Narrow-band Random Excitation Research Articles

Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations

The effect of the narrow-band random excitation on the non-linear response of sandwich plates with an incompressible viscoelastic core is investigated. To model the core, both the transverse shear strains and rotations are assumed to be moderate and the displacement field in the thickness direction is assumed to be linear for the in-plane components and quadratic for the out-of-plane components. In connection to the moderate shear strains considered for the core, a non-linear single-integral viscoelastic model is also used for constitutive modeling of the core. The fifth-order perturbation method is used together with the Galerkin method to transform the nine partial differential equations to a single ordinary integro-differential equation. Converting the lower-order viscoelastic integral term to the differential form, the fifth-order method of multiple scale is applied together with the method of reconstitution to obtain the stochastic phase-amplitude equations. The Fokker–Planck–Kolmogorov equation corresponding to these equations is then solved by the finite difference method, to determine the probability density of the response. The variation of root mean square and marginal probability density of the response amplitude with excitation deterministic frequency and magnitudes are investigated and the bimodal distribution is recognized in narrow ranges of excitation frequency and magnitude.

Nonlinear liquid sloshing in square tanks subjected to horizontal random excitation

hing dynamics in a square tank are numerically investigated when the tank is subjected to horizontal, narrowband random ground excitation. The natural frequencies of the two predominant sloshing modes are identical and therefore 1:1 internal resonance may occur. Galerkin’s method is applied to derive the modal equations of motion for nonlinear sloshing including higher modes. The Monte Carlo simulation is used to calculate response statistics such as mean square values and probability density functions (PDFs). The two predominant modes exhibit complex phenomena including “autoparametric interaction” because they are nonlinearly coupled with each other. The mean square responses of these two modes and the liquid elevation are found to differ significantly from those of the corresponding linear model, depending on the characteristics of the random ground excitation such as bandwidth, center frequency and excitation direction. It is found that the direction of the excitation is a significant factor in predicting the mean square responses. The frequency response curves for the same system subjected to equivalent harmonic excitation are also calculated and compared with the mean square responses to further explain the phenomena. Changing the liquid level causes the peak of the mean square response to shift. Furthermore, the risk of the liquid overspill from the tank is discussed by showing the three-dimensional distribution charts of the mean square responses of liquid elevations.

Nonlinear Responses of Sloshing in Square Tanks Subjected to Horizontal Random Ground Excitation

Nonlinear responses of the predominant two sloshing modes in a square tank have been investigated when the tank is subjected to horizontal, narrow-band random ground excitation. Galerkin’s method is applied to derive the modal equations of motion for nonlinear sloshing. Then the Monte Carlo simulation is used to calculate the mean square responses of these two modes. These to modes are nonlinearly coupled with each other, known as ‘autoparametric interaction’. The responses differ significantly from those of the corresponding linear model, depending on the characteristics of the narrow-band ground excitation such as the bandwidth, center frequency and the intensity. In addition, it is found that the direction of the excitation is a significant factor in predicting the mean square responses.

601 複素ウェーブレット変換を用いたDuffing系の確率論的ジャンプ現象の各種統計量 (続報,二つの狭帯域励振モデルの比較)(機械力学・計測制御II)

601 複素ウェーブレット変換を用いたDuffing系の確率論的ジャンプ現象の各種統計量 (続報,二つの狭帯域励振モデルの比較)(機械力学・計測制御II)

State transitions in the Duffing resonator excited by narrow band random noise

The Dufffing hardening spring resonator has the status of a canonical model for nonlinear vibrations. Over a limited range of excitation frequencies and depending on the degree of nonlinearity, it has three states of response to sinusoidal excitation, one of which is unstable. The system will remain stably in either of the other two states depending on the history of excitation: in a higher energy state for the frequency ascending and in the lower energy state for the frequency descending. When the sinusoidal excitation is replaced by narrow band random excitation, the author showed in a 1961 paper experimentally that the system could make transitions between these two states and argued that the fluctuating phase of the excitation would allow the source to inject or draw energy from the resonator allowing a transition from one state to the other. This presentation develops a dynamic model for the system that allows energy transmission between the source and resonator and an indicator of the state of response based on instantaneous impedance.

