Nonlinear Dynamics Of Mechanical Systems Research Articles

“Nonlinear Dynamics of Mechanical Systems” dedicated to 75th birthday and 52 years scientific contribution of Prof. Katica R. (Stevanovic) Hedrih

“Nonlinear Dynamics of Mechanical Systems” dedicated to 75th birthday and 52 years scientific contribution of Prof. Katica R. (Stevanovic) Hedrih

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams.

Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.

Computational analysis of nonlinearities within dynamics of cable-based driving systems

This paper deals with computational nonlinear dynamics of mechanical systems containing some flexural parts within the actuating scheme, and, especially, the situations of the cable-based driving systems were treated. It was supposed both functional nonlinearities and the real characteristic of the power supply, in order to obtain a realistically computer simulation model being able to provide very feasible results regarding the system dynamics. It was taken into account the transitory and stable regimes during a regular exploitation cycle. The authors present a particular case of a lift system, supposed to be representatively for the objective of this study. The simulations were made based on the values of the essential parameters acquired from the experimental tests and/or the regular practice in the field. The results analysis and the final discussions reveal the correlated dynamic aspects within the mechanical parts, the driving system, and the power supply, whole of these supplying potential sources of particular resonances, within some transitory phases of the working cycle, and which can affect structural and functional dynamics. In addition, it was underlines the influences of computational hypotheses on the both quantitative and qualitative behaviour of the system. Obviously, the most significant consequence of this theoretical and computational research consist by developing an unitary and feasible model, useful to dignify the nonlinear dynamic effects into the systems with cable-based driving scheme, and hereby to help an optimization of the exploitation regime including a dynamics control measures.

Multi-body contact in non-linear dynamics of real mechanical systems

The considerable number of real mechanical systems operates in conditions of multi-body contact. Very often, elastic properties in contact areas couldn’t be neglected, so the elastic deformations exist in contact area of each two deformable bodies in contact. When there are two or more contacts at the same time and rotations and/or translations of contact bodies, these conditions cause the continuous changes of contact areas’ geometry, friction, load distribution and other parameters. Therefore, the problem of non-linear dynamics of mechanical system becomes very complex.In this paper, the new approach for analyzing the non-linear dynamics of mechanical systems with multi-body contacts will be presented. In this approach, the contact bodies in mechanical systems are simulated with basic geometric deformable bodies coupled with elements with time-varying stiffness and damping. The main principles of mathematical phenomenological mapping are used for multi-body contact reduction to simple single-degree of few-degrees of freedom systems with total stiffness as main characteristic. The cases of two and three basic geometric bodies in contact are modeled to enable the investigation of influence of main parameters (stiffness, damping, external load, friction coefficient) in non-linear dynamics of multi-body contacts. The Finite Element Analysis is used to calculate the time-varying contact deformations, total stiffness and external load distribution.A particular attention is paid to the presentation of new approach application on real mechanical systems with multi-body contacts, such as gears and ball bearings. The analyses of non-linear dynamics of a particular gear pair and a particular ball bearing will illustrate the capabilities and advantages of presented new approach for comfortable and qualitative analysis of nonlinear dynamic behavior of real complex mechanical systems.

Increasing Practical Safety of Von Mises Truss via Control of Dynamic Escape

This paper investigates the nonlinear dynamics and stability of the shallow von Mises truss, which is a prototype for buckling analysis of several planar and spatial truss systems and shallow lattice shell structures, including the geodesic dome, and which has a theoretical and practical interest in many engineering fields. These structural systems are liable to limit point instability and may fail at load levels well below the theoretical limit point load due to complex nonlinear phenomena that reduce the safety and the dynamic integrity of the structure; this often occurs as a consequence of the erosion of the basins of attraction of the safe pre-buckling solutions. So, it is vital to increase the safety of the structure in a dynamic environment. In this paper a method for controlling the global nonlinear dynamics of mechanical systems is applied to the shallow von Mises truss. The method consists of the (optimal) elimination of homoclinic intersection by properly adding superharmonic terms to a given harmonic excitation. By means of the solution of an appropriate optimization problem, it is possible to select the amplitudes and the phases of the added superharmonics in such a way that the manifolds distance is as large as possible. This methodology is here applied to increase the integrity of the basins of attraction of the system and consequently its practical safety.

