Mean Square Response Research Articles

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

Dynamical responses of a Gaussian colored noise-driven shape memory alloy oscillator with a periodic force

For shape memory alloy (SMA) oscillator, the stochastic transitions between low-amplitude and undesirable high-amplitude oscillations will affect the safety and normal use of the system. Therefore, the nonlinear dynamical responses of a SMA oscillator with both periodic and Gaussian colored noise excitations are investigated in this paper. Firstly, to reveal the influences of random factors, a stochastic SMA model with a Gaussian colored noise is introduced. Subsequently, the approximate analytical solution of the steady-state amplitude response under the deterministic case is given by the averaging method, and the time-averaging mean-square response of the stochastic SMA system is obtained by the stochastic averaging method. The deduced approximated analytical solutions are verified by Monte Carlo numerical results, and a good consistency is observed. Finally, the effects of the random excitation and temperature on the steady-state responses of the SMA system are explored. We uncover that the temperature influences the frequency of bifurcation points, which can disrupt the stable structure of the SMA oscillator. In addition, noise intensity and correlation time of the colored noise can induce the occurrence of stochastic transitions between low-amplitude and high-amplitude oscillations, which are visible through the steady-state probability density and time history diagrams. All these obtained results may provide theoretical guidance and valuable insights for avoiding catastrophic stochastic transitions in SMA materials.

A computational framework for mean square responses of bidirectional nonlinear systems under correlated stochastic excitation

A mathematical framework using Ito–Taylor expansion based stochastic integration scheme for planar structural systems subjected to correlated bidirectional excitation at the base is presented. The bidirectional excitation is modeled using correlated Wiener processes. Instead of simple lumped mass representation of structural systems, a more realistic planar multidirectional model is adopted here. Such assumption adequately captures the behavior of systems under real ground excitation which are often multidimensional in nature. The bidirectional excitation model is developed using a mathematically rigorous framework of correlated Wiener derivatives, exploiting the relationship between Wiener and white noise processes, something that is rarely adopted in literature. An Ito–Taylor expansion based Taylor 1.5 integration scheme is developed first for generating system responses, and, the same Ito–Taylor based scheme is utilized to create an automated framework for the evolution of mean-square responses, a manually difficult exercise otherwise. The proposed framework facilitates the development and simultaneous evaluation of n+nC2 (where, n is the number of system states) mean square equations after obtaining the mean square terms using Kronecker product. Extensive simulation studies are carried out for a torsionally coupled linear bidirectional frame, stochastic bidirectional duffing system and a Bouc–Wen hysteretic system at the base. The results for each of the case studies show that non-accounting of correlation in the bidirectional excitation leads to as much as 30% error in the estimation of displacement mean square responses.

Pendulum tuned mass damper: optimization and performance assessment in structures with elastoplastic behavior

Different types of tuned mass dampers (TMD) have been applied to reduce wind and seismic induces vibrations in buildings. We analyze a pendulum tuned mass damper (PTMD) to reduce vibrations of structures that exhibit elastoplastic behavior subjected to ground motion excitation. Using a simple dynamic model of the primary structure with and without the PTMD and a random process description of the ground acceleration, the performance improvement of the structure is assessed using statistical linearization. The Liapunov equation is used to estimate the mean-square response in the stationary condition of the random process and optimize PTMD parameters. The optimum values of the PTMD frequency and damping ratio are defined as PTMD design values for a specific maximum seismic intensity design criterion. The results show that: (1) The values of the PTMD effectiveness criterion and the optimal design values of the frequency ratio are higher when the damping ratio of the primary structure decreases. (2) The performance of the optimized PTMD is higher when the structure exhibits a linear hysteresis loop (low seismic intensity). (3) The optimized PTMD controls the development of structural plasticity reducing vulnerability. (4) There is a strong dependence of the optimum PTMD parameters on the dynamic soil properties of the building foundation. (5) The PTMD performance improves as its mass increases. The optimum frequency ratio decreases, and the damping ratio increases as the mass of the pendulum increases. The PTMD designed and optimized with the proposed methodology reduces vibrations, controls the development of plasticity, and protects the primary structure, particularly in low and medium-intensity earthquakes.

