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In this study, nonlinear vibration and stability of a pipe conveying fluid, with axially moving supports, immersed in an external fluid is studied. The equation of motion is derived with the aid of the Newtonian method via employing the Euler–Bernoulli beam theory and plug flow assumption. Considering the stretching effect, the nonlinear equation of motion is obtained, and it is discretized via the Galerkin method. Afterward, the stability of the system is investigated using two different approaches, and their results are compared. Numerical simulations show the system has complex dynamic behavior. To unveil the dynamics of the system, various analysis techniques such as bifurcation diagram of Poincaré map, phase portrait, power spectrum, and time trajectories are used. Limit cycle, period-n, quasi-periodic, and chaotic regime are observed. It is found that pipe can flutter around the trivial solution or post-buckling equilibrium points. Moreover, the influences of some system parameters on the chaotic behavior of the pipe are discussed. [ABSTRACT FROM AUTHOR]

10.3390/app12052333 23 Hong J., Jiang L., Wang Y., Su Z., Ma Y. Nonlinear Dynamics of an Elastic Stop System and Its Application in a Rotor System. Nonlinear phenomena occur in engineering structures, biological systems, social processes, and in economics. Particle weight adjustment can ensure the stability, tracking efficiency, and continuity of the particle filter control process. Xu et al. [[8]] develop a particle filter design scheme for a robust nonlinear control system of an uncertain heat exchange process against noise and communication delay. [Extracted from the article]

Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system that correlates with two devices, namely a piezoelectric device and an electromagnetic one, to produce two novel models. These devices are connected to a nonlinear damping spring pendulum with two degrees of freedom. The damping spring pendulum is supported by a point moving in a circular orbit. Lagrange's equations of the second kind were utilized to obtain the equations of motion. The asymptotic solutions of these equations were acquired up to the third approximation using the approach of multiple scales. The comparison between the approximate and the numerical solutions reveals high consistency between them. The steady-state solutions were investigated, and their stabilities were checked. The influences of excitation amplitudes, damping coefficients, and the different frequencies on energy-harvesting device outputs are examined and discussed. Finally, the nonlinear stability analysis of the modulation equations is discussed through the stability and instability ranges of the frequency response curves. The work is significant due to its real-life applications, such as a power supply of sensors, charging electronic devices, and medical applications. [ABSTRACT FROM AUTHOR]

Results of the simulations of vibrations of collecting electrode system of dry electrostatic precipitator (ESP) are reported. The system of nine electrodes 16m long, suspension beam (upper beam) and anvil beam (bottom beam) is analyzed. First, the variants regarding differ position of the anvil beam are analyzed. Second, the differ position of a point of the impulse force as well as direction of the impulse force are investigated (in this case the anvil beam is in a 'classic' position). Discretization of both beams and electrodes is carried out employing the modified rigid finite element method (RFEM). It allows to study an influence of the mass distribution and geometric properties of the analyzed system. Moreover, the method is convenient for the introduction of additional lumped mass parameters modeling the joints that fasten the electrodes to the suspension beam, distance-marking bushes, riveted and screw joints, etc. Position of the anvil beam and position/direction of the point of impulse force application can have essential influence on tangent and normal accelerations at different points of the plates, and thus on the effectiveness of the dust removal process. We aim to find optimal position of an anvil beam, which guaranties the maximum amplitudes of vibrations and their proper distribution in the plates. [ABSTRACT FROM AUTHOR]

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA). [ABSTRACT FROM AUTHOR]

In this work, nonlinear longitudinal vibrations of a finite composite rod are studied including geometric and physical nonlinearities. An original boundary value problem for a heterogeneous rod yielded by the macroscopic approximation obtained earlier by the higher-order asymptotic homogenization method is used. The effects of internal resonances and modes coupling are predicted, validated and analyzed. The defined novel continuous problem governed by PDEs is solved using space-discretization and the method of multiple time scales. We are aimed at understanding and analyzing how the presence of the microstructure influences the processes of mode interaction. It is shown that, depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell, different scenarios of the modes coupling can be realized. Additionally to the asymptotic solution, numerical simulation of the modes coupling is performed by means of the Runge–Kutta fourth-order method. The obtained numerical and analytical results demonstrate good qualitative agreement. [ABSTRACT FROM AUTHOR]

