle. Its additional property is co...

This study is concerned with oscillatory systems with pure cubic nonlinearity, which are systematically presented, starting from a one degree of freedom oscillator, then focusing on chains with two and multi degrees of freedom, and ending with elastic systems (systems having an infinite number degrees of freedom). A one degree of freedom pure cubic...

This study is concerned with a chain of oscillators in which discrete masses are connected mutually and with a base with purely nonlinear springs whose power of nonlinearity is assumed to be any real (non-integer or integer) number higher than unity. Similar nonlinear normal modes are determined and found to be associated with different bifurcation...

This study is concerned with the motion and displacement of points on branches of a bio-inspired sympodial tree-like structure of a different hierarchy. First, displacements of points of a sympodial-like tree are recorded in pull-and-release experiments to get a qualitative insight into their behaviour. Then, a sympodial-type structure is analyzed,...

This work presents a theoretical concept for obtaining exact solutions for a forced response of a wide class of externally excited nonlinear oscillators. This includes Duffing-type (hardening, softening, bistable, pure cubic) oscillators and purely nonlinear oscillators whose power of nonlinearity can be any positive real number higher than unity....

A backbone curve is a graphical presentation of the relationship between the natural frequency of an oscillator and its amplitude. A linear oscillator has a straight-line backbone curve as its natural frequency is amplitude-independent. However, nonlinear oscillators, in general have the frequency that changes with their amplitude, which implies th...

This study deals with tree-like structures that mimic a trunk with either first-order branches or both first and second-order branches of a sympodial tree. The corresponding mechanical model comprises physical pendula coupled with torsional springs and viscous dampers. Natural frequencies and modal shapes are obtained analytically and the effects o...

This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so tha...

This work is concerned with bursting oscillations – a kind of mixed-mode oscillations in which fast flows occur along periodically changeable slow flows. The corresponding governing equation stems from a non-autonomous oscillator, with a harmonic external excitation that has a low-valued angular frequency. Two cases regarding the existence and infl...

This study is concerned with free and forced undamped purely nonlinear oscillators. First, the exact closed-form solution for free vibrations given in terms of the Ateb function is discussed. An insight is provided with respect to the period of vibrations and the harmonic content of the response. Then, forced purely nonlinear oscillators with an At...

This article is concerned first with free vibrations in a chain of two-mass oscillators with purely nonlinear springs whose power of nonlinearity can be any real number higher than unity. Similar normal modes are obtained by uncoupling the equations of motion. The corresponding time responses are determined in exact analytical forms in terms of Ate...

The scientific community working on the effects of excitations on trees is quite diverse: from botany and forestry to meteorology and biomimetics. From a mechanical point of view, trees have been recognized as blurring the boundary between a structure, material and mechanism. As such, they represent inspiring concept generators for improved or inno...

Experimental investigations of the dynamic response of potted trees were carried out by using the Vicon 3D motion capture system, which is a leading state-of-the-art infrared marker-tracking system. Reflective markers were arranged along the trunk of a young trunk-dominated tree (Aesculus hippocastanum) and along the trunk and branches of a young b...

This review article is concerned with the applications of Jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and/or quadratic nonlinearity. First, geometric interpretations of three basic Jacobi elliptic functions are given and their characteristics are discussed. It is shown...

Free vibrations of a linear, single-degree-of-freedom oscillator with a periodic piecewise-defined time-varying mass are studied. Two different cases of this variation are investigated: first, the mass increases and then decreases linearly in time, i.e. it changes triangularly, and the second, when the mass changes trapezoidically, which, unlike th...

This study is concerned with certain mechanical systems that comprise discrete masses moving along slowly rotating objects. The corresponding equation of relative motion is derived, with the rotating motion creating slowly varying external excitation. Depending on the system parameters, two cases are distinguished: two-well and single-well potentia...

This study is concerned with a general perturbation method for the quantitative analysis of one degree-of-freedom nonlinear oscillators governed by the equations of motion that contain a coefficient of the nonlinear term which need not be small. Four classes of conservative oscillators with odd restoring forces are considered simultaneously: purely...

This work concerns the application of certain non-linear phenomena – jump frequencies in a base-excited Duffing oscillator – to the estimation of the parameters of the system. First, approximate analytical expressions are derived for the relationships between the jump-up and jump-down frequencies, the damping ratio and the cubic stiffness coefficie...

