Resonant Excitation of a Spinning, Nutating Plate

1979 ◽  
Vol 46 (1) ◽  
pp. 132-138 ◽  
Author(s):  
J. W. Klahs ◽  
J. H. Ginsberg
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Resonant Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Principal Coordinates ◽  
Method Of Solution ◽  
The Galerkin Method ◽  
Plane Displacement

The equations of motion governing the three-dimensional finite-amplitude response of a plate in arbitrary space motion are derived and shown to lead to dynamic coupling between the transverse and in-plane displacement. A general method of solution for such problems is demonstrated in an example involving a simply supported rectangular plate spinning about an axis parallel to an edge and nutating through a small angle. The method involves an asymptotic expansion using the derivative expansion version of the method of multiple time scales, in conjunction with the Galerkin method. A critical spin rate leading to the loss of stability in divergence is determined. Then, a numerical example of resonant excitation of one principal coordinate demonstrates that the nonlinear response resembling the one obtained from linear theory may lose stability in favor of a second response in which several principal coordinates are mutually excited. Consideration of the interaction between in-plane and transverse displacements is shown to be crucial to the prediction of this “unusual” response.

scholarly journals Active Vibration Control of a Nonlinear Beam with Self- and External Excitations

2013 ◽  
Vol 20 (6) ◽  
pp. 1033-1047 ◽  
Author(s):  
J. Warminski ◽  
M. P. Cartmell ◽  
A. Mitura ◽  
M. Bochenski
Keyword(s):  
Analytical Solutions ◽  
Order Approximation ◽  
Equations Of Motion ◽  
Active Vibration Control ◽  
Harmonic Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Beam ◽  
Approximate Analytical Solutions ◽  
Full Structure

An application of the nonlinear saturation control (NSC) algorithm for a self-excited strongly nonlinear beam structure driven by an external force is presented in the paper. The mathematical model accounts for an Euler-Bernoulli beam with nonlinear curvature, reduced to first mode oscillations. It is assumed that the beam vibrates in the presence of a harmonic excitation close to the first natural frequency of the beam, and additionally the beam is self-excited by fluid flow, which is modelled by a nonlinear Rayleigh term for self-excitation. The self- and externally excited vibrations have been reduced by the application of an active, saturation-based controller. The approximate analytical solutions for a full structure have been found by the multiple time scales method, up to the first-order approximation. The analytical solutions have been compared with numerical results obtained from direct integration of the ordinary differential equations of motion. Finally, the influence of a negative damping term and the controller's parameters for effective vibrations suppression are presented.

Nonlinear Dynamic Response of a Thin Plate in a Fractional Viscoelastic Medium under Internal Resonance 1:1:2

2014 ◽  
Vol 518 ◽  
pp. 60-65 ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Nonlinear Equations ◽  
Time Scales ◽  
Viscoelastic Medium ◽  
Equations Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Nonlinear Dynamic Response ◽  
New Approach ◽  
New Type

Dynamic behaviour of a nonlinear plate embedded in a fractional derivative viscoelastic medium and subjected to the conditions of the internal resonances two-to-one has been studied by Rossikhin and Shitikova in [1]. Nonlinear equations, the linear parts of which occur to be coupled, were solved by the method of multiple time scales. A new approach proposed in this paper allows one to uncouple the linear parts of equations of motion of the plate, while the same method, the method of multiple time scales, has been utilized for solving nonlinear equations. The new approach enables one to find a new type of the internal resonanse, i.e., one-to-one-to-two, as well as to solve the problems of vibrations of thin bodies more efficiently.

scholarly journals ANALYSIS OF NONLINEAR FORCED VIBRATIONS OF FRACTIONALLY DAMPED SUSPENSION BRIDGES SUBJECTED TO THE ONE-TO-ONE INTERNAL RESONANCE

2020 ◽  
Vol 16 (2) ◽  
pp. 113-131
Author(s):  
Marina Shitikova ◽  
Aleks Katembo
Keyword(s):  
Internal Resonance ◽  
Fractional Power ◽  
Suspension Bridge ◽  
Multiple Time Scales ◽  
Suspension Bridges ◽  
Multiple Time ◽  
Method Of Solution ◽  
Nonlinear Force ◽  
One To One ◽  
The One

Nonlinear force driven coupled vertical and torsional vibrations of suspension bridges, when the frequency of an external force is approaching one of the natural frequencies of the suspension system, which, in its turn, undergoes the conditions of the one-to-one internal resonance, are investigated. The method of multiple time scales is used as the method of solution. The damping features are described by the fractional derivative, which is interpreted as the fractional power of the differentiation operator. The influence of the fractional parameters (orders of fractional derivatives) on the motion of the suspension bridge is investigated.

scholarly journals Approximate Analytical Three-Dimensional Multiple Time Scales Solution to a Circular Restricted Three-Body Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fabao Gao ◽  
Yongqing Wang
Keyword(s):  
Time Scales ◽  
Body Problem ◽  
Three Dimensional ◽  
Classical Problem ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Chaotic Nature ◽  
Three Body

We illustrate the chaotic nature of the circular restricted three-body problem from the perspective of the bifurcation diagram with respect to the mass ratio parameter. Moreover, it is shown that when the frequency ratio in different directions of the classical problem is irrational, it has the quasiperiodic characteristics. In addition, a three-dimensional approximate solution to this problem under two time scales is proposed by using the multiple time scales method.

