Formulation of Statistical Linearization for M-D-O-F Systems Subject to Combined Periodic and Stochastic Excitations

2019 ◽  
Vol 86 (10) ◽  
Author(s):  
Pol D. Spanos ◽  
Ying Zhang ◽  
Fan Kong
Keyword(s):  
Harmonic Balance ◽  
Equation Of Motion ◽  
Periodic Component ◽  
Statistical Moments ◽  
System Response ◽  
Statistical Linearization ◽  
Ensemble Averaging ◽  
Stochastic Component ◽  
Stochastic Excitations ◽  
Linear Elements

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts. The steady-state system response is expressed as the sum of a periodic (deterministic) component and of a zero-mean stochastic component. Next, the equation of motion leads to a nonlinear vector stochastic ordinary differential equation (ODE) for the zero-mean component of the response. The nonlinear term contains both the zero-mean component and the periodic component, and they are further equivalent to linear elements. Furthermore, due to the presence of the periodic component, these linear elements are approximated by averaging over one period of the excitation. This procedure leads to an equivalent system whose elements depend both on the statistical moments of the zero-mean stochastic component and on the amplitudes of the periodic component of the response. Next, input–output random vibration analysis leads to a set of nonlinear equations involving the preceded amplitudes and statistical moments. This set of equations is supplemented by another set of equations derived by ensuring, in a harmonic balance sense, that the equation of motion of the M-D-O-F system is satisfied after ensemble averaging. Numerical examples of a 2-D-O-F nonlinear system are considered to demonstrate the reliability of the proposed technique by juxtaposing the semi-analytical results with pertinent Monte Carlo simulation data.

Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

2021 ◽  
Vol 88 (5) ◽  
Author(s):  
Fan Kong ◽  
Pol D. Spanos
Keyword(s):  
Harmonic Balance Method ◽  
Linearization Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Single Degree Of Freedom ◽  
Stochastic Component ◽  
Linear Differential

Abstract A statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

Dynamic Model and Simulation of Flag Vibrations Modeled as a Membrane

2014 ◽  
Author(s):  
Gary Frey ◽  
Ben Carmichael ◽  
Joshua Kavanaugh ◽  
S. Nima Mahmoodi
Keyword(s):  
Constant Velocity ◽  
Cross Flow ◽  
Vertical Direction ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
System Response ◽  
Wind Flow ◽  
Pressure Function ◽  
Model And Simulation ◽  
Reasonable Model

A flag is modeled as a membrane to investigate the two-dimensional characteristics of the vibration response to an uniform wind flow. Both the affecting tension and pressure functions for the wind flow with constant velocity are introduced and utilized in the modeling. In this case, the tension is caused by the weight of the flag. The pressure function is a function describing the pressure variations caused on the flag when in uniform flow. The pressure function is found by assuming that the air flow is relatively slow and that the flag is wide enough to minimize cross flow at the boundaries. An analysis of the downstream motion of the flag is necessary as well. Hamilton’s principle is employed to derive the partial differential equation of motion. The flag is oriented in the vertical direction to neglect the effect of the flag’s weight on the system’s response. Galerkin’s method is used to solve for the first four mode shapes of the system, and the system response is numerically solved. Simulations reveal a very reasonable model when the flag is modeled as a membrane.

Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

2016 ◽  
Vol 849 ◽  
pp. 76-83
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
Harmonic Balance ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
System Response ◽  
Algebraic Equations ◽  
Single Degree Of Freedom ◽  
Nonlinear Dynamical ◽  
The Difference ◽  
Eigen Frequency

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Pol D. Spanos ◽  
Alberto Di Matteo ◽  
Yezeng Cheng ◽  
Antonina Pirrotta ◽  
Jie Li
Keyword(s):  
Fractional Derivative ◽  
Survival Probability ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
System Response ◽  
Statistical Linearization ◽  
Van Der Pol ◽  
Galerkin Scheme ◽  
Confluent Hypergeometric Functions

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

scholarly journals Flutter and parametric stability analysis of axially moving composite graphene sheets

2019 ◽  
Vol 286 ◽  
pp. 01008
Author(s):  
A. Azrar ◽  
N. Fakri ◽  
A.A. Aljinaidi ◽  
L. Azrar
Keyword(s):  
Harmonic Balance ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Excitation Frequency ◽  
Equation Of Motion ◽  
Graphene Sheets ◽  
Parametric Stability ◽  
Various Boundary Conditions ◽  
Axially Moving ◽  
Fibers Orientation

The dynamic analysis instability of axially moving rectangular composite graphene sheets with visco elastic foundation is modeled and numerically simulated for various boundary conditions based on the differential quadrature method (DQM). The partial differential equation of motion based on the nonlocal elasticity and the Kirchhoff plate theories is given. The Galerkin and harmonic balance methods are used for the linear and parametric vibration analysis. The influences of nonlocal parameter, the fibers orientation and the viscoelastic foundation effects on the dynamic behaviors of the rectangular graphene sheet as well as the instabilities induced by the time dependent axial speed and its excitation frequency are investigated.