This article provides an overview of the exact closed-form solutions for purely nonlinear oscillators. These solutions comprise the period of vibration and free and forced responses for one-degree-of-freedom systems. The use of special function for this purpose is demonstrated, which includes: beta function, gamma function, hypergeometric function,...

This study is concerned with a new generalised mathematical model for single degree of freedom bistable oscillators with harmonic excitation of low frequency, linear viscous damping and a restoring force that contains a negative linear term and a positive nonlinear term which is a power-form function of the generalised coordinate. Comprehensive num...

It is shown in this paper that the harmonically excited two-well bistable oscillator for which the excitation frequency is definitionally low exhibits bursting oscillations for low values of damping and relaxation oscillations for high damping values. It is also shown that the region for mixed-mode oscillations can extend for increasing value of th...

Oscillators with an antisymmetric constant restoring force are considered in this paper. Their general characteristics - the corresponding amplitude and period of motion, are discussed first. Then, the exact solutions for acceleration, velocity and motion of such conservative oscillators are presented. One of them includes the use of the square wav...

This chapter describes the dynamic behaviour of a coupled system where a nonlinear oscillator is attached and driven harmonically by an electro-dynamic shaker. The shaker is modelled as a linear single degree-of-freedom oscillator and the nonlinear attachment is modelled as a hardening Duffing oscillator. The attachment consists of four elastic wir...

This work is concerned with nonlinear springs in series with the aim of obtaining the equivalent spring and its characteristics. The case of two linear springs in series is presented first as a basis for the extension to the cases of two purely nonlinear springs in series and two or more equal Duffing springs in series, which both allow the exact d...

This study is concerned with nonlinear oscillators whose equations of motion contain a linear stiffness term with a positive, zero or negative coefficient and a power-form nonlinear term whose coefficient is not necessarily small. The case of free conservative oscillations and forced non-conservative oscillations are treated separately. A new small...

This study is concerned with the use of energy methods to find the equivalent stiffness properties of certain vibration models. The first class of models are one-degree-of-freedom oscillatory systems that include several combinations of a main spring and an additional spring. The stiffness properties of the equivalent system as well as the conditio...

This study provides two theorems defining two classes of conservative nonlinear oscillators that have some characteristics of the linear harmonic oscillators, one of which is an amplitude-independent period. These theorems also define their response in terms of initial conditions. The first class of these nonlinear oscillators is also characterized...

This study is concerned with a planar tether containing payloads in which two lumped masses are fitted whose mutual distance can be contrived to change periodically in time. This periodical variation in distance of the two masses is symmetrical and results in a time-varying coefficient of the kinetic energy, which in turn introduces parametric exci...

This study is concerned with a planar tether containing payloads in which two lumped masses are fitted whose mutual distance can be contrived to change periodically in time. This periodical variation in distance of the two masses is symmetrical and results in a time-varying coefficient of the kinetic energy, which in turn introduces parametric exci...

Base excited vibration isolation systems with a purely nonlinear restoring force and a velocity nth power damper are considered. The restoring force has a single-term power form with the exponent that can be any non-negative real number. Approximations for the steady-state response at the frequency of excitation are obtained by using the Jacobi ell...

In this work, concurrent linear springs placed in the system that performs small in-plane oscillations around the stable equilibrium position are considered. New theorems defining how they can be replaced by two mutually orthogonal springs are provided. The same concept is applied to find two mutually orthogonal linear viscous dampers that can repl...

Concurrent linear springs belonging to systems that perform small out-of-plane oscillations around a stable equilibrium position are considered with a view to obtaining equivalent systems of three mutually orthogonal linear springs. Theorems defining their stiffness coefficients as well as their position, i.e. the position of the principal stiffnes...

Nonlinear oscillators with hardening and softening cubic Duffing nonlinearity are considered. Such classical conservative oscillators are known to have an amplitude-dependent period. In this work, we design oscillators with the Duffing-type restoring force but an amplitude-independent period. We present their Lagrangians, equations of motion, conse...

This work is concerned with free conservative oscillators with a single-term power-form nonlinearity. An overview of a variety of special functions that one can use to express the period as well as an exact or approximate solution for motion of these oscillators is given. First, it is demonstrated that several approaches with different special func...

