# Nonlinear Dynamics

## List of Papers (Total 137)

#### Chaotic vibrations of flexible shallow axially symmetric shells

In this work, chaotic dynamics of flexible spherical axially symmetric shallow shells subjected to sinusoidal transverse load is studied with emphasis put on the vibration modes. Chaos reliability is verified and validated by solving the implemented mathematical model by partial nonlinear equations governing the dynamics of flexible spherical shells and by estimating the signs of ...

#### Nonlinear dynamics of a spinning shaft with non-constant rotating speed

Research on spinning shafts is mostly restricted to cases of constant rotating speed without examining the dynamics during their spin-up or spin-down operation. In this article, initially the equations of motion for a spinning shaft with non-constant speed are derived, then the system is discretised, and finally a nonlinear dynamic analysis is performed using multiple scales ...

#### Asymptotical stability of the motion of mechanical systems with partial energy dissipation

We consider a linear mechanical system under the action of potential, gyroscopic and dissipative (partial) forces. The classical Kelvin–Chetaev theorems are not applicable here, and another approach, which is based on Barbashin–Krasovskii theorem, is suggested. This approach is based on decomposition of the whole system and is convenient for systems of high dimension or with ...

#### Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

The dynamical behavior of a single-degree-of-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening–strengthening friction law. The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the ...

#### Fractal patterns from the dynamics of combined polynomial root finding methods

Fractal patterns generated in the complex plane by root finding methods are well known in the literature. In the generation methods of these fractals, only one root finding method is used. In this paper, we propose the use of a combination of root finding methods in the generation of fractal patterns. We use three approaches to combine the methods: (1) the use of different ...

#### A coupled-field model of a rotating composite beam with an integrated nonlinear piezoelectric active element

The dynamics of a system consisting of a rotating rigid hub and a thin-walled composite beam with embedded active element is presented. The beam comprises of a generally orthotropic host made of an arbitrary laminate and an additional layer of a transversely isotropic piezoceramic material. The higher-order constitutive relations for piezoelectric are used to properly model its ...

#### The planning of optimal motions of non-holonomic systems

A new method to the planning of optimal motions of the non-holonomic systems is presented. It is based on a non-classical formulation of the Pontryagin Maximum Principle given in variational form, which handles efficiently various control and/or state-dependent constraints. They arise naturally due to both physical limits of the actuators of the non-holonomic systems and potential ...

#### Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium

One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like ...

#### Analysis of a remarkable singularity in a nonlinear DDE

We investigate the dynamics of the nonlinear DDE (delay-differential equation) $\frac{\hbox {d}^2x}{\hbox {d}t^2}(t)+x(t-T)+x(t)^3=0$, where T is the delay. For $T=0$, this system is conservative and exhibits no limit cycles. For $T>0$, no matter how small T is, an infinite number of limit cycles exist, their amplitudes going to infinity in the limit as T approaches zero. We ...

#### Stability of a convex order one periodic solution of unilateral asymptotic type

In this paper, we consider semi-continuous dynamical systems with linear impulsive conditions, which have a convex order one periodic solution of unilateral asymptotic type. By constructing a sequence of switched systems and using the square approximation of the order one periodic solution, some stability criteria of the order one periodic solution are obtained. Compared with the ...

#### A robust fixed point transformation-based approach for type 1 diabetes control

Modeling and control of diabetes mellitus (DM) are difficult due to the highly nonlinear attitude, time-delay effects, the impulse kind input signals and the lack of continuously available blood glucose (BG) level to be regulated. Regarding the mentioned problems, identification of DM model is crucial. Furthermore, due to the lack of information about the internal states (which ...

#### A new equation and exact solutions describing focal fields in media with modular nonlinearity

Brand-new equations which can be regarded as modifications of Khokhlov–Zabolotskaya–Kuznetsov or Ostrovsky–Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting ...

#### Bifurcation analysis of 4-axle rail vehicle models in a curved track

The article presents authors’ recent results on nonlinear lateral stability of rail vehicles in a curved track. The theories of self-exciting vibrations and bifurcation are the key elements here. The general objective is presentation of extended use of the earlier worked out authors’ method to more complex rail vehicle models. Two 4-axle vehicle models were created. The first one ...

#### A comparative study of the vibro-impact capsule systems with one-sided and two-sided constraints

This paper studies the dynamics of the vibro-impact capsule systems with one-sided and two-sided soft constraints under variations of various system and control parameters, including mass ratio, stiffness ratio, gap of contact, and amplitude and frequency of external excitation. The aim of this study is to optimise the progression speed and energy consumption of the capsule and ...

#### Assessment of predictive control performance using fractal measures

This paper presents novel approach to the task of control performance assessment. Proposed approach does not require any a priori knowledge on process model and uses control error time series data using nonlinear dynamical fractal persistence measures. Notion of the rescaled range R/S plots with estimation of Hurst exponent is applied. Crossover phenomenon is observed in data being ...

#### Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

The stability and convergence of state, disturbance and parametric estimates of a robot have been analyzed using the Lyapunov method in the existing literature. In this paper, we analyze the problem of stochastic stability and also prove some results regarding behavior of statistically averaged Lyapunov energy function in the presence of jerk noise modeled as the sum of independent ...

#### Dynamical behavior and exact solution in invariant manifold for a septic derivative nonlinear Schrödinger equation

In this paper, we consider a pulse dynamics in nonlinear optics (fiber-optic communications) in the presence of both self-steepening and septic nonlinear effects. Propagating profiles of the septic derivative nonlinear Schrödinger model which are isolated via coupled integrable invariants of motion, that admits exact solution, are investigated by a method of dynamical systems. By ...

#### Application of dynamic optimisation to stabilise bending moments and top tension forces in risers

The study discusses the problem of determining vertical displacements of a riser’s ends, which, despite its horizontal displacements induced by waves, mitigate stresses. A spatial model of riser dynamics is presented that considers the geometric nonlinearity due to large deflections. The Rigid Finite Element Method was used for riser discretisation. Analyses are reported that ...

#### Inertia forces and shape integrals in the floating frame of reference formulation

Modeling and analysis of complex dynamical systems can be effectively performed using multibody system (MBS) simulation software. Many modern MBS packages are able to efficiently and reliably handle rigid and flexible bodies, often offering a wide choice of different formulations. Despite many advances in modeling of flexible systems, the most widely used formulation remains the ...

#### Egalitarian versus prioritarian approach in multiple task motion planning for nonholonomic systems

In this paper, two different concepts of multiple task motion planning algorithm for nonholonomic systems are considered. The egalitarian approach treats all the tasks equivalently and tries to solve all the tasks simultaneously. In contrast, the prioritarian approach arranges the tasks with decreasing priorities in such a way that the solution of the lower order task should not ...