Deterministic Harmonic Excitation Research Articles

Coupled Parametric Vibration Model and Response Analysis of Single Beam and Double Cable Under Deterministic Harmonic and Random Excitation

In order to reveal the coupling parametric vibration characteristics of stay cables under combined excitation, considering the effects of cable geometric nonlinearity, inclination and the cooperative vibration of adjacent cables and bridge deck beams, a single-beam–double-cable coupled parametric vibration model excited by Gaussian white noise and deterministic harmonic excitation is established, and the coupled motion equations are derived. The Milstein–Platen method is used to directly obtain the coupled random vibration time history of the single-beam–double-cable structure, and an iterative method is proposed to counter the influence of the parametric diffusion coefficient on the numerical format. By comparing with the finite element method and the Monte Carlo numerical simulation method, the accuracy of the Milstein–Platen method in solving the vibration time history of cable–beam coupling parameters under combined excitation is first verified. Then the random displacement, power spectral density and probability density variation of the cable and beam under the combined excitation of different intensities are analyzed from the angle of random orbit. The results show that under the joint action of deterministic harmonics and random excitation, the time–domain, frequency, and probability characteristics of the single-beam–double-cable coupling system are greatly affected by the Gaussian white noise excitation proportion coefficient, and the degree of influence is different. In addition, compared with the single cable model, the vibration analysis results of the coupling model considering multiple stay cables are more reasonable.

Operational modal analysis under harmonic excitation using Ramanujan subspace projection and stochastic subspace identification

The presence of deterministic harmonic excitation, such as that induced by rotating machinery, violates the classical operational modal analysis (OMA) assumption that the system output is strictly ergodic. This paper proves that the presence of harmonic excitation has no effect on the identification of structural modes except in the case where some of the harmonic excitation frequencies are close to the structural frequencies. Therefore, the harmonic component must be removed from the mixed random and harmonic system output before further processing. This paper proposes a Ramanujan subspace projection (RSP) method for harmonic removal, which is realized by projecting the raw system output onto the complex conjugate division (CCD) of the Ramanujan subspace. The novelty of this study is that it reveals the relationship between the frequencies defined in the period and frequency domains, allowing the RSP method to directly extract the harmonic component from the specific CCD without any frequency domain analysis. In addition, an energy indicator is proposed to select the underlying CCDs with the most robust harmonic feature. Using the indicator, one only needs to project the raw system output onto a subset of the CCDs rather than all of them, thereby significantly reducing the computational effort. After removing the harmonic components from the raw output, the remainder can be fed into the covariance-driven stochastic subspace identification (Cov-SSI) method for OMA. The numerical, experimental and field test results show that the proposed RSP method is not only resistant to random noise but also capable of precisely extracting the weak harmonic component from the raw output with less computation. Furthermore, the modal parameters of the structures subjected to mixed random and harmonic excitation can be accurately identified by combining the Cov-SSI and RSP methods.

Piezoelectric vortex induced vibration energy harvesting in a random flow field

Vibration-based energy harvesters have significant potential for sustainable energy generation from ambience for micro-scale systems like wireless sensor networks and similar low power electronic devices. Vortex-induced vibration (VIV) is one of the richest sources for such power generation for devices installed within a fluid environment. However, uncertainty in the direction and magnitude of the free stream velocity could affect the performance of such systems. We have first developed the mathematical model of a piezoelectric cantilever beam with end mass vibrating under the influence of VIV. The piezo patch is assumed to be in the unimorph and bimorph configurations. From the unimodal dynamic response of the system, an equivalent single degree of freedom mechanical model is developed. This is further integrated with the electrical model of the piezoelectric system without and with an inductor. The energy harvested from the deterministic harmonic excitation is estimated against the non-dimensional velocity parameter. A random process model is developed considering the excitation force due to vortex shedding to be a bounded, weakly stationary and narrowband random process. The power spectral density of the random process is obtained using the Fourier transform of the auto-correlation function. The dynamic response of the energy harvester is obtained against such random excitations. The expressions of the mean power are obtained in closed-form corresponding to the cases without and with the inductor integrated to the electrical circuit. It is observed that while for cases without the inductor, the average harvested power monotonically decreases with increase in damping ratio and decrease in the coupling factor; for models with the inductor, an optimal inductor constant exists corresponding to the maximum mean-power condition. The extensive analytical modelling and initial representative results are expected to pave the way for the practical design of VIV based piezoelectric energy harvesting system subjected to stochastic excitation.