Deterministic and Random Response Evaluation of a Straight Beam with Nonlinear Boundary Conditions

Deterministic and random responses of one-dimensional continuous structures with nonlinear boundary conditions are evaluated in this article. Dynamic behaviors (i.e., the time-domain response for the deterministic case and mean-square response for the stochastic case) of a straight beam with a cubic nonlinear elastic boundary is investigated numerically. The nonlinear vibration problem is discretized by a new global spatial discretization method, and its responses are compared with those from the finite element method and assumed mode method. The effects of selected various types of linear homogeneous boundary conditions and the number of expansion modes on the accuracy of the responses under harmonic excitation are discussed. Detailed discussions on the influence of nonlinear boundary conditions on dynamic behaviors uncover a counter-intuitive phenomenon: with the increase of the nonlinear spring stiffness, the stationary mean-square displacement of the mid-point descends first and then ascends slightly. Numerical results show that the present procedure possesses high precision for evaluating harmonic responses for both low-frequency and high-frequency excitations. Furthermore, it can be directly generalized to two-dimensional continuous systems and various types of nonlinear boundary conditions.

Studies on vibration control effects of a semi-active impact damper for seismically excited nonlinear building

The semi-active impact damper (SAID) is proposed to improve the damping efficiency of traditional passive impact dampers. In order to investigate its damping mechanism and vibration control effects on realistic engineering structures, a 20-story nonlinear benchmark building is used as the main structure. The studies on system parameters, including the mass ratio, damping ratio, rigid coefficient, and the intensity of excitation are carried out, and their effects both on linear and nonlinear indexes are evaluated. The damping mechanism is herein further investigated and some suggestions for the design in high-rise buildings are also proposed. To validate the superiority of SAID, an optimal passive particle impact damper (PIDopt) is also investigated as a control group, in which the parameters of the SAID remain the same, and the optimal parameters of the PIDopt are designed by differential evolution algorithm based on a reduced-order model. The numerical simulation shows that the SAID has better control effects than that of the optimized passive particle impact damper, not only for linear indexes (e.g., root mean square response), but also for nonlinear indexes (e.g., component energy consumption and hinge joint curvature).

Laplace domain method for evaluating mean-square response of simple oscillators to nonstationary excitation

Evaluating mean-square response of linear systems to nonstationary excitation has been extensively studied, but exact closed-form solutions under various assumptions have been rarely available. This paper derives exact closed-form solutions for the mean-square response of simple oscillators subjected to nonstationary excitation which is formulated as the multiplication of a stationary excitation characterized by an arbitrary spectral density function (PSD) and an envelope function being the sum of several exponential functions. Special attention is given to the envelope function as the sum of two exponential functions because of its practical importance. While analytically evaluating the nonstationary mean square response of a SDOF (single-degree-of-freedom) system subjected to a nonstationary stochastic excitation requires carrying out a triple integral, traditional methods would conduct tedious calculus in the time, frequency or time–frequency domain. In contrast, the proposed method – much more efficient than traditional methods – primarily operates in the Laplace domain (complex plane) based on pole–residue formulations. Another advantage of the proposed pole–residue method is that meaningful physical and mathematical insights could be gained in the solution procedure. For demonstrating the effectiveness and versatility of the proposed method, two cases of the input PSD function are studied: (1) a modulated white noise process, and (2) a modulated correlated process. The correctness of the exact closed-form solutions is verified numerically by Monte Carlo simulations.

Stochastic response analysis for nonlinear vibration systems with adjustable stiffness property under random excitation.

Nonlinear vibration systems with adjustable stiffness property have attracted considerable attentions for their prominent broadband performances. In the present manuscript, we consider the stochastic dynamical systems with adjustable stiffness and proposed a numerical method for the random responses analysis of the Gaussian white noise excited systems. A multi-dimensional Fokker-Plank-Kolmogorov equation governing the joint probability density of the mechanical states is derived according to the theory of diffusion processes. We solve the multi-dimensional equation using a splitting method and obtained the stationary probability densities and the mean-square responses directly. Two classical nonlinear vibration systems with adjustable stiffness, including the energy harvesting system and the Duffing system with Dahl friction, are presented as examples. Their comparisons with the results from Monte-Carlo simulations illustrate the effectiveness of the proposed procedure for both monostable and bistable cases, even for cases with strong excitation. In addition, the splitting method is efficient for higher-dimensional problem and has advantages of simple implementation, less storage of intermediate values and so on. Hence, in terms of the application scope, the proposed procedure is superior to the current mainstream methods for the random response evaluation of nonlinear vibration systems with adjustable stiffness.

Laplace Domain Approach for Computing Transient Response of Simple Oscillators to Stationary Excitation

AbstractEvaluating the mean-square response at the transient state of a single degree of freedom (SDOF) system to stationary random excitation has been extensively studied in the past, but the expl...