Subharmonic response of single-degree-of-freedom linear vibroimpact system to narrow-band random excitation

The subharmonic response of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to the narrow-band random excitation is investigated. The analysis is based on a special Zhuravlev transformation, which reduces the system to the one without impacts or velocity jumps, and thereby permits the applications of asymptotic averaging over the period for slowly varying the inphase and quadrature responses. The averaged stochastic equations are exactly solved by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset. The effects of damping, detuning, and bandwidth and magnitudes of the random excitations are analyzed. The theoretical analyses are verified by the numerical results. The theoretical analyses and numerical simulations show that the peak amplitudes can be strongly reduced at the large detunings.

Stochastic jump and bifurcation of a slender cantilever beam carrying a lumped mass under narrow-band principal parametric excitation

A detailed theoretical investigation into the single-mode approximate response of a slender cantilever beam carrying a lumped mass subjected to base narrow-band random excitation is presented for the first time. The method of multiple scales is used and the stochastic jump and bifurcation have been investigated for the principal parametric resonance of the system using the stationary joint probability. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency-response domain, if the excitation central frequency is a variable and others keep constant, the basic phenomena imply that the higher the frequency, the more probable the jump from the stationary non-trivial branch to the stationary trivial one once the frequency exceeds a certain value. If the bandwidth is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the non-trivial branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the trivial one when the bandwidth becomes higher. For the force-response domain, there is a region of excitation acceleration within which the joint probability density has two peaks: an outer flabellate peak and a central volcano peak. Results show that the outer flabellate peak decreases while the central volcano peak increases as the value of the excitation acceleration decreases.

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over “fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.

Nonlinear Dynamics of a Parametrically Excited Laminated Beam: Narrow-Band Random Excitation

The first mode parametric resonance of a laminated beam subject to narrow-band random excitation is taken into consideration. The method of multiple scales is used and the stochastic jump and bifurcation have been investigated aiming at the stationary joint probability of the response of the system by using finite difference method. Results show that stochastic jump occurs mainly in the region of triple valued solution. The higher is the frequency, the more probable is the jump from the stationary nontrivial branch to the trivial one, whereas the most probable motion gradually approaches the trivial one when the band width becomes higher.

Computation of the Optimal Normal Load for a Mistuned and Frictionally Damped Bladed Disk Assembly Under Different Types of Excitation

Using nondimensional variables, the performances of friction dampers of a mistuned bladed disk assembly are examined for different types of excitation: white noise excitation, independent narrow band random excitation, and sinusoidal excitation with unknown amplitudes. Based on the harmonic balance method, an analytical technique is developed to compute the statistics of response for sinusoidal excitation with unknown amplitudes. The performances of blade-to-blade and blade-to-ground dampers are compared under different types of excitation. It is found that the nondimensional optimal normal loads of friction dampers are almost independent of the nature of excitation. Therefore, optimal normal loads of friction dampers can be chosen without any knowledge of the nature of excitation.

Stochastic and chaotic sub- and superharmonic response of shallow cables due to chord elongations

The paper deals with the non-linear response of shallow cables driven by stochastically varying chord elongations caused by random vibrations of the supported structure. The chord elongation introduces parametric excitation in the linear stiffness terms of the modal coordinate equations, which are responsible for significant internal subharmonic and superharmonic resonances. Under harmonically varying support motions coupled ordered or chaotic in-plane and out-of-plane subharmonic and superharmonic periodic motions may take place. If the harmonically varying chord elongation is replaced by a zero-mean, stationary narrow-band random excitation with the same standard deviation and center frequency, qualitatively and quantitatively completely different modes of vibration are registered no matter how small the bandwidth of the excitation process is. Additionally, the stochastic excitation process tends to enhance chaotic behavior. Based on Monte Carlo simulation on a reduced non-linear two-degree-of freedom system the indicated effects have been investigated for stochastic subharmonic resonance of order 2:1, and stochastic superharmonic resonances of orders 1:2 and 2:3. By analyzing the responses for two chord elongation processes with almost identical auto-spectral density function, but completely different amplitudes, it is shown that the indicated qualitative and quantitative changes of the subharmonic resonance primarily are caused by the slowly varying phase of the stochastic excitation. The superharmonic stochastic responses are dominated by random jumps between a single mode in-plane and a coupled mode attractor, which are caused by the variation of the amplitude of the random excitation. Such jumps do not occur in the subharmonic response, because the single mode in-plane attractor is unstable.