Equivalent linearization technique in nonlinear stochastic dynamics of a cable-mass system with time-varying length

In this paper the transverse vibrations of a vertical cable carrying a concentrated mass at its lower end and moving slowly vertically within the host structure are considered. It is assumed that longitudinal inertia of the cable can be neglected, with the longitudinal motion of the concentrated mass coupled with the lateral motion of the cable. An expansion of the lateral displacement of a cable in terms of approximating functions is used. The excitation acting upon the cable-mass system is a base-motion excitation due to the sway motion of a host tall structure. Such a motion of a structure often results due to action of the wind, hence it may be adequately idealized as a narrow-band random process. The narrow-band process is represented as the output of a system of two linear filters to the input in a form of a Gaussian white noise process. The non-linear problem is dealt with by an equivalent linearization technique, where the original non-linear system is replaced with an equivalent linear one, whose coefficients are determined from the condition of minimization of a mean-square error between the non-linear and the linear systems. The mean value and variance of the transverse displacement of the cable as well as those of the longitudinal motion of the lumped mass are determined with the aid of an equivalent linear system and compared with the response of the original non-linear system subjected to the deterministic harmonic excitation.

A fast random method for three-dimensional analysis of train-track-soil dynamic interaction

A fast computation method that can be used for efficient analysis of full scale three-dimensional random vibrations induced by railway traffic is presented in this paper. The method uses the pseudo-excitation method (PEM) for random analysis and the proposed multi-point synchronous algorithm (MPSA) for solution of the large sparse linear equations of the train-track-soil coupled system (TTSCS). A mixed two- and three-dimensional TTSCS model is established firstly. Based on the linear Hertzian wheel/rail contact relationship, the time-dependent equations of motion of the TTSCS are deduced. This formulation leads to a global system of equations that can be solved in a directional manner without the need for iterative processes. By means of the PEM, the self-excitation induced by the random track irregularity is transformed into a series of deterministic harmonic excitation vectors. To accelerate the computation, a fast computation strategy is proposed. In the numerical example, the proposed method is validated through comparison with field measured results. Comparison of the results of the MPSA with those of the triangular factorization algorithm and ANSYS is undertaken to evaluate the efficiency of the MPSA. It is confirmed that the MPSA can result in a five- to tenfold increase in computation efficiency.

Random vibration analysis for coupled vehicle-track systems with uncertain parameters

Purpose The purpose of this paper is to present a new random vibration-based assessment method for coupled vehicle-track systems with uncertain parameters when subjected to random track irregularity. Design/methodology/approach The uncertain parameters of vehicle are described as bounded random variables. The track is regarded as an infinite periodic structure, and the dynamic equations of the coupled vehicle-track system, under mixed physical coordinates and symplectic dual coordinates, are established through wheel-rail coupling relationships. The random track irregularities at the wheel-rail contact points are converted to a series of deterministic harmonic excitations with phase lag by using the pseudo excitation method. Based on the polynomial chaos expansion of the pseudo response, a chaos expanded pseudo equation is derived, leading to the combined hybrid pseudo excitation method - polynomial chaos expansion method Findings The impact of uncertainty propagation on the random vibration analysis is assessed efficiently. According to GB5599-85, the reliability analysis for the stability index is implemented, which can grade the comfort level by the probability. Comparing to the deterministic analysis, it turns out that neglect of the parameter uncertainty will lead to potentially risky analysis results. Originality/value The proposed method is compared with Monte Carlo simulations, achieving good agreement. It is an effective means for random vibration analysis of uncertain coupled vehicle-track systems and has good engineering practicality

Theoretical analysis of linear and nonlinear piezoelectric vibrational energy harvesters for human walking

The role of nonlinear stiffness in the performance of the piezoelectric vibrational energy harvester (pVEH) was discussed. Harmonic balance and numerical methods are applied to characterize the electromechanical response of pVEHs based on Duffing oscillator at a deterministic harmonic excitation of fundamental vibration characteristics (2 Hz, 1 m·s−2), which corresponds to human walking. Then, the response to a vibration with two harmonic waves, which has a fixed fundamental frequency (2 Hz, 1 m·s−2) and a frequency varied from 1.5 to 2.5 Hz. The numerical results obtained in this study indicate that nonlinearity does not have a significant advantage on the energy harvesting from human walking.

