Responses of a non-linearly coupled pitch-roll ship model under harmonic excitation

Stationary responses of nonlinearly coupled pitch and roll ship modes are studied using a modified averaging method, along with two second order multiple time scale (MTS) methods for comparison. Stability of the solutions is also studied. In the case of harmonic excitation all the three methods give fairly accurate results to the original problem but the averaging method is the most efficient. Analytic solutions are obtained from the averaged equations, which can be used to predict stationary responses both for small and for large excitations. From the averaging method several qualitatively different phenomena which cannot be addressed by the first order theory have been obtained: (i) the saturation phenomenon is lost, (ii) the bifurcation points are altered and (iii) a drift term is present which, although small, appears to have a significant effect on the accuracy of the solutions.     

Nonlinear responses of a cable-beam coupled system under parametric and external excitations

This paper analytically investigates the nonlinear responses of a cable-beam coupled system under the combined effects of internal and external resonance. The cable is considered a geometric nonlinearity, and the beam is considered as Euler–Bernoulli model, but it is coupled by fixing it at one end. The coupled nonlinear differential equations are formulated by using the Hamilton principle. The spatial problem is solved by using Galerkin’s method to simplify the governing equations to a set of ordinary differential equations. Applying the multiple time scales method to the ordinary differential equations, the first approximate solutions and solvability condition are derived. The effects of the cable sag to span ratio, mass ratio, and stiffness ratio on the nonlinear responses are investigated. The results show good agreement between the analytical and numerical solutions especially near the external resonance frequency.     

Analytical parametric study of bi-linear hysteretic model of dry friction under harmonic,impulse and random excitations

Hysteretic behavior due to some nonlinear sources is a common phenomenon in many dynamical systems. One of the sources of this behavior in mechanical systems is dry friction. Dry friction in bolted or riveted joints of mechanical structures makes their dynamic behavior hysteretic. Bi-linear hysteresis is one of the models that can be used to study these systems which is used in this paper. A SDOF system containing a bi-linear hysteretic element called Jenkins element under harmonic, impulse and random excitations is considered. For all three types of excitations, the effects of system and excitation parameters on the defined equivalent system parameters and the response specifications are studied. Harmonic balance method is employed for harmonic excitation studies, and optimum friction threshold for minimizing response amplitude is obtained versus other system parameters and response amplitude. Energy balance method is used for impulse excitation through which the desired decaying ratio can be achieved by tuning the friction threshold, depending on stiffness ratio. System under random excitation is investigated by equivalent linearization technique in two steps. At the first step, equivalent properties are obtained versus instantaneous amplitude of response. In this step, the paper contains the parametric study of system in which the variations of equivalent parameters are described when physical parameters of system or input intensity vary. Overall variance of system response is determined in the second step, and optimum sliding threshold is obtained to have minimum overall variance of system response.     

Complex response analysis of a non-smooth oscillator under harmonic and random excitations

It is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo(MC) simulation.Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.     

Vibration absorption of parallel-coupled nonlinear energy sink under shock and harmonic excitations

Nonlinear energy sink(NES) can passively absorb broadband energy from primary oscillators. Proper multiple NESs connected in parallel exhibit superior performance to single-degree-of-freedom(SDOF) NESs. In this work, a linear coupling spring is installed between two parallel NESs so as to expand the application scope of such vibration absorbers. The vibration absorption of the parallel and parallel-coupled NESs and the system response induced by the coupling spring are studied. The results show that the responses of the system exhibit a significant difference when the heavier cubic oscillators in the NESs have lower stiffness and the lighter cubic oscillators have higher stiffness. Moreover, the efficiency of the parallel-coupled NES is higher for medium shocks but lower for small and large shocks than that of the parallel NESs. The parallel-coupled NES also shows superior performance for medium harmonic excitations until higher response branches are induced. The performance of the parallel-coupled NES and the SDOF NES is compared. It is found that, regardless of the chosen SDOF NES parameters, the performance of the parallel-coupled NES is similar or superior to that of the SDOF NES in the entire force range.     