Analysis of Free Nonlinear Vibration Behavior for Curved Embedded Carbon Nanotubes on Elastic Foundation

2010 ◽  
Author(s):  
M. Rezaee ◽  
H. Fekrmandi
Keyword(s):  
Carbon Nanotubes ◽  
Dynamic Behavior ◽  
Continuum Mechanics ◽  
Nonlinear Term ◽  
Nonlinear Oscillations ◽  
Equation Of Motion ◽  
System Response ◽  
Response Curves ◽  
Vibration Behavior ◽  
The Continuum

Carbon nanotubes (CNTs) are expected to have significant impact on several emerging nanoelectromechanical (NEMS) applications. Vigorous understanding of the dynamic behavior of CNTs is essential for designing novel nanodevices. Recent literature show an increased utilization of models based on elastic continuum mechanics theories for studying the vibration behavior of CNTs. The importance of the continuum models stems from two points; (i) continuum simulations consume much less computational effort than the molecular dynamics simulations, and (ii) predicting nanostructures behavior through continuum simulation is much cheaper than studying their behavior through experimental verification. In numerous recent papers, CNTs were assumed to behave as perfectly straight beams or straight cylindrical shells. However, images taken by transmission electron microscopes for CNTs show that these tiny structures are not usually straight, but rather have certain degree of curvature or waviness along the nanotubes length. The curved morphology is due to process-induced waviness during manufacturing processes, in addition to mechanical properties such as low bending stiffness and large aspect ratio. In this study the free nonlinear oscillations of wavy embedded multi-wall carbon nanotubes (MWCNTs) are investigated. The problem is formulated on the basis of the continuum mechanics theory and the waviness of the MWCNTs is modeled as a sinusoidal curve. The governing equation of motion is derived by using the Hamilton’s principle. The Galerkin approach was utilized to reduce the equation of motion to a second order nonlinear differential equation which involves a quadratic nonlinear term due to the curved geometry of the beam, and a cubic nonlinear term due to the stretching effect. The system response has been obtained using the incremental harmonic balanced method (IHBM). Using this method, the iterative relations describing the interaction between the amplitude and the frequency for the single-wall nanotube and double-wall nanotube are obtained. Also, the influence of the waviness, elastic medium and van der Waals forces on frequency-response curves is researched. Results present some useful information to analyze CNT’s nonlinear dynamic behavior.

Analysis on Bistable Response of a Disk-Rod-Fastening Rotor

2011 ◽  
Vol 199-200 ◽  
pp. 983-987
Author(s):  
Li Cheng ◽  
Zheng Wen Qian ◽  
Wei Chen
Keyword(s):  
Contact Surface ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Equation Of Motion ◽  
Simplified Model ◽  
Nonlinear Stiffness ◽  
Contact Effect ◽  
Bending Resistance ◽  
Rod Fastening Rotor ◽  
Rotor Stiffness

In the view of the fault of bistable response appeared in the disk-rod-fastening rotor, the peculiar structure of this kind rotor was taken into account in this paper. By considering the influence of contact effect of discontinuous interfaces on the rotor stiffness, the tie-rod and the contact surface between the wheel disks were equated to a bending resistance spring with nonlinear stiffness. Then the equation of motion of the disk-rod-fastening rotor was put forward according to this simplification. The vibration characteristics of rotor was calculated by the harmonic balance method that combined with the predict-correct and homotopy arithmetic. The bistable response behaviors recurred in this simplified model was much closer to the measured results in flying. The study showed that the nonlinear factor of stiffness, resulted from the contact effect of discontinuous interfaces, was the main reason which led to the appearance of bistable response in the disk-rod-fastening rotor.

Stability Analysis of an Articulated Loading Platform in Regular Sea

2007 ◽  
Vol 3 (1) ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Harmonic Balance ◽  
Harmonic Wave ◽  
Harmonic Balance Method ◽  
Equation Of Motion ◽  
Damping Term ◽  
Incremental Harmonic Balance Method ◽  
Left Hand ◽  
Hydrodynamic Damping ◽  
Incremental Harmonic Balance ◽  
The Right

Stability of the response of an articulated loading platform under regular wave, modeled as a SDOF nonlinear oscillator, is investigated. Relative velocity square drag force for harmonic wave appearing in the right hand side of the equation of motion is mathematically treated to bring the velocity dependent nonlinear hydrodynamic damping term to the left hand side of the equation of motion. Use of these two techniques makes the equation of motion amenable to the application of method IHBC. In order to trace different branches of the response curve and investigate different instability phenomena that may exist, the commonly used incremental harmonic balance method (IHB) is modified and integrated with an arc-length continuation technique to develop into incremental harmonic balance continuation (IHBC) method. Further, a technique for treating the nonlinear hydrodynamic damping term using a concept of distribution theory has been developed. The stability of the solution is investigated by the Floquet theory using Hsu’s scheme. The stable solutions obtained by the IHBC method are compared with those obtained by the numerical integration of equation of motion wherever applicable.

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