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

Recently, nonlinearities have been shown to play an important role in increasing the extracted energy of vibration-based energy harvesting systems. In this paper, we study the dynamical behavior of a piecewise linear (PWL) spring-mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. Different configurations of the PWL model and their corresponding state-space regions are derived. Then, from this PWL model, extensive numerical simulations are carried out by computing time-domain waveforms, state-space trajectories and frequency responses under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Filippov method, Poincaré map modeling and finite difference method (FDM). The Floquet multipliers are calculated using these three approaches and a good concordance is obtained among them. The performance of the system in terms of the harvested energy is studied by considering both purely harmonic excitation and a noisy vibrational source. A frequency-domain analysis shows that the harvested energy could be larger at low frequencies as compared to an equivalent linear system, in particular, for relatively low excitation intensities. This could be an advantage for potential use of this system in low frequency ambient vibrational-based energy harvesting applications.

Stability Analysis of Periodic Orbits in a Class of Duffing-Like Piecewise Linear Vibrators

In this paper, we study the dynamical behavior of a Duffing-like piecewise linear (PWL) spring- mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. From this PWL model, numerical simulations are carried out by computing frequency response and bifurcation diagram under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Fillipov method.

Strip Rolling Mill Random Vibration Analysis Based on Pseudo-Excitation Method

The five-octave mill stochastic dynamics model was established, and then constructed the virtual rolling force excitation, converted random excitation into the rolling force deterministic harmonic excitation by taking advantage of pseudo-excitation method, finally the variance and power spectral density dynamic random changes in the roll gap was Obtained . The results show that the emergence degree of rolled-strip oscillation marks is proportional to random rolling force power spectral density, and in over the five-octave frequency, the roll gap’s dynamic variance no longer change.

An Efficient Sensitivity Analysis Method for Optimization of Vehicle Random Vibrations

An efficient and accuracy sensitivity analysis method for optimal analysis of random vibration of vehicle-bridge coupled system is purposed. The pseudo-excitation method is used to transform random road surface roughness into a series of deterministic harmonic excitations, and then the precise integration method is adopted to compute vehicle/bridge system response. The pseudo-excitation method and the precise integration method are both accurate and efficient, so that the first and second order sensitivity information of the responses can be obtained very conveniently. Taking ride comfort as the objective function, an optimal analysis for a vehicle/bridge system is performed.

Symplectic random vibration analysis of a vehicle moving on an infinitely long periodic track

Based on the pseudo-excitation method (PEM), symplectic mathematical scheme and Schur decomposition, the random responses of coupled vehicle–track systems are analyzed. The vehicle is modeled as a spring–mass–damper system and the track is regarded as an infinitely long substructural chain consisting of three layers, i.e. the rails, sleepers and ballast. The vehicle and track are coupled via linear springs and the “moving-vehicle model” is adopted. The latter assumes that the vehicle moves along a static track for which the rail irregularity is further assumed to be a zero-mean valued stationary Gaussian random process. The problem is then solved efficiently as follows. Initially, PEM is used to transform the rail random excitations into deterministic harmonic excitations. The symplectic mathematical scheme is then applied to establish a low degree of freedom equation of motion with periodic coefficients. In turn these are transformed into a linear equation set whose upper triangular coefficient matrix is established using the Schur decomposition scheme. Finally, the frequency-dependent terms are separated from the load vector to avoid repeated computations for different frequencies associated with the pseudo-excitations. The proposed method is subsequently justified by comparison with a Monte-Carlo simulation; the fixed-vehicle model and the moving-vehicle model are compared and the influences of vehicle velocity and class of track on system responses are also discussed.

Global analyses of crisis and stochastic bifurcation in the hardening Helmholtz-Duffing oscillator

Crisis and stochastic bifurcation of the hardening Helmholtz-Duffing oscillator are studied by means of the generalized cell mapping method using digraph. For the system subject to a single deterministic harmonic excitation, our study reveals that a series of crisis phenomena can occur when the system parameter passes through different critical values, including chaotic boundary crisis, regular boundary crisis and interior crisis. A chaotic boundary crisis due to the collision of regular attractor with chaotic saddle embedded in a fractal basin boundary and an interior crisis due to the collision of regular attractor with chaotic saddle of its attraction basin are discovered. A new phenomenon, namely the global properties of dynamical system show symmetric as system parameter is varied, can be also revealed according to our analysis. For the system subject to a combination of a deterministic harmonic excitation and a random excitation, it is found that stochastic bifurcation, defined as a sudden change in character of a stochastic attractor, can occur one after another when the noise intensity passes through different critical values. This kind of stochastic bifurcation corresponds to stochastic crisis essentially. Our study also reveals that the generalized cell mapping method using digraph is a powerful tool not only for the crisis behavior analysis of deterministic system, but also for the global property analysis of stochastic bifurcation.