An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

This paper deals with the idea of the orthogonal functions in the equivalent linearization of the nonlinear systems. Block Pulse (BP) function gives effective tools to approximate complex problems. The aim of this work is on using properties of the BP function as an orthogonal function in process of linearization. The BP functions have been used to propose an equivalent linearization method in the time domain to determine the unknown linearization coefficients. The accuracy of the proposed method compared with the other equivalent linearization approaches, including the regulation linearization and the dual criterion linearization methods. This study exploited the nonlinear Van der Pol oscillator system under stationary random excitation to demonstrate the feasibility of the proposed method. The validity of the analytical method is verified by applying different values of nonlinearity and intensity of excitation. Besides, by comparing the mean-square responses and frequency response functions of the linearized systems for a wide range of nonlinearity depicted the present method is in agreement with other methods.

Transient Response of MDOF Systems With Inerters to Nonstationary Stochastic Excitation

Explicit, closed-form, exact analytical expressions are derived for the covariance kernels of a multi degrees-of-freedom (MDOF) system with arbitrary amounts of viscous damping (not necessarily proportional-type), that is equipped with one or more auxiliary mass damper-inerters placed at arbitrary location(s) within the system. The “inerter” is a device that imparts additional inertia to the vibration damper, hence magnifying its effectiveness without a significant damper mass addition. The MDOF system is subjected to nonstationary stochastic excitation consisting of modulated white noise. Results of the analysis are used to determine the dependence of the time-varying mean-square response of the primary MDOF system on the key system parameters such as primary system damping, auxiliary damper mass ratio, location of the damper-inerter, inerter mass ratio, inerter node choices, tuning of the coupling between the damper-inerter and the primary system, and the excitation envelope function. Results of the analysis are used to determine the dependence of the peak transient mean-square response of the system on the damper/inerter tuning parameters, and the shape of the deterministic intensity function. It is shown that, under favorable dynamic environments, a properly designed auxiliary damper, encompassing an inerter with a sizable mass ratio, can significantly attenuate the response of the primary system to broad band excitations; however, the dimensionless “rise-time” of the nonstationary excitation substantially reduces the effectiveness of such a class of devices (even when optimally tuned) in attenuating the peak dynamic response of the primary system.

Approximate Analytical Mean-Square Response of an Impacting Stochastic System Oscillator With Fractional Damping

Abstract The paper deals with the stochastic dynamics of a vibroimpact single-degree-of-freedom system under a Gaussian white noise. The system is assumed to have a hard type impact against a one-sided motionless barrier, located at the system's equilibrium. The system is endowed with a fractional derivative element. An analytical expression for the system's mean squared response amplitude is presented and compared with the results of numerical simulations.

Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance

In this paper, we investigate nonlinear dynamical responses of two-degree-of-freedom airfoil (TDOFA) models driven by harmonic excitation under uncertain disturbance. Firstly, based on the deterministic airfoil models under the harmonic excitation, we introduce stochastic TDOFA models with the uncertain disturbance as Gaussian white noise. Subsequently, we consider the amplitude–frequency characteristic of deterministic airfoil models by the averaging method, and also the stochastic averaging method is applied to obtain the mean-square response of given stochastic TDOFA systems analytically. Then, we carry out numerical simulations to verify the effectiveness of the obtained analytic solution and the influence of harmonic force on the system response is studied. Finally, stochastic jump and bifurcation can be found through the random responses of system, and probability density function and time history diagrams can be obtained via Monte Carlo simulations directly to observe the stochastic jump and bifurcation. The results show that noise can induce the occurrence of stochastic jump and bifurcation, which will have a significant impact on the safety of aircraft.

Transient Response of a SDOF System With an Inerter to Nonstationary Stochastic Excitation

An analytical study is presented of the covariance kernels of a damped, linear, two-degrees-of-freedom (2DOF) system which resembles a primary system that is provided with an auxiliary mass damper (AMD), in addition to an “inerter” (a device that imparts additional inertia to the vibration damper, hence magnifying its effectiveness without a significant damper mass addition). The coupled 2DOF system is subjected to nonstationary stochastic excitation consisting of a modulated white noise. An exponential function, resembling the envelope of a typical earthquake, is considered. Results of the analysis are used to determine the dependence of the peak transient mean-square response of the system on the damper/inerter tuning parameters, and the shape of the deterministic intensity function. It is shown that, under favorable dynamic environments, a properly designed auxiliary damper, encompassing an inerter with a sizable mass ratio, can significantly attenuate the response of the primary system to broad band excitations; however, the dimensionless “rise-time” of the nonstationary excitation substantially reduces the effectiveness of such a class of devices (even when optimally tuned) in attenuating the peak dynamic response of the primary system.

