Chaotic Vibrations Research Articles

Chaotic vibrations of a harmonically excited carbon nanotube with consideration of thermomagnetic filed and surface effects

In the studies of the dynamic response of carbon nanotubes, the stability, predictable, and unpredictable chaotic vibrations are fundamental characteristics. In this paper, we investigate the chaotic and periodic vibrations of a single-walled carbon nanotube resting on the viscoelastic foundation, based on the nonlocal Euler–Bernoulli beam model. It is assumed that the single-walled carbon nanotube is subjected to an external harmonic excitation. The axial thermomagnetic field and the surface effect on the governing equation of single-walled carbon nanotube are taken into account. We also solve the nonlinear governing equation by using the Galerkin decomposition method along with the fourth-order Rung–Kutta numerical integration scheme. Furthermore, we analyze the effects of amplitude and frequency of excitation on the formation of chaotic and periodic regions using bifurcation diagrams and largest Lyapunov exponents. Moreover, we present the phase portrait, Poincare maps, and time history to observe the periodic and chaotic responses of the system. The results show that the nonlinear dynamic response of single-walled carbon nanotube is much more sensitive to both amplitude and frequency of excitation.

Chaotic motion of piezoelectric material hyperbolic shell under thermoelastic coupling

Piezoelectric material, which exhibits excellent electro-mechanical conversion properties, is widely used in smart sensors and structures for sonar systems, weather detection and remote sensing. Hyperbolic shell structure made of piezoelectric material is liable to break down when it is used in high temperature environment, which is caused by the unexpected chaotic dynamic motion under the coupling effect of thermal filed and force field. Therefore, the chaotic nonlinear dynamic vibration of simply-supported piezoelectric material hyperbolic shell is studied under the combined action of temperature field and simple harmonic excitation. Based on the theory of finite deformation, the non-linear vibration equation and coordination equation of the hyperbolic shell are established. The non-linear dynamic equation of the structure is obtained by the Bubnov-Galerkin principle. The corresponding undisturbed Hamilton system has a homoclinic orbit. Using Melnikov function, the chaotic motion condition of the dynamic system under the criterion of Smale-horseshoe transformation is obtained. Furthermore, the mathematical model is established by Simulink software and the numerical simulations are performed by the fourth-order Runge-Kutta method. The simulation results accord well with those from the Melnikov method. The bifurcation diagram, the Lyapunov exponent diagram, the phase diagram and Poincaré section diagram are acquired to analyze the influence of temperature field on the non-linear characteristic of piezoelectric material hyperbolic shell system. When the temperature is close to 32℃ and 41℃, the Lyapunov index is less than 0 and the corresponding movement of the system is in the periodic zone, which is the same as that for a temperature range from 36℃ to 37℃. When the Lyapunov index is greater than 0, the corresponding movement of the system is in chaos zone. Therefore, the change of temperature has an additional effect on the stiffness of the system which affects the vibration of the system. The chaos and periodic zones of the system alternate with the increase of temperature and the vibration characteristics of the system can be controlled by changing the temperature field. Therefore, adjusting the temperature field can control the motion state of the system, which helps to improve reliability of the structure.

Identification of Chaotic Vibration Based on Continuous Wavelet Transform

Identification of Chaotic Vibration Based on Continuous Wavelet Transform

Analysis of chaotic vibration of Shilnikov-type in rotor with asymmetric and non-linear stiffness

We study the chaotic vibration of a rotor with an asymmetric and non-linear stiffness. The system of motion equations of the rotor in the fixed coordinate system is derived and converted to the normal form by the method of multiple scales. On the basis of the normal form, we derive the conditions under which the chaotic vibration of the Shilnikov-type in the rotor can occur. Finally, the numerical simulations are presented to verify the above-mentioned conditions.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

Analyzing Complex Dynamic Modes in Different Modifications of Relativistic Generators on a Virtual Cathode

Features of chaotic vibrations generated in one-beam and multibeam vircators are studied. The transition to the chaotic generation is found to pass in accordance with a complex quasi-periodical dynamics, which is confirmed by the calculated Lyapunov exponents.