A novel numerical/analytical approach to study geometrically nonlinear vibrations of shells with variable thickness of layers is proposed. It enables investigation of shallow shells with complex forms and different boundary conditions. The proposed method combines application of the R-functions theory, variational Ritz’s method, as well as hybrid Bubnov–Galerkin method and the fourth-order Runge–Kutta method. Mainly two approaches, classical and first-order shear deformation theories of shells are used. An original scheme of discretization regarding time reduces the initial problem to the solution of a sequence of linear problems including those related to linear vibrations with a special type of elasticity, as well as problems governed by non-linear system of ordinary differential equations. The proposed method is validated by the investigation of test problems for shallow shells with rectangular planform and applied to new vibration problems for shallow shells with complex planforms and variable thickness of layers. [ABSTRACT FROM AUTHOR]

The dynamic response of a nonlinear system with three degrees of freedom, which is excited by nonideal excitation, is investigated. In the considered system the role of a nonideal source is played by a direct current motor, where the central axis of the rotor is not coincident with the axis of rotation. This translation generates a torque whose magnitude depends on the angular velocity. During the system operation a general coordinate assigned to the nonideal source grows rapidly as a result of rotation. We propose the decomposition of the equations of motion in such a way to extract the solution which is directly related to the rotation of an unbalanced rotor. The remaining part of the solution describes pure oscillation depending on the dynamical behaviour of the whole system. The decomposed equations are solved numerically. The influence of selected system parameters on the rotor vibration is examined. The presented approach can be applied to separate vibration and rotation of motions in many other engineering systems. [ABSTRACT FROM AUTHOR]

The present formulation of the analysed problem is based on Donell's nonlinear shallow shell theory, which adopts Kirchhoff's hypothesis. Transverse shear deformations and rotary inertia of a shell are neglected. According to this theory, the non-linear strain-displacement relations at the shell midsurface has been proposed. The validity and reliability of the proposed approach has been illustrated and discussed, and then a few examples of either linear or non-linear dynamics of shells with variable thickness and complex shapes have been presented and discussed. [ABSTRACT FROM AUTHOR]

A weakly nonlinear 2-DOF system, parametrically and externally excited, is studied. An intensive energy transfer between modes of vibrations is discovered. Multiple scales method is used for recognizing resonances occurring in the system. The amplitude response functions for some chosen cases of resonances are obtained and studied. Many of the engineering systems can be investigated in the presented way. [ABSTRACT FROM AUTHOR]

Abstract: The paper is dedicated to mathematical analysis of the idealized model of a string moving at a variable speed. Resonance case is studied and it is shown that an infinite system of equations cannot be reduced. Some exact solutions are obtained. [Copyright &y& Elsevier]

Abstract: The paper focuses on 1D continuous models derived from a discrete micro-structure. A new continualization procedure that takes into account the nonlocal interaction between variables of the discrete media is proposed. The proposed procedure mainly contains an application of two-point Padé approximations and allows obtaining continuous models suitable to analyze 1D lattice vibrations with arbitrary frequencies. [Copyright &y& Elsevier]

The paper introduces the constructed prototype of the hexapod robot designed based on the biomechanics of insects for inspection and operation applications as well as for different research investigations related to the walking robots. A detailed discussion on the design and realization of mechanical construction, electronic control system and devices installed on the robot body are presented. Moreover, the control problem of the robot legs is studied in detail. In order to find the relationship between movements commonly used by insects legs and stable trajectories of mechanical systems, first we analyze different previous papers and leg movements of real insects. Next, we are focus on the control the robot leg with several oscillators working as a so-called Central Pattern Generator (CPG) and we propose other model of CPG based on the oscillator describing stick-slip induced vibrations. Some advantages of the proposed model are presented and compared with other previous applied mechanical oscillators with help of numerical simulations performed for both single robot leg and the whole robot. In order to confirm the mentioned numerical simulations, the conducted real experiments are described and some interesting results are reported. Both numerical and experimental results indicate some analogies between the characteristics of the simulated walking robot and animals met in nature as well as the benefits of the proposed stick-slip vibrations as a CPG are outlined. Our research work has been preceded by a biological inspiration, scientific literature review devoted to the six-legged insects met in nature as well as various prototypes and methods of control hexapod robots which can be found in engineering applications. [ABSTRACT FROM AUTHOR]