Vibration Analysis of a Piecewise-Smooth System with Negative Stiffness under Delayed Feedback Control

The principal resonance of a delayed piecewise-smooth (DPWS) system with negative stiffness under narrow-band random excitation is investigated in aspects of multiscale analysis, design methodology of the controller, and response properties. The amplitude-frequency response and steady-state moments together with the corresponding stability conditions of the controlled stochastic system are derived, in which the degradation case is also under consideration. Then, from the perspective of the equivalent damping, the comparisons of the response characteristics of the controlled system to the uncontrolled system, such as the phenomenon of frequency island, are fulfilled. Furthermore, sensitivity of the system response to feedback gain and time delay is studied and interesting dynamic properties are found. Meanwhile, the classification of the steady-state solution is also discussed. To control the maximum amplitude, the feedback parameters are determined by the frequency response together with stability boundaries which must be utilized to exclude the combinations of the unstable parameters. For the case with small noise intensity, mean-square responses present the similar characteristics to what is discussed in the deterministic case.

Generalization of Seide’s problem by the regulated stochastic linearization technique

In this paper, the nonlinear random vibration of a uniform beam clamped at both of its ends is investigated by the regulated stochastic linearization technique. A closed system of nonlinear algebraic equations for linearization coefficients is obtained using the frequency–response function matrix. Root mean-square response of maximum deflection of the beam that is obtained from the present technique is subsequently compared with that furnished by the conventional linearization. The numerical results show that the regulated stochastic linearization technique constitutes an excellent alternative to the classical linearization scheme for analyzing responses of the clamped–clamped beam.

On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation

AbstractThis paper applied the idea of block pulse (BP) transform in the equivalent linearization of a nonlinear system. The BP transform gives effective tools to approximate complex problems. The main goal of this work is on using BP transform properties in process of linearization. The accuracy of the proposed method compared with the other equivalent linearization including the stochastic equivalent linearization and the regulation linearization methods. Numerical simulations are applied to the nonlinear Van der Pol oscillator system under Gaussian white noise excitation to demonstrate the feasibility of the present method. Different values of nonlinearity are considered to show the effectiveness of the present method. Besides, by comparing the mean-square responses for divers values of nonlinearity and excitation intensity depicted the present method is able to approximate the behavior of nonlinear system and is in agreement with other methods.

Semi-analytical solution of random response for nonlinear vibration energy harvesters

Due to the prominent broadband performance of nonlinear vibration energy harvester, theoretical evaluations for the mean-square response to random excitations and the associated mean output power are of great interest. By employing the generalized harmonic transformation and equivalent nonlinearization technique, established here is a semi-analytical solution of random response for nonlinear vibration energy harvesters subjected to Gaussian white noise excitation. The semi-analytical solution for stationary probability density of the system response is obtained by two iterative processes. Numerical results for a Duffing-type harvester demonstrate rapid convergence of the iterative processes and high evaluation accuracy for the mean-square response and the mean output power. Furthermore, the influence of harvesting circuit on the mechanical subsystem can be converted to modified quasi-linear damping and stiffness with energy-dependent coefficients, which is different from the traditional viewpoint on the equivalence of constant-coefficient damping and provides more comprehensive explanation on the influence of harvesting circuit.

Concurrent design of composite macrostructure and cellular microstructure under random excitations

In this paper, a method for the concurrent topology optimization of macrostructural material distribution and periodic microstructure under random excitations is proposed. The sensitivity analysis of dynamic response with respect to design variables in two scales, i.e., macro and micro scales, is presented. The corresponding concurrent topology optimization of macrostructure and microstructure is established, where the objective function is to minimize the displacement response mean square (RMS) of the prescribed degree of freedom while volume constraints are applied to the macromaterial distribution and phase materials. The optimization problem is solved using a bi-directional evolutionary structural optimization (BESO) method. Several examples are presented to demonstrate the effectiveness of the proposed method.

Multi-valued responses of a nonlinear vibro-impact system excited by random narrow-band noise

We investigate the multi-valued responses of a non-linear vibro-impact oscillator with a one-sided barrier subject to random narrow-band excitation. The frequency response of the system is obtained using the Krylov-Bogoliubov averaging method. Meanwhile, the backbone curve and the critical equation of unstable region are also derived for the deterministic case. Then, the method of moment is applied to obtain the iterative calculation equation for the mean-square response amplitude under the stochastic case. Excitation frequency, nonlinearity intensity, damping parameters, especially the distance between the system’s static equilibrium position and the barrier can lead to triple-valued response under certain case. In some conditions the impact system may have two or four steady-state solutions, which is an interesting phenomenon for impact system. The unstable region is one uniform part while under smaller nonlinearity intensity it is divided into two parts. Moreover, we also find that as random noise intensity increases, the pervasion of the phase trajectories strengthens, and then destroys the topological property of the phase trajectories.