Non-Stationary Random Vibration of FE Structures Subjected to Moving Loads

An efficient and accurate FEM based method is proposed for studying non-stationary random vibration of structures subjected to moving loads. The loads are assumed to be a stationary process with constant mean value. The non-stationary power spectral densities (PSD) and the time dependent standard deviations of dynamic response are derived by using the pseudo excitation method (PEM) to transform this random excitation problem into a deterministic harmonic excitation one. The precise integration method (PIM) is extended to solve the equation of motion of beams under moving harmonic loads by enhancing the very recent consistent decomposition procedure, in order to simulate the movement of the loads. Six numerical examples are given to show the very high efficiency and accuracy of the method and also to deduce some useful preliminary conclusions from investigation of the dynamic statistical characteristics of a simply supported beam and of a symmetrical three span beam with its centre span unequal to the outer ones.

Non-Stationary Random Vibration of FE Structures Subjected to Moving Loads

An efficient and accurate FEM based method is proposed for studying non-stationary random vibration of structures subjected to moving loads. The loads are assumed to be a stationary process with constant mean value. The non-stationary power spectral densities (PSD) and the time dependent standard deviations of dynamic response are derived by using the pseudo excitation method (PEM) to transform this random excitation problem into a deterministic harmonic excitation one. The precise integration method (PIM) is extended to solve the equation of motion of beams under moving harmonic loads by enhancing the very recent consistent decomposition procedure, in order to simulate the movement of the loads. Six numerical examples are given to show the very high efficiency and accuracy of the method and also to deduce some useful preliminary conclusions from investigation of the dynamic statistical characteristics of a simply supported beam and of a symmetrical three span beam with its centre span unequal to the outer ones.

Symplectic analysis of vertical random vibration for coupled vehicle–track systems

A computational model for random vibration analysis of vehicle–track systems is proposed and solutions use the pseudo excitation method (PEM) and the symplectic method. The vehicle is modelled as a mass, spring and damping system with 10 degrees of freedom (dofs) which consist of vertical and pitching motion for the vehicle body and its two bogies and vertical motion for the four wheelsets. The track is treated as an infinite Bernoulli–Euler beam connected to sleepers and hence to ballast and is regarded as a periodic structure. Linear springs couple the vehicle and the track. Hence, the coupled vehicle–track system has only 26 dofs. A fixed excitation model is used, i.e. the vehicle does not move along the track but instead the track irregularity profile moves backwards at the vehicle velocity. This irregularity is assumed to be a stationary random process. Random vibration theory is used to obtain the response power spectral densities (PSDs), by using PEM to transform this random multiexcitation problem into a deterministic harmonic excitation one and then applying symplectic solution methodology. Numerical results for an example include verification of the proposed method by comparing with finite element method (FEM) results; comparison between the present model and the traditional rigid track model and; discussion of the influences of track damping and vehicle velocity.

GLOBAL ANALYSIS OF STOCHASTIC BIFURCATION IN DUFFING SYSTEM

Stochastic bifurcation of a Duffing system subject to a combination of a deterministic harmonic excitation and a white noise excitation is studied in detail by the generalized cell mapping method using digraph. It is found that under certain conditions there exist two stable invariant sets in the phase space, associated with the randomly perturbed steady-state motions, which may be called stochastic attractors. Each attractor owns its attractive basin, and the attractive basins are separated by boundaries. Along with attractors there also exists an unstable invariant set, which might be called a stochastic saddle as well, and stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. This definition applies equally well either to randomly perturbed motions, or to purely deterministic motions. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence and evolution of stochastic bifurcation can be explored clearly and vividly.

Iterative schemes for stability problems with non-singular Fokker-Planck equations

Stability investigations of perturbed dynamical systems are considerably refined by utilizing the invariant measures of the systems, which can be calculated by the stationary solutions of the associated Fokker-Planck equations. The paper presents iterative schemes in order to obtain these solutions as limit cycles and check them by means of Monte-Carlo simulations. Both methods are applied to parametrically excited oscillators including the limiting cases of deterministic harmonic excitations and stochastic white noise perturbations.