Energy Harvesting From Vibrations With a Nonlinear Oscillator

In this paper, we present a nonlinear electromagnetic energy harvesting device that has a broadly resonant response. The nonlinearity is generated by a particular arrangement of magnets in conjunction with an iron-cored stator. We show the resonant response of the system to both pure-tone excitation and narrow-band random excitation. In addition to the primary resonance, the superharmonic resonances of the harvester are also investigated and we show that the corresponding mechanical upconversion of the excitation frequency may be useful for energy harvesting. The harvester is modeled using a Duffing-type equation and the results are compared with the experimental data.

Subharmonic response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random excitation

The subharmonic response of single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to narrow-band random excitation is investigated. The narrow-band random excitation used here is a filtered Gaussian white noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the applications of asymptotic averaging over the "fast" variables. The averaged stochastic equations are solved exactly by the method of moments for the mean-square response amplitude for the case of linear system with zero offset. A perturbation-based moment closure scheme is proposed and the formula of the mean-square amplitude is obtained approximately for the case of linear system with nonzero offset. The perturbation-based moment closure scheme is used once again to obtain the algebra equation of the mean-square amplitude of the response for the case of nonlinear system. The effects of damping, detuning, nonlinear intensity, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detunings or large nonlinear intensity.

Subharmonic response of a single-degree-of-freedom nonlinear vibroimpact system to a randomly disordered periodic excitation

The subharmonic resonant response of a single-degree-of-freedom nonlinear vibroimpact oscillator with a one-sided barrier to narrow-band random excitation is investigated. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts or velocity jumps, thereby permitting the applications of asymptotic averaging over the period for slowly varying random processes. The averaged equations are solved exactly and algebraic equation of the amplitude of the response is obtained in the case without random disorder. A perturbation-based moment closure scheme is proposed and an iterative calculation equation for the mean square response amplitude is derived in the case with random disorder. The effects of damping, nonlinear intensity, detuning, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detuning or large nonlinear stiffness, and when intensity of the random disorder increase, the steady-state solution may change from a limit cycle to a diffused limit cycle, and even change to a chaos one.

Explanation on the importance of narrow-band random excitation characters in the response of a cantilever beam

A detailed theoretical investigation into the first- and second-mode response of a parametrically excited slender cantilever beam, where, the narrow-band random excitation characters are taken into consideration, is presented. The method of multiple scales is used to determine a uniform first-order expansion of the solution of the nonlinear integro-differential equations. The first-order moment frequency- and force–response data (curves) of a specimen beam tested by other investigators are obtained. Further comparisons have been made and results show that whether the first-order moment frequency–response data (curves) or the first-order moment force–response data (curves) of the first two modes are all in agreement with other investigators’ experimental results. Furthermore, the stochastic jump and bifurcation have been investigated for the first modal parametric principal resonance by using the stationary joint probability of amplitude and phase. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency–response domain, if the bandwidth γ is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the higher branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the lower one when the bandwidth becomes higher; if the excitation central frequency f is a variable and others keep constant, the basic phenomena imply that the higher is f, the more probable is the jump from the higher branch to the lower one once f exceeds an certain value. For the force–response domain, there is a region of excitation acceleration a within which the joint probability density has two peaks: an upper peak and a lower peak. Results show that the upper peak decreases while the lower peak increases as the value of a decreases.