Nonlinear dynamics of mechanical systems with friction contacts: Coupled static and dynamic Multi-Harmonic Balance Method and multiple solutions

Real applications in structural mechanics, where the dynamic behavior is linear, are rare. Usually, structures are made of components assembled together by means of joints whose behavior maybe highly nonlinear. Depending on the amount of excitation, joints can dramatically change the dynamic behavior of the whole system, and the modelling of this type of constraint is therefore crucial for a correct prediction of the amount of vibration. The solution of the nonlinear equilibrium equations by means of the Harmonic Balance Method (HBM) is widely accepted as an effective approach to calculate the steady state forced response in the frequency domain, in spite of Direct Time Integration (DTI). The state of the art contact element used to model the friction forces at the joint interfaces is a nodal contact element, where the local contact compliance is modelled by means of linear springs and Coulomb law is used to govern the friction phenomena. In the literature, when the HBM is applied to vibrating systems with joint interfaces and the state-of-the-art contact model is used, an uncoupled approach is mostly employed: the static governing equations are solved in advance to compute the pre-stress effects and then the dynamic governing equations are solved to predict the vibration amplitude of the system. As a result, the HBM steady-state solution may lead to a poor correlation with the DTI solution, where static and dynamic loads are accounted for simultaneously. In this paper, the HBM performances are investigated by comparing the uncoupled approach to a fully coupled static/dynamic approach. In order to highlight the main differences between the two approaches, a lumped parameter system, characterized by a single friction contact, is considered in order to show the different levels of accuracy that the proposed approaches can provide for different configurations.

On the computation of the slow dynamics of nonlinear modes of mechanical systems

A novel method for the numerical prediction of the slowly varying dynamics of nonlinear mechanical systems has been developed. The method is restricted to the regime of an isolated nonlinear mode and consists of a two-step procedure: In the first step, a multiharmonic analysis of the autonomous system is performed to directly compute the amplitude-dependent characteristics of the considered nonlinear mode. In the second step, these modal properties are used to construct a two-dimensional reduced order model (ROM) that facilitates the efficient computation of steady-state and unsteady dynamics provided that nonlinear modal interactions are absent.The proposed methodology is applied to several nonlinear mechanical systems ranging form single degree-of-freedom to Finite Element models. Unsteady vibration phenomena such as approaching behavior towards an equilibrium point or limit cycles, and resonance passages are studied regarding the effect of various nonlinearities such as cubic springs, unilateral contact and friction. It is found that the proposed ROM facilitates very fast and accurate analysis of the slow dynamics of nonlinear systems. Moreover, the ROM concept offers a huge parameter space including additional linear damping, stiffness and near-resonant forcing.

Controlling practical stability and safety of mechanical systems by exploiting chaos properties

In this paper, a method for controlling the global nonlinear dynamics of mechanical systems is applied to two models: the model of Augusti and an inverted guyed pendulum. These simplified models represent a large class of structures liable to buckling exhibiting interacting buckling phenomena. These structures may fail at load levels well below the theoretical buckling load due to complex nonlinear phenomena that decrease the safety and the dynamic integrity of the structure; this often occur as a consequence of imperfections and of the erosion of the basins of attraction of the safe pre-buckling solutions. So, it is of paramount practical importance to increase the safety of these structures in a dynamic environment. This can be achieved by increasing the integrity of the basins of attraction of the safe solutions, a goal that is attained by a control method which consists of the (optimal) elimination of homoclinic (or heteroclinic) intersection by properly adding superharmonic terms to a given harmonic excitation. By means of the solution of an appropriate optimization problem, it is possible to select the amplitudes and the phases of the added superharmonics in such a way that the manifolds distance is as large as possible. The results show that this methodology increases the integrity of the basins of attraction of the system and, consequently, the practical safety of the structure.