Analysis of the stick-slip vibration of a new brake pad with double-layer structure in automobile brake system

Abstract This study establishes a dynamic model of a new brake system with two-layer structure to explore the mechanism of vibration and noise reduction of a brake pad with new structure. The model is used to analyze the effects of parameters of the double-layer pad on the stability and stick-slip vibration characteristics of the brake system. The stability of the brake system is optimized by brake parameters with respect to the stability diagram of brake pressure under a relatively wide range. System vibration modes can be changed from period-doubling bifurcation to chaos based on the variation of brake pressure and the parameters of double-layer pad via numerical analysis. The chaotic vibration region is optimized in this study using the correlation coefficient, which consists of friction-layer pad mass mb2 and connection stiffness k2. Results show that the range of chaotic vibration region can be reduced when the friction-layer pad mass and connection stiffness are large.

Chaotic Characteristics Analysis and Sliding Mode Control for Hydraulic Turbine Governing Systems

A nonlinear mathematical model for hydro-turbine governing system (HTGS) has been proposed. Using the proposed model, chaotic characteristics of the system with practical system parameters are studied extensively and presented in forms of Lyapunov exponent, chaotic attractors, sensitivity to initial conditions and the Poincare map. And then a mathematical control scheme of the system is deduced based on the integral sliding mode control. Numerical simulations show that the chaotic vibration is eliminated and the system could be controlled to any fixed points or any periodic orbits strictly and smoothly with a short transition time. The simulation results verify the feasibility of the method, and also provide a reasonable reference for relevant chaos control of HTGS in future.

Lyapunov exponent of friction-induced vibration under smooth friction curve

This paper studies the nonlinear behavior of the friction-induced vibration by using spring-mass model subject to the smooth frictionvelocity curve. The nonlinearity and instability of the friction may produce the chaotic vibration depending on the friction curve. In order to show this, the Lyapunov exponents are calculated for a variety of the slope and magnitude in the smooth friction curve. In turn, the dependency of the friction curve on the chaotic attractor is illustrated.

Multi-scale simulation of nonlinear thin-shell sound with wave turbulence

Thin shells --- solids that are thin in one dimension compared to the other two --- often emit rich nonlinear sounds when struck. Strong excitations can even cause chaotic thin-shell vibrations, producing sounds whose energy spectrum diffuses from low to high frequencies over time --- a phenomenon known as wave turbulence. It is all these nonlinearities that grant shells such as cymbals and gongs their characteristic "glinting" sound. Yet, simulation models that efficiently capture these sound effects remain elusive. We propose a physically based, multi-scale reduced simulation method to synthesize nonlinear thin-shell sounds. We first split nonlinear vibrations into two scales, with a small low-frequency part simulated in a fully nonlinear way, and a high-frequency part containing many more modes approximated through time-varying linearization. This allows us to capture interesting nonlinearities in the shells' deformation, tens of times faster than previous approaches. Furthermore, we propose a method that enriches simulated sounds with wave turbulent sound details through a phenomenological diffusion model in the frequency domain, and thereby sidestep the expensive simulation of chaotic high-frequency dynamics. We show several examples of our simulations, illustrating the efficiency and realism of our model.

On the contact interaction of a two-layer beam structure with clearance described by kinematic models of the first, second and third order approximation

Abstract In this article, the contact interaction of two geometrically non-linear beams with a small clearance is studied by employing various approximations of the kinematic beam models. Our investigation concerns two-case studies, i.e. when the upper beam is governed by the second order Timoshenko model (i) or by third order Pelekh-Sheremetev-Reddy-Levinson model (ii), whereas in both cases the lower beam is described by the kinematic (Bernoulli-Euler) model of the first approximation. The upper beam is subjected to the transversal, uniformly distributed harmonic load, whereas the beam interaction follows the classical Kantor’s model. The problem is highly non-linear due to occurrence of the geometric von Karman non-linearity and the contact interaction between beams (structural non-linearity). The governing PDEs are reduced to ODEs by the method of finite differences (FDM) of the second order. The obtained ODEs are solved by a few Runge-Kutta type methods of different orders. Results of convergence versus the number of partition points along the spatial co-ordinate and time steps are investigated. New non-linear phenomena of the studied structural package are detected, illustrated and discussed with emphasis on the “true” chaotic vibrations. In particular, three qualitatively different algorithms for computation of largest Lyapunov exponents are employed, and the differences between geometrically linear versus non-linear problems are reported.