狭帯域不規則励振を受けるDuffing系の確率論的ジャンプ現象の各種統計量(機械力学,計測,自動制御)

Statistics of stochastic jump phenomena in the random responses of a Duffing oscillator subjected to narrow band random excitation are investigated. The deterministic jump phenomena of the Duffing oscillator which is subjected to harmonic excitation correspond to the existence of two stationary responses in some frequency range. These responses differ in the phase angle between the excitation and the response, as well as the amplitude. In the case of the stochastic excitation, there are two states of the response, and the jump between two states occurs suddenly and multiple times in the long sample function. These two states correspond to two stationary responses to the harmonic excitation. In the time histories, two states of the response are quite different when the stochastic excitation is narrow band, but the difference becomes obscurer when the band width of the excitation becomes broader. In our previous research, the product of each wavelet transforms of the excitation and the response is proposed as the frequency decomposed phase angle. It was shown that the product successfully identified the states of the responses in the stochastic jump phenomena. In this paper, the rate of the stochastic jump, the ratio and the duration of each state are chosen as the representative statistics of the stochastic jump, and the relation between these statistics and parameters of the narrow band random excitation is studied.

Response to bounded noise excitation of stochastic Mathieu-Duffing system with time delay state feedback

We investigate the principal parametric resonance of Mathieu-Duffing Equation under a narrow-band random excitation with time delay feedback. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The bifurcation of the system is discussed. We find that the bifurcation can be influenced by the detuning parameter, time delay, and the intensity of the non-linear term, and an appropriate choice of these parameters can change the response of bifurcation. In addition the stability of nontrivial solution is studied. The nontrivial solution of necessary and sufficient condition for stability is obtained. Moreover, we find that when the bandwidth of the random excitation is smaller, the multi-solution phenomenon still exists, and bifurcation and jumping phenomenon will occur. Theoretical analysis is verified by numerical results.

Resonance response of a single-degree-of-freedom nonlinear dry system to a randomly disordered periodic excitation

The resonance response of a single-degree-of-freedom nonlinear dry oscillator of Coulomb type to narrow-band random parameter excitation is investigated. The analysis is based on the Krylov-Bogoliubov averaging method. The averaged equations are solved exactly and the algebraic equation of the amplitude of the response is obtained in the case without random disorder. Linearization method and moment method are used to obtain the mean square response amplitude for the case with random disorder. The effects of damping, nonlinear intensity, detuning, bandwidth, dry intensity, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced and the bifurcation of the system will be delayed when intensity of the nonlinearity increases. The peak amplitudes will also be reduced and the bifurcation of the system will be delayed when damping and dry intensity of the system increases.

狭帯域不規則励振を受けるDuffing系のジャンプ現象を伴う応答の確率密度の解析(機械力学,計測,自動制御)

An approximate analytical method is developed for calculating the probability density of the response of a Duffing oscillator subjected to narrow band random excitation when the stochastic jump phenomenon occurs. The stochastic jump phenomenon is characterized by the existence of multiple states of amplitude and sudden jumps between the states in the sample function of response. In this paper, the trigger for the stochastic jump is considered, and the probabilities of large and small amplitude states of the response are evaluated. The local probability density function of each state is separately assumed using the equivalent linearization method, and the probability density function of the overall response is derived by sum of these local probability density functions weighted with the probability of each state. In the illustrative example, the calculated probability density functions are compared with the Monte Carlo simulation results. It is shown that the present method well expresses the significant characteristic of the stochastic jump phenomenon.

NUMERICAL SOLUTION OF NONLINEAR EQUATION TO COMBINED DETERMINISTIC AND NARROW-BAND RANDOM EXCITATION

The Duffing oscillator to combined deterministic and narrow-band random excitation, which is a nonlinear equation, is studied and solved numerically using three numerical methods based on finite difference schemes. Method 1, the well-known Euler method, is an explicit method; Method 2 is an implicit first-order method which does not bring contrived chaos into the solution; and Method 3 is based on two first-order methods which is second-order method and is chaos-free. In a series of numerical experiments, it is seen that the proposed methods have superior stability properties to those of the well-known Euler and fourth-order Runge-Kutta methods to which they are compared. When extended to the numerical solution of Duffing oscillator to combined deterministic and narrow-band random excitation, the developed methods give the correct steady-state solutions compared with the literature.