Controlling Nonlinear Dynamics of Systems Liable to Unstable Interactive Buckling

A method for controlling the global nonlinear dynamics of mechanical systems is applied to an archetypal model of a guyed tower. This structure has coincident or nearly coincident buckling modes, and may fail at load levels well below the theoretical buckling load due to complex nonlinear phenomena that decrease the safety and the dynamic integrity of the structure; this often occurs as a consequence of imperfections and of the erosion of the basins of attraction of the safe pre-buckling solutions. So, it is of paramount practical importance to increase the safety of these structures in a dynamic environment. This can be achieved by increasing the integrity of the basin of attraction of the safe solutions, that in the present investigation is attained by a control method that consists in the optimal elimination of homoclinic intersections by properly adding superharmonic terms to a given harmonic excitation. By means of the solution of an appropriate optimization problem, it is possible to select the amplitudes and the phases of the added superharmonics in such a way that the manifolds distance is as large as possible. The results show that this methodology increases the integrity of the basins of attraction of the system and, consequently, the practical safety of the structure.

A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008

In this article, we provide an overview of a selection of topics of current interest in nonlinear dynamics and vibrations of mechanical systems. Specifically, we cover the traditional topics of structural dynamics, rotating systems, vibration control, vehicle dynamics, and machining, as well as some topics that have emerged more recently, namely micro- and nano-electromechanical systems, piecewise smooth systems, and structural health monitoring and system identification. In the introductory and closing sections, we offer our perspective on the state of affairs in nonlinear dynamics and its utility in the study of mechanical systems.

Nonlinear Dynamics of Mechanical Systems with Sectional Frictions

This paper presents mainly the nonlinear dynamics of the mechanical system with sectional frictions and combination friction coefficients. It is clear that the nonlinear dynamic characteristics of mechanical systems with sectional frictions are quite different from those of classical machinery, and have more precise and valuable in many practical projects. The expressions of various nonlinear forces are given firstly and the approximation solutions of the system are found with asymptotic method in nonlinear theory, and the combination friction coefficients and damping coefficients of the materials are obtained. Then some nonlinear dynamic characteristics of the system with sectional frictions are also discussed. The results are very important for designers of these machines.

Tensile-compressive asymmetry influence on shape memory alloy system dynamics

The remarkable properties of shape memory alloys (SMAs) are attracting much technological interest in several science and engineering fields, varying from medical to aerospace applications. Hysteretic response of these systems is one of their essential characteristics being related to the martensitic phase transformation. The dynamical response of systems with SMA actuators presents a rich behavior due to their intrinsic nonlinear characteristic. Since experimental results show that SMAs present an asymmetric behavior when subjected to tensile or compressive loads, it is important to evaluate the influence of this kind of behavior in the nonlinear dynamics of mechanical systems with SMA devices. This article discusses the nonlinear dynamics of shape memory alloy systems, considering the influence of tensile-compressive asymmetry in the thermomechanical behavior of SMAs. An iterative numerical procedure based on the operator split technique, the orthogonal projection algorithm and the fourth-order Runge–Kutta method is developed to deal with nonlinearities in the formulation. A numerical investigation is carried out showing some qualitative results such as chaotic-like response and multi-stability behavior for a single degree of freedom SMA oscillator.

Preface: Recent advances in nonlinear dynamics of mechanical systems

Preface: Recent advances in nonlinear dynamics of mechanical systems

Nonlinear Dynamics of Mechanical Systems Under Complicated Forcing

This paper considers the problem of simple linear mechanical oscillators forced by complicated forcing functions, such as those induced by fluids or humans. Such problems arise in civil engineering in the design of bridges, grandstands and towers. The paper questions whether low dimensional models have much to contribute to the understanding of such problems, given the complicated possibilities of high-dimensional forcing.