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations is studied. The motion equation of the system is reduced to a set of averaged Itô stochastic differential equations by stochastic averaging in the case of resonance. Then, the backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function and the conditional probability density and mean first-passage time are obtained by solving the backward Kolmogorov equation and Pontryagin equation with suitable initial and boundary conditions. The procedure is applied to Duffing–van der Pol system in resonant case and the analytical results are verified by Monte Carlo simulation.     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Prediction of the response of non-linear oscillators under stochastic parametric and external excitations

The method of equivalent external excitation is derived to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations. The oscillator is interpreted as one which is excited solely by an external zero-mean stochastic process. The Fokker-Planck-Kolmogorov equation is then applied to solve for the density functions and match the stationary variances of the states. Four examples which include polynomial, non-polynomial, and Duffing type non-linear oscillators are used to illustrate this approach. The validity of the present approach is compared with some exact solutions and with Monte Carlo simulations.     

Stochastic linearization of MDOF systems under parametric excitations

The stochastic linearization approach is examined for non-linear systems subjected to parametric type excitations. It is shown that, for these systems too, stochastic linearization and Gaussian closure are two equivalent approaches if the former is applied to the coefficients of the Itô differential rule. A critical review of other stochastic linearization approaches is also presented and discussed by means of simple examples.     

Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations

This paper is mainly dealing with the stochastic responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise. By the aid of approximate conversion for the viscoelastic force, the original stochastic VI system is transformed into an equivalent stochastic system without viscoelastic term. Then, the equations of the converted system are simplified by non-smooth transformation, and the stochastic averaging method is employed to solve the above simplified system. A Van der Pol VI oscillator coupled with viscoelastic force is worked out in detail to illustrate the application of the mentioned method, and therewith the analytical solutions fit the numerical simulation results based on the original system. Therefore, the present analytical means of investigating this system is proved to be feasible. Additionally, the exploration of stochastic P-bifurcation by two different ways is also demonstrated in this paper through varying the value of the certain system parameters. Besides, it shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.     

Vibration reduction of a nonlinear spring pendulum under multi external and parametric excitations via a longitudinal absorber

The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method. The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation results are achieved using MATLAB 7.0 programs. Results are compared to previously published work.     

Nonlinearly damped systems under simultaneous broad-band and harmonic excitations

The probability distribution of the response of a nonlinearly damped system subjected to both broad-band and harmonic excitations is investigated. The broad-band excitation is additive, and the harmonic excitations can be either additive or multiplicative. The frequency of a harmonic excitation can be either near or far from a resonance frequency of the system. The stochastic averaging method is applied to obtain the Itô type stochastic differential equations for an averaged system described by a set of slowly varying variables, which are approximated as components of a Markov vector. Then, a procedure based on the concept of stationary potential is used to obtain the exact stationary probability density for a class of such averaged systems. For those systems not belonging to this class, approximate solutions are obtained using the method of weighted residuals. Application of the exact and approximate solution procedures are illustrated in two specific cases, and the results are compared with those obtained from Monte Carlo simulations.     

Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations

The resonant resonance response of a single-degree-of-freedom non-linear vibro-impact oscillator, with cubic non-linearity items, to combined deterministic harmonic and random excitations is investigated. The method of multiple scales is used to derive the equations of modulation of amplitude and phase. The effects of damping, detuning, and intensity of random excitations are analyzed by means of perturbation and stochastic averaging method. The theoretical analyses verified by numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under certain conditions, impact system may have two steady-state responses. One is a non-impact response, and the other is either an impact one or a non-impact one.     

Stability of elastic systems under a stochastic parametric excitation

An efficient method to investigate the stability of elastic systems subjected to the parametric force in the form of a random stationary colored noise is suggested. The method is based on the simulation of stochastic processes, numerical solution of differential equations, describing the perturbed motion of the system, and the calculation of top Liapunov exponents. The method results in the estimation of the almost sure stability and the stability with respect to statistical moments of different orders. Since the closed system of equations for moments of desired quantities y _j (t) cannot be obtained, the statistical data processing is applied. The estimation of moments at the instant t _n is obtained by statistical average of derived from the solution of equations for the large number of realizations. This approach allows us to evaluate the influence of different characteristics of random stationary loads on top Liapunov exponents and on the stability of system. The important point is that results found for filtered processes, are principally different from those corresponding to stochastic processes in the form of Gaussian white noises.