In this work we propose a class of problems in nonlinear vibrations related to avoiding undesirable hysteresis and jump phenomena and offer sample conservative systems for which the backbone curve is a straight vertical line.

This work is concerned with single-degree-of-freedom conservative nonlinear oscillators that have a fixed restoring force, which comprises a linear term and an odd-powered nonlinear term with an arbitrary magnitude of the coefficient of nonlinearity. There are two cases of interest: i) non-isochronous, when the system has an amplitude-dependent fre...

In this paper, a pendulum parametrically excited by the excitation which has the form of the Jacobi cn elliptic function is considered. Three cases related to the value of the elliptic parameter are distinguished: the case when it is smaller than zero, when it ranges between zero and unity, and when it is higher than unity. First, interpretations o...

We propose a class of problems in nonlinear vibrations related to avoiding undesirable hysteresis and jump phenomena by designing an oscillator for which the backbone curve is a straight vertical line. In particular we consider the class of conservative oscillators of the form: Display Formulax..+xx.2+fx=0 and we choose f(x) so that the frequency o...

This work is concerned with nonlinear oscillators that have a fixed, amplitude-independent frequency. This characteristic, known as isochronicity/isochrony, is achieved by establishing the equivalence between the Lagrangian of the simple harmonic oscillator and the Lagrangian of conservative oscillators with a position-dependent coefficient of the...

Harmonically excited generalized van der Pol oscillators with power-form non-linearities in the restoring and damping-like force are investigated from the viewpoint of the occurrence of harmonic entrainment. Locked periodic motion is obtained by adjusting the averaging method. The influence of the powers of the restoring and damping-like force on t...

The method for finding approximate expressions for the deflection and the potential energy of linear and certain nonlinear springs is presented. These expressions are given in terms of the constant static deflection, which can be zero or non-zero, and the deflection considered in two mutually orthogonal directions and the corresponding deflections,...

Harmonically excited oscillators with a purely non-linear non-negative real-power restoring force are considered in this paper. The solution for motion is assumed in the form of a Jacobi cn elliptic function, the frequency and the parameter of which are obtained from the exact period of motion calculated from the energy conservation law. The parame...

In this study, parametrically excited purely nonlinear oscillators are considered. Instabilities associated with 2:1, 3:1, and 4:1 subharmonics resonances are analyzed by assuming the solution for motion in the form of a Jacobi elliptic function, the elliptic parameter, and the frequency of which are calculated based on the energy conservation law...

A pure cubic oscillator with a constant and a harmonic force acting on it, which represents a nonlinear asymmetric system, is considered. Building on previous studies on the matter, analytical and numerical approaches are used to examine and illustrate its dynamics related to the phenomenon of period-doubling bifurcations and their development into...

This study is concerned with autoparametric interaction in a four degree of freedom damped mechanical system consisting of two identical pendula fitted onto a horizontal massive rod which can oscillate vertically and rotationally. One pendulum is harmonically excited. The equations of motion indicate that autoparametric interaction is possible by m...

This article investigates a four-degree-of-freedom mechanical model comprising a horizontal bar onto which two identical pendula are fitted. The bar is suspended from a pair of springs and the left-hand-side pendulum is excited by means of a harmonic torque. The article shows that autoparametric interaction is possible by means of typical external...

In this paper free oscillators with a power-form restoring force and with a fractional derivative damping term are considered. An analytical approach based on the averaging method is adjusted to derive analytical expressions for the amplitude and phase of oscillations. Effects of the fractional-order derivative on the amplitude and frequency of osc...

A generalized van der Pol type oscillator is considered, with power-form nonlinearities in the restoring and damping force. The amplitude of the limit cycle is obtained by applying the generalized Krylov–Bogoliubov method for purely nonlinear oscillators. The influence of the values of the powers of the damping and the restoring force on this ampli...

The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple s...

Harmonically excited oscillators with non-negative real-power geometric nonlinearities and no linear term in the restoring force are considered. Perturbation approaches are developed for the cases of weak and strong nonlinearity. Frequency–amplitude equations are derived for an arbitrary value of the non-negative real power of the restoring force a...

Oscillators with a non-negative real-power restoring force and quadratic damping are considered in this paper. The equation of motion is transformed into a linear first-order differential equation for the kinetic energy. The expressions for the energy-displacement function are derived as well as the closed form exact solutions for the relationship...