Non-stationary effects in the friction-induced dynamics of a two-degree-of-freedom brake model are examined in this paper. The belt–spring–block model is designed to take into account variations of the normal load during the braking process. It is shown that due to the adiabatically slowing down velocity of the belt, the system response experiences specific qualitative transitions that can be viewed as simple mechanical indicators for the onset of squeal phenomenon. In particular, the creep-slip leading to a significant widening of the spectrum of the dynamics is observed at the final phase of the process. [ABSTRACT FROM AUTHOR]

A new method of consideration and reduced analysis of long-time series is suggested. The method is based on the scaling (fractal) properties of the beta-distribution function that allows finding the stable parameters and reducing the long-time series containing 5×10 5 –10 6 data points to analysis of 10–20 stable parameters. The new procedure of clusterization with the usage of the generalized Pearson correlation function allows to take into account the influence of external factor and combine/separate different parameters into a cluster with respect to the influence of the qualitative external parameter considered. The method is rather flexible and can be applied to wide set of large data. As an example the triple pendulum data are considered. This method opens new possibilities in creation of the reduced database (specific fingerprints) of different long-time series for their comparison and subsequent analysis. [ABSTRACT FROM AUTHOR]

Control of vibrations in fluid transmitting pipes has emerged as a major engineering challenge. For higher fluid flow rate, support vibration, and excitation load, the vibrations in pipes that convey fluid can be complex and, at times, catastrophic. Recent works have demonstrated that Nonlinear Energy Sink (NES) can be effectively employed for passive vibration control of fluid-conveying pipes. The present work explores the possibility of using NES with nonlinear damping to overcome the limitations of conventional NES-based vibration control in fluid-transmitting pipes. Geometric nonlinear damping (velocity-displacement dependent damping), whose physical realization is possible by a suitable geometric configuration of the linear viscous damper, is considered in this study. Euler–Bernoulli beam theory models the fluid conveying pipe under an external harmonic load. The reduced order model obtained through the Galerkin procedure is investigated analytically using the complex averaging method. The occurrence of strongly modulated and weakly modulated responses are demonstrated, and their effect on vibration mitigation is shown. The frequency range exhibiting strongly modulated response in the system is significantly increased with the addition of nonlinear damping to NES. This assists in mitigating pipe vibrations by initiating targeted energy transfer from pipe to NES. The Saddle–Node and Hopf bifurcation boundaries in the model are identified using the slow-flow dynamics obtained by complex averaging. The effect of nonlinear damping, external excitation, the location of NES, and the flow rate on the bifurcation boundaries are investigated in detail. It is shown that the external excitation needed to trigger Saddle–Node bifurcation is largely reduced for NES having nonlinear damping. Furthermore, a study of Hopf bifurcation in the frequency domain shows that using NES with nonlinear damping significantly decreases the frequency range in which high-amplitude isolated solutions occur. The method of multiple scales is used to derive the analytical expression for the slow invariant manifold of the pipe-NES system. The vibration suppression mechanism and the occurrence of strongly modulated responses are explained with the help of flow on slow invariant manifolds. A comparison of the topology of the slow invariant manifolds shows that using nonlinear damping with NES substantially reduces the vibration amplitude of the pipe structure. The percentage of energy transferred to the NES is quantified to show the effectiveness of the proposed absorber over normal NES for different flow speeds. [ABSTRACT FROM AUTHOR]