Bifurcation and chaos analysis of a gear pair system with multi-clearance

In order to investigate the characteristics of bifurcation and chaos for a spur gear pair system, a three-degree-of-freedom nonlinear dynamic model with multi-clearance is established, in which time-varying meshing stiffness, static transmission error, gear backlash and bearing clearance are comprehensively taken into account. Through introducing a relative generalized coordinate, the dimensionless dynamic equations of motion of system are derived and then solved by using Runge-Kutta numerical integration method. And the bifurcation and chaos features of gear pair are systematically analyzed and discussed from bifurcation diagrams with meshing frequency, gear backlash, bearing clearance and damping ratio as control parameters under different loaded conditions. Meantime, with the help of Poincaré map and phase diagram, the motion forms of system are accurately identified. The analysis results reveal that as meshing frequency increases, the system shows various types of motion states which contain periodic motion, quasi-periodic motion and chaotic motion. Similarly, with the increasing of gear backlash, the system undergoes complex motion forms under lightly loaded condition, whereas it is only in period-one motion state under heavily loaded condition. Furthermore, the system motion state is gradually switched from chaos to periodic or quasi-periodic motion under lightly loaded condition when bearing clearance changes. However, under heavily loaded condition, the bearing clearance has a weak effect on dynamic behavior of the gear system. Apparently, the system tends to be more stable under heavily loaded condition than that under lightly loaded condition. In addition, the growing damping ratio can effectively suppress the chaotic behavior and control nonlinear vibration of gear system. The research results provide useful guidance for dynamic design and vibration control for gear set.

Experimental verification of the effectiveness of elastic cross-ties in suppressing wake-induced vibrations of staggered stay cables

In the present study, two pairs of cable models were specially designed and tested to reproduce wake induced vibration (WIV) of stay cables in a wind tunnel. The interaction between multi-modes and the flexibility of the stay cables were considered. The downwind cables experienced large vibrations beyond a critical wind speed, similar to classical galloping. The response characteristics in terms of vibration amplitude, frequency, and trajectory were investigated in detail. The observed tendency of the increase in vibration frequency with increasing wind speed is adequately consistent with the previous studies. Finally, elastic cross-ties were vertically installed and connected two neighboring cable models in an attempt to suppress WIV. The wind tunnel tests demonstrated that the elastic cross-tie modified the single-mode dominated WIV into a type of chaotic multi-mode vibration and successfully reduced the vibration amplitude by ∼74%. The reason why the elastic cross-tie suppressed RWIV was qualitatively discussed. The scale ratio of the cross-tie stiffness was derived for engineers. Consequently, for the first time, the elastic cross-tie was proposed and verified to be effective to suppress the WIV of stay cables.

An accurate method for the dynamic behavior of tensegrity structures

PurposeThe purpose of this paper is to propose an accurate and efficient numerical method for determining the dynamic responses of a tensegrity structure consisting of bars, which can work under both compression and tension, and cables, which cannot work under compression.Design/methodology/approachAn accurate time-domain solution is obtained by using the precise integration method when there is no cable slackening or tightening, and the Newton–Raphson scheme is used to determine the time at which the cables tighten or slacken.FindingsResponses of a tensegrity structure under harmonic excitations are given to demonstrate the efficiency and accuracy of the proposed method. The validation shows that the proposed method has higher accuracy and computational efficiency than the Runge–Kutta method. Because the cables of the tensegrity structure might be tense or slack, its dynamic behaviors will exhibit stable periodicity, multi-periodicity, quasi-periodicity and chaos under different amplitudes and frequencies of excitation.Originality/valueThe steady state response of a tensegrity structure can be obtained efficiently and accurately by the proposed method. Based on bifurcation theory, the Poincaré section and phase space trajectory, multi-periodic vibration, quasi-periodic vibration and chaotic vibration of the tensegrity structures are predicted accurately.

Nonlinear dynamics of contact interaction of a size-dependent plate supported by a size-dependent beam.