Nonlinear dynamics of mechanical system with external excitation

Abstract This paper attends to the problem of vibration of a self-excited mechanical system with a relatively complicated, history dependent dry friction model. The experimentally identified friction model allows description of various cases of stationary and transient motion. The system is composed of a lumped mass that interacts with a belt by means of dry friction. The system is additionally subjected to an external harmonic excitation through elastic element. The main objective of the experimental research has been focused on the analysis of the system behaviour for various values of the excitation frequency. This paper includes also computer simulation of the vibration of the considered system and comparison between the results of experimental and theoretical analysis. The comparison enables the assumed friction model for steel-polyester pair to be verified. It was another goal of the investigations.

Piecewise analysis of oblique vibro-impacting systems

A great number of studies have shown the complex nonlinear dynamics of mechanical systems with repeated normal impacts. An oblique frictional impact introduces even more complicated dynamics such as stick-slip motions to those systems. Hence, the dynamics of oblique vibro-impacting systems with possible sliding motion is an open problem. Based on a hybrid analysis of vibro-impact dynamics, kinematics and complementary conditions, a piecewise analysis method is developed in the paper to describe the sliding motion during an oblique impact. Thereby, a parametrically excited planar pendulum between two parallel rigid walls is studied as an illustrative example. The example, together with the corresponding numerical results, shows that the sliding impacts occur in such a system with a set of properly selected parameters.

Nonlinear dynamics of mechanical systems: Foreword

Nonlinear dynamics of mechanical systems: Foreword

Nonlinear dynamics of milling processes

In this article, dynamics and stability of milling operations with cylindrical end mills are investigated. A unified–mechanics–based model, which allows for both regenerative effects and loss–of–contact effects, is presented for study of partial–immersion, high–immersion and slotting operations. Reduced–order models that can be used for certain milling operations such as full–immersion operations and finishing cuts are also presented. On the basis of these models, the loss of stability of periodic motions of the workpiece–tool system is assessed by using Poincare sections and the numerical predictions of stable and unstable motions are found to correlate well with the corresponding experimental observations. Bifurcations experienced by periodic motions of the workpiece–tool system with respect to quasi–static variation of parameters such as axial depth of cut are examined and discussed. For partial–immersion operations, consideration of both time–delay effects and loss–of–contact effects is shown to have a significant influence on the structure of the stability boundaries in the space of spindle speed and axial depth of cut. The sensitivity of system dynamics to multiple–regenerative effects, mode–coupling effects and feed rate is also discussed.

Dynamic analysis of rigid and deformable multibody systems with penalty methods and energy–momentum schemes

A multibody formulation for the nonlinear dynamics of mechanical systems composed of both rigid and deformable bodies is proposed in this work, focusing on its conservation properties for basic magnitudes such as total energy and momentum. The approach is based on the use of dependent variables (cartesian coordinates of selected points) and the enforcement of the constraints through the penalty method. This choice has the advantage of providing a simple overall structure that allows the inclusion of both rigid bodies (discrete model) and elastic bodies (continuum model discretised with the finite element method) under the same framework, in order to build a single set of ordinary differential equations. The elastic bodies are represented by general hyperelastic models and may undergo large displacements, rotations and strains. An energy–momentum time integration method has been employed, achieving remarkable stability and robustness with exact conservation of total energy. This approach effectively overcomes drawbacks associated with penalty formulations in other time integration algorithms. This important result in fact proves to be the main conclusion of this work. Some representative numerical simulations are presented for mechanical systems comprised of rigid and deformable bodies.

Nonlinear Position-Dependent Circuits: Another Language for Describing Robot Motions and Mechatronic Systems

A class of nonlinear position-dependent circuits is introduced in order to give a physical understanding of nonlinear dynamics of mechanical systems and facilitate the performance analysis of robot controllers such as PD and SP-ID with or without regressors. The robustness of such simple controllers against uncertainty of dynamic parameters is analyzed on the basis of "passivity" of an input-output pair of a circuit, which is considered to be a generalization of impedance concept. It is shown that the passivity implies the H∞-tuning in a sense of disturbance attenuation. In addition, a necessary and sufficient condition for such a nonlinear system to maintain the passivity for all structural disturbances within a given margin is obtained.