scholarly journals Robustness Analysis of the Collective Dynamics of Nonlinear Periodic Structures Under Parametric Uncertainty

2016 ◽  
Author(s):  
Khaoula Chikhaoui ◽  
Diala Bitar ◽  
Najib Kacem ◽  
Noureddine Bouhaddi
Keyword(s):  
Multiple Scales ◽  
Periodic Structures ◽  
Uncertainty Propagation ◽  
Time Integration ◽  
Robustness Analysis ◽  
Parametric Uncertainty ◽  
Latin Hypercube Sampling ◽  
Algebraic Equations ◽  
Collective Dynamics ◽  
Model Combining

In order to ensure more realistic design of nonlinear periodic structures, the collective dynamics of a coupled pendulums system is investigated under parametric uncertainties. A generic discrete analytical model combining the multiple scales method, the perturbation theory and a standing-wave decomposition is proposed and adapted to the presence of uncertainties. These uncertainties are taken into account through a probabilistic modeling implying that the stochastic parameters vary according to random variables of chosen probability density functions. The proposed model leads to a set of coupled complex algebraic equations written according to the number and positions of the uncertainties in the structure and numerically solved using the time integration Runge-Kutta method. The uncertainty propagation through the established model is finally ensured using the Latin Hypercube Sampling method. The analysis of the dispersion, in term of variability of the frequency and amplitude intervals of the multistability domain shows the effects of uncertainties on the stability and nonlinearity of a three coupled pendulums structure. The nonlinear aspect is strengthened, the multistability domain is wider, more stable branches are obtained and thus the multimode solutions are enhanced.

Transient growth in the near wake region of the flow past a finite span wing

2019 ◽  
Vol 866 ◽  
pp. 399-430 ◽  
Author(s):  
Navrose ◽  
V. Brion ◽  
L. Jacquin
Keyword(s):  
Periodic Structures ◽  
Time Integration ◽  
Tip Vortex ◽  
Wake Region ◽  
Wind Tunnel Experiments ◽  
Near Wake ◽  
Optimal Perturbation ◽  
Tip Vortices ◽  
Naca0012 Airfoil ◽  
Flow Past

We investigate optimal perturbation in the flow past a finite aspect ratio ($AR$) wing. The optimization is carried out in the regime where the fully developed flow is steady. Parametric study over time horizon ($T$), Reynolds number ($Re$), $AR$, angle of attack and geometry of the wing cross-section (flat plate and NACA0012 airfoil) shows that the general shape of linear optimal perturbation remains the same over the explored parameter space. Optimal perturbation is located near the surface of the wing in the form of chord-wise periodic structures whose strength decreases from the root towards the tip. Direct time integration of the disturbance equations, with and without nonlinear terms, is carried out with linear optimal perturbation as initial condition. In both cases, the optimal perturbation evolves as a downstream travelling wavepacket whose speed is nearly the same as that of the free stream. The energy of the wavepacket increases in the near wake region, and is found to remain nearly constant beyond the vortex roll-up distance in nonlinear simulations. The nonlinear wavepacket results in displacement of the tip vortex. In this situation, the motion of the tip vortex resembles that observed during vortex meandering/wandering in wind tunnel experiments. Results from computation carried out at higher $Re$ suggest that, even beyond the steady flow regime, a perturbation wavepacket originating near the wing might cause meandering of tip vortices.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

Nonlinear interphase effects on plastic hardening of nylon 6/clay nanocomposites: A computational stochastic analysis

2019 ◽  
Vol 54 (6) ◽  
pp. 753-763
Author(s):  
Vahid Yaghoubi ◽  
Mohammad Silani ◽  
Hossein Zolfaghari ◽  
Mostafa Jamshidian ◽  
Timon Rabczuk
Keyword(s):  
Stochastic Analysis ◽  
Uncertainty Propagation ◽  
Clay Content ◽  
Latin Hypercube Sampling ◽  
Information Criterion ◽  
Nylon 6 ◽  
Clay Nanocomposites ◽  
Chi Square ◽  
Hardening Modulus ◽  
Plastic Hardening