A mathematical model of complex vibrations exhibited by contact dynamics of size-dependent beam-plate constructions was derived by taking the account of constraints between these structural members. The governing equations were yielded by variational principles based on the moment theory of elasticity. The centre of the investigated plate was supported by a beam. The plate and the beam satisfied the Kirchhoff/Euler-Bernoulli hypotheses. The derived partial differential equations (PDEs) were reduced to the Cauchy problems by the Faedo-Galerkin method in higher approximations, whereas the Cauchy problem was solved using a few Runge-Kutta methods. Reliability of results was validated by comparing the solutions obtained by qualitatively different methods. Complex vibrations were investigated with the help of methods of nonlinear dynamics such as vibration signals, phase portraits, Fourier power spectra, wavelet analysis, and estimation of the largest Lyapunov exponents based on the Rosenstein, Kantz, and Wolf methods. The effect of size-dependent parameters of the beam and plate on their contact interaction was investigated. It was detected and illustrated that the first contact between the size-dependent structural members implies chaotic vibrations. In addition, problems of chaotic synchronization between a nanoplate and a nanobeam were addressed.

Closed form solutions of two time fractional nonlinear wave equations

Abstract In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized ( G ′ / G ) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.

Vibration Characteristics Analysis of Integrally Shrouded Turbine Blade Considering Separation and Reattachment

The nonlinear forced vibration responses of integrally shrouded turbine blade considering different contact state are investigated in this paper. A lumped mass model of integrally shrouded turbine blade considering separation and reattachment of adjacent shrouds and centrifugal stiffening effects is established and its dynamic equation is deduced. Collision force is approximated by linear and cubic springs together which is more realistic, and friction force is approximated by an exponential-type velocity-dependent model. Numerical results indicate that integrally shrouded turbine blade displays very complex nonlinear phenomena and it can experience periodic 1, periodic 2, periodic 3, periodic 4 and chaotic vibration. Gap asymmetry leads to more complicated motions of the shrouded blade.

Bifurcation and chaotic vibration of frictional chatter in turning process

A comprehensive 3-degree-of-freedom dynamic model of turning process is proposed to overcome the shortage of consideration on the feed velocity in Rusinek’s work; nonlinear dynamics behaviors of th...

Dynamic analysis of integrally shrouded group blades with rubbing and impact

Forced vibration responses of integrally shrouded group blades are investigated in this paper. A lumped mass model of integrally shrouded group blades considering centrifugal stiffening of the blade and rubbing and impact between adjacent shrouds is established. In the proposed model, collision force is approximated by linear springs, and friction force is approximated by an exponential-type velocity-dependent model. Stick-slip-separation transition boundaries are determined. Runge–Kutta algorithm is used to compute vibration responses and study the effects of stiffness ratio, rotating speed and aerodynamic excitation amplitude on the vibration responses of integrally shrouded group blades. Vibration reduction effect of the shroud on the reference blade at the first dynamic resonance speed is illustrated. Numerical results indicate that stiffness ratio, initial gap and contact angle have a great effect on the normalised energy density (the value of which can represent vibration reduction effect of the shroud), and the reference blade can experience periodic, quasi-periodic and chaotic vibration due to nonsmooth behaviour at the shroud contact interfaces.

Non-symmetric forms of non-linear vibrations of flexible cylindrical panels and plates under longitudinal load and additive white noise

Parametric non-linear vibrations of flexible cylindrical panels subjected to additive white noise are studied. The governing Marguerre equations are investigated using the finite difference method (FDM) of the second-order accuracy and the Runge-Kutta method. The considered mechanical structural member is treated as a system of many/infinite number of degrees of freedom (DoF). The dependence of chaotic vibrations on the number of DoFs is investigated. Reliability of results is guaranteed by comparing the results obtained using two qualitatively different methods to reduce the problem of PDEs (partial differential equations) to ODEs (ordinary differential equations), i.e. the Faedo-Galerkin method in higher approximations and the 4th and 6th order FDM. The Cauchy problem obtained by the FDM is eventually solved using the 4th-order Runge-Kutta methods. The numerical experiment yielded, for a certain set of parameters, the non-symmetric vibration modes/forms with and without white noise. In particular, it has been illustrated and discussed that action of white noise on chaotic vibrations implies quasi-periodicity, whereas the previously non-symmetric vibration modes are closer to symmetric ones.