Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

Recent Developments on Nonstationary Parametric Responses of Rectangular Plates

1993 ◽  
Author(s):  
Germain L. Ostlguy ◽  
Patrice Lavigne
Keyword(s):  
Initial Conditions ◽  
Lateral Displacement ◽  
Asymptotic Series ◽  
Excitation Frequency ◽  
Continuous System ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Rectangular Plates ◽  
Stationary Response

Abstract The non-stationary parametric response of a rectangular plate during a logarithmic sweep of the excitation frequency through a system resonance is studied using five different techniques of solution. Considering only the case of principal parametric resonance, the continuous system is spatially discretized by means of a single-term modal approximation for the lateral displacement. The general form of the resulting nonlinear temporal equation of motion for the damped parametric vibrations in any spatial mode is analyzed using the multiple time scales method and the method of asymptotic series expansion developed by Mitropolsky, in the first and second approximation. The non-stationary response of the plate during transition through parametric resonances is also evaluated by direct integration of the temporal equation of motion and the results obtained by the different techniques are compared. The non-stationary response displays several phenomena depending on the conditions of in-plane loading, the amount of damping, the initial conditions, and the rate as well as the direction of the sweep. The validity of these results is ascertained experimentally.

Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations

2017 ◽  
Vol 24 (13) ◽  
pp. 2894-2911 ◽  
Author(s):  
Zia Saadatnia ◽  
Hassan Askari ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Governing Equation ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
External Excitation ◽  
Nonlinear Parameters ◽  
Multiple Time Scales Method ◽  
Frequency Excitation ◽  
Geometrical Factors

The multi-frequency excitation of a microbeam, resting on a nonlinear foundation, is investigated and the governing equation of motion of the microbeam system is developed. The viscoelastic-type foundation is considered by assuming nonlinear parameters for both Pasternak and Winkler coefficients. The well-known Galerkin approach is utilized to discretize the governing equation of motion and to obtain its nonlinear ordinary differential equations. The multiple time-scales method is employed to study the multi-frequency excitation of the microbeam. Furthermore, the resonant conditions due to the external excitation as well as the combination resonances for the first two modes are investigated. The influences of different parameters, namely the Pasternak and Winkler coefficients, the position of the applied force and the geometrical factors on the frequency response of the system are examined.

scholarly journals On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

2019 ◽  
Vol 14 (4) ◽  
Author(s):  
Saad Ilyas ◽  
Feras K. Alfosail ◽  
Mohammad I. Younis
Keyword(s):  
Analytical Solution ◽  
Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Equation Of Motion ◽  
Higher Order ◽  
Resonance Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Electrostatically Actuated

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

scholarly journals INTEREST GROUP SESSION—MEASUREMENT, STATISTICS, AND RESEARCH DESIGN: ANALYSIS OF DEVELOPMENTAL AND COMPLEX SYSTEMS: UNDERSTANDING THE DYNAMICS OF AGING

2019 ◽  
Vol 3 (Supplement_1) ◽  
pp. S376-S376
Author(s):  
Xiao Yang ◽  
Nilam Ram ◽  
Nilam Ram
Keyword(s):  
Complex Systems ◽  
Dynamic Models ◽  
Initial Conditions ◽  
Developmental Trajectories ◽  
Dynamic Change ◽  
System Thinking ◽  
Multiscale Entropy ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Regulation Model

Abstract Aging is the product of numerous dynamic processes that span multiple domains of functioning (e.g., biological, psychological, social), multiple levels of analysis, and multiple time-scales. Scientific inquiry in many fields has benefited from articulation and analysis of complex systems. This symposium brings together a collection of papers that illustrate how dynamical systems modeling is contributing to both theory and understanding of aging. Yang and colleagues apply Boolean network approach to intensive longitudinal data to identify sequences of emotion and behavior that lead to a stable equilibrium, and suggest how that information can be used to design interventions that push individuals toward a healthier equilibrium. Rector and colleagues illustrate use of dynamic indicators and multiscale entropy measures as indicators of resilience and explain how those measures may be used in prediction of physical recovery. Brick highlights how sequence mining methods can be used to identify commonalities and differences in dynamic change, and how those patterns characterize and distinguish groups with respect to aging trajectories. Moulder and colleagues demonstrate how latent maximum Lyapunov exponents can be used to study sensitivity of individuals’ developmental trajectories to initial conditions. Boker and colleagues provide a general overview of how dynamic models, including an adaptive equilibrium regulation model, distinguish resilience to acute versus chronic stressors and patterns of regulation. Together these papers highlight the value complex system thinking can add to our understanding and optimization of aging.