In this paper, the nonlinear effect of interphase properties on the macroscopic plastic response of nylon 6/clay nanocomposites is investigated by applying a stochastic analysis on a multiscale computational model of nanocomposites. The mechanical behavior of interphase is described with respect to that of the matrix by a weakening coefficient. The interphase thickness and properties are considered as the stochastic inputs and the hardening modulus and hardening exponent describing the plastic hardening characteristics of the nanocomposite are the random outputs. The stochastic analysis consists of three procedures including (i) model selection using Akaike information criterion, (ii) uncertainty propagation using Latin Hypercube sampling in conjunction with chi-square test, and (iii) sensitivity analysis using Sobol indices. The results indicate that the exponential hardening model best describes the flow stress–plastic strain response of the nanocomposite. It is also shown that increasing the clay content generally increases the plastic hardening rate of the nanocomposite up to 4% clay content. Besides, the hardening characteristics of the nanocomposite are more sensitive to the weakening coefficient than the interphase thickness.

scholarly journals Dynamics of gravity–capillary solitary waves in deep water

2012 ◽  
Vol 708 ◽  
pp. 480-501 ◽  
Author(s):  
Zhan Wang ◽  
Paul A. Milewski
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Periodic Structures ◽  
Time Integration ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Full Potential ◽  
Numerical Time Integration ◽  
The Stability ◽  
Nonlinear Focusing

AbstractThe dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Convergence of topological patterns of optimal periodic structures under multiple scales

2011 ◽  
Vol 46 (1) ◽  
pp. 41-50 ◽  
Author(s):  
Yi Min Xie ◽  
Zhi Hao Zuo ◽  
Xiaodong Huang ◽  
Jian Hua Rong
Keyword(s):  
Multiple Scales ◽  
Periodic Structures

scholarly journals Wavenumber-space band clipping in nonlinear periodic structures

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210052
Author(s):  
Weijian Jiao ◽  
Stefano Gonella
Keyword(s):  
Dispersion Relation ◽  
Multiple Scales ◽  
Periodic Structures ◽  
Dispersion Correction ◽  
Cubic Nonlinearity ◽  
Frequency Shifts ◽  
Weakly Nonlinear ◽  
Practical Applications ◽  
Correction Effect ◽  
Main Effect

In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as an initial condition or as a boundary condition, respectively. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. We then extend the framework to locally resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.

A Multiple-Scales Analysis of Nonlinear, Localized Modes in a Cyclic Periodic System

1993 ◽  
Vol 60 (2) ◽  
pp. 388-397 ◽  
Author(s):  
A. Vakakis ◽  
T. Nayfeh ◽  
M. King
Keyword(s):  
Vibration Isolation ◽  
Multiple Scales ◽  
Algebraic Equations ◽  
Localized Modes ◽  
Mode Localization ◽  
Nonlinear Mode ◽  
Weakly Coupled ◽  
Passive Vibration Isolation ◽  
Coupling Stiffness ◽  
Multiple Scales Analysis

In this work the nonlinear localized modes of an n-degree-of-freedom (DOF) nonlinear cyclic system are examined by the averaging method of multiple scales. The set of nonlinear algebraic equations describing the localized modes is derived and is subsequently solved for systems with various numbers of DOF. It is shown that nonlinear localized modes exist only for small values of the ratio (k/μ), where k is the linear coupling stiffness and μ is the coefficient of the grounding stiffness nonlinearity. As (k/μ) increases the branches of localized modes become nonlocalized and either bifurcate from “extended” antisymmetric modes in inverse, “multiple” Hamiltonian pitchfork bifurcations (for systems with even-DOF), or reach certain limiting values for large values of(k/μ) (for systems with odd-DOF). Motion confinement due to nonlinear mode localization is demonstrated by examining the responses of weakly coupled, perfectly periodic cyclic systems caused by external impulses. Finally, the implications of nonlinear mode localization on the active or passive vibration isolation of such structures are discussed.

Bifurcation Analysis for Brake Squeal

2010 ◽  
Author(s):  
Hartmut Hetzler
Keyword(s):  
Bifurcation Analysis ◽  
Sliding Friction ◽  
Multiple Scales ◽  
Time Integration ◽  
Perturbation Approach ◽  
Disk Brake ◽  
System A ◽  
Bifurcation Behaviour ◽  
Multiple Scales Technique ◽  
Tribological Parameters

This article presents a perturbation approach for the bifurcation analysis of MDoF vibration systems with gyroscopic and circulatory contributions, as they naturally arise from problems involving moving continua and sliding friction. Based on modal data of the underlying linear system, a multiple scales technique is utilized in order to find equations for the nonlinear amplitudes of the critical mode. The presented method is suited for an algorithmic implementation using commercial software and does not involve costly time-integration. As an engineering example, the bifurcation behaviour of a MDoF disk brake model is investigated. Sub- and supercritical Hopf-bifurcations are found and stationary nonlinear amplitudes are presented depending on operating parameters of the brake as well as of tribological parameters of the contact.

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