Singular perturbation analysis of a two-site model for an epidemic in age-structured population

2018 ◽  
Vol 11 (07) ◽  
pp. 1850087
Author(s):  
Rodrigue Yves M'pika Massoukou ◽  
Hermane Mambili-Mamboundou
Keyword(s):  
Singular Perturbation ◽  
Time Scales ◽  
Perturbation Analysis ◽  
Linear Models ◽  
Initial Conditions ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Site Model ◽  
Singular Perturbation Analysis ◽  
Perturbed Model

We formulate a model describing the dynamics for the spatial propagation of an SIS epidemic within a population, with age structure, living in an environment divided into two sites. An analysis of the model is given. We prove the existence of a unique disease free equilibrium (DFE) and its (local and global) stability. Further, we assume that fast infection processes and fast migration processes take place in the above-mentioned model; i.e. such processes last only a few days (less than a week). In opposition to such processes, demographic processes such as birth, death and maturation last quite a lot of years. Such a gap between the time scales gives rise to a multiple time scales model, in particular a singularly perturbed model. Through a singular perturbation analysis, based on Tikhonov theorem, we prove that for certain classes of initial conditions the nonlinear perturbed model can be approximated with very good accuracy by lower-dimensional linear models.

A beam–ring circular truss antenna restrained by means of the negative speed feedback procedure

2021 ◽  
pp. 107754632110036
Author(s):  
Ashraf T EL-Sayed Taha ◽  
Hany S Bauomy
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Nonlinear Partial Differential Equations ◽  
Time Frame ◽  
Ring Structure ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Averaged Equations ◽  
Fractional Differential

The present article contemplates the nonlinear powerful exhibitions of affecting dynamic vibration controller over a beam–ring structure for demonstrating the circular truss antenna exposed to mixed excitations. The dynamic controller comprises the included negative speed input added to the framework’s idea. By using the statue, Hamilton, the nonlinear fractional differential administering conditions of movement and the limit conditions have inferred for the shaft ring structure. Through Galerkin’s method, the nonlinear partial differential equations referred to overseeing the movement of the shaft ring structure have diminished to a coupled normal differential equations extending the nonlinearities square terms. Multiple time scales have helped in acquiring (getting) the four-dimensional averaged equations for measuring the primary and 1:2 internal resonances. This article’s controlled assessment is useful for controlling the nonlinear vibrations of the considered framework. Likewise, the controller dispenses with the framework’s oscillations in a brief time frame. The demonstrations of the numerous coefficients and the framework directed at the examined resonance case have been determined. Using MATLAB 7.0 programs has aided in completing the simulation results. At last, the numerical outcomes displayed an admirable concurrence with the methodical ones. A comparison with recently available articles has also indicated good results through using the presented controller.

Parametric Excitation of Pre-Stressed Graphene Sheets under Magnetic Field: Nonlinear Vibration and Dynamic Instability

2019 ◽  
Vol 19 (11) ◽  
pp. 1950135
Author(s):  
Majid Ghadiri ◽  
S. Hamed S. Hosseini
Keyword(s):  
Magnetic Field ◽  
Parametric Excitation ◽  
Dynamic Instability ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Graphene Sheets ◽  
Multiple Time ◽  
Unstable Solutions ◽  
Instability Regions ◽  
Voigt Model

Motivated by the lack of sufficient accuracy in investigation of nonlinear dynamics of graphene sheets (GS), nonlinear dynamic instability and frequency response of the pre-stressed single layered GS (SLGS) are investigated in the present paper. To achieve this aim, in the first step, SLGS embedded on a visco-Pasternak foundation is modeled while it is under an initial stress and subjected to a parametric axial force and magnetic field. Then, based on Eringen’s theory, nonlinear von Karman relations and Kelvin–Voigt model, the nonlinear governing equation of motion is derived. In the next step, Galerkin technique and multiple time scales method are employed to analyze and solve the equation of motion. Emphasizing the effect of parametric excitation, for considering the instability regions, bifurcation points are discussed. As a result, a parametric study is conducted to show the importance of damping coefficient and parametric excitation in dynamic instability of the system. Numerical examples are also treated which show various discontinuous bifurcations. Also, infinitely stable and unstable solutions are addressed.

Interaction of electromagnetic fields with warm slightly ionized magneto-plasmas

1967 ◽  
Vol 1 (4) ◽  
pp. 473-481 ◽  
Author(s):  
O. De Barbieri ◽  
C. Maroli
Keyword(s):  
Electromagnetic Fields ◽  
Maxwell Equations ◽  
Initial Conditions ◽  
Electron Distribution Function ◽  
Collision Integral ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Initial Value ◽  
Initial Field ◽  
Longitudinal Fields

The problem concerning the interaction of electromagnetic fields with a warm, slightly ionized magneto-plasma is analysed by means of the multiple time scales asymptotic technique. Only initial value problems in an infinite plasma are here considered in detail. It is shown how the full set of Vlasov–Maxwell equations, modified through a standard Boltzmann collision integral to take account of the elastic electron-neutral encounters, can be reduced, in the basic limit assumed, to a set of two coupled non-linear integro-difl�erential equations for the dominant terms of the longitudinal fields and of the electron distribution function. Once the initial conditions are specified, the behaviour of the transverse initial field is also described. A particular example has been given for the so-called progressing-wave initial conditions.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

Sign in / Sign up

Export Citation Format

Share Document