Nonlinear Forced Vibration of Curved Carbon Nanotube Resonators

2016 ◽  
Author(s):  
Hassan Askari ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Carbon Nanotube ◽  
Forced Vibration ◽  
Primary Resonance ◽  
Beam Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Forced Vibration ◽  
Resonator System ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam

Nonlinear forced vibration of the nonlocal curved carbon nanotubes is investigated. The governing equation of vibration of a nonlocal curved carbon nanotube is developed. The nonlinear Winkler and Pasternak type foundations are chosen for the nanotube resonator system. Furthermore, the shape of the carbon nanotube system is assumed to be of a sinusoidal curvature form and different types of the boundary conditions are postulated for the targeted system. The Euler-Bernoulli beam theory in conjunction with the Eringen theory are implemented to obtain the partial differential equation of the system. The Galerkin method is applied to obtain the nonlinear ordinary differential equations of the system. For the sake of obtaining the primary resonance of the considered system the multiple time scales method is utilized. The influences of different parameters, namely, the position of the applied force, different forms of boundary condition, amplitude of curvature, and the coefficient of the Pasternak foundation, on the frequency response of the system were fully investigated.

Nonlinear evolution of waves on a vertically falling film

1993 ◽  
Vol 250 ◽  
pp. 433-480 ◽  
Author(s):  
H.-C. Chang ◽  
E. A. Demekhin ◽  
D. I. Kopelevich
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Capillary Waves ◽  
Stationary Waves ◽  
Falling Film ◽  
Linear Mode

Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ1travels slower than infinitesimally small waves of the same wavelength while the other family γ2and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ1with wavenumber αs(or α2) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber αmand approaches the wave on γ1with the highest flow rate at low Reynolds numbers. On the fast γ2family, however, multiple bands of near-solitary waves bounded below by αfare found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α2of the slow γ1family. They then approach the long stable waves on the γ2family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.

Nonlinear Dynamic Response of a Thin Plate Embedded in a Fractional Viscoelastic Medium under Combinational Internal Resonances

2014 ◽  
Vol 595 ◽  
pp. 105-110 ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Nonlinear Equations ◽  
Time Scales ◽  
Viscoelastic Medium ◽  
Equations Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Nonlinear Dynamic Response ◽  
New Approach ◽  
Difference Type

Dynamic behaviour of a nonlinear plate embedded in a fractional derivative viscoelastic medium and subjected to the conditions of the combinational internal resonances of the additive and difference types has been studied by Rossikhin and Shitikova in [1]. Nonlinear equations, the linear parts of which occur to be coupled, were solved by the method of multiple time scales. A new approach proposed in [2] allows one to uncouple the linear parts of equations of motion of the plate, while the same method, the method of multiple time scales, has been utilized for solving nonlinear equations. The new approach enables one to find an additional combinational resonance of the additive-difference type, as well as to solve the problems of vibrations of thin bodies more efficiently.

Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation

2016 ◽  
Vol 23 (7) ◽  
pp. 1106-1114 ◽  
Author(s):  
Saim Kural ◽  
Erdoğan Özkaya
Keyword(s):  
Elastic Foundation ◽  
Couple Stress Theory ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Approximate Solutions ◽  
Modified Couple Stress Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Classical Beam Theory

In this study, fluid conveying continuous media was considered as micro beam. Unlike the classical beam theory, the effects of shear stress on micro-structure's dynamic behavior not negligible. Therefore, modified couple stress theory (MCST) were used to see the effects of being micro-sized. By using Hamilton's principle, the nonlinear equations of motion for the fluid conveying micro beam were obtained. Micro beam was considered as resting on an elastic foundation. The obtained equations of motion were became independence from material and geometric structure by nondimensionalization. Approximate solutions of the system were achieved with using the multiple time scales method (a perturbation method). The effects of micro-structure, spring constant, the occupancy rate of micro beam, the fluid velocity on natural frequency and solutions were researched. MCST compared with classical beam theory and showed that beam models that based on classical beam theory are not capable of describing the size effects. Comparisons of classical beam theory and MCST were showed in graphics and these graphics also proved that obtained mathematical model suitable for describe the behavior of normal sized beams.

DYNAMICS OF RELAXATION OSCILLATIONS

2001 ◽  
Vol 11 (05) ◽  
pp. 1471-1482 ◽  
Author(s):  
PAUL E. PHILLIPSON ◽  
PETER SCHUSTER
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Time Span ◽  
Singular Perturbation Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Relaxation Oscillations ◽  
Van Der Pol

Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker–Haag piecewise-linear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker–Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to three-dimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos.

scholarly journals CLASSIFICATION OF INTERNAL RESONANCES IN NONLINEAR FRACTIONALY DAMPED UFLYAND-MINDLIN PLATES

2020 ◽  
Vol 16 (3) ◽  
pp. 60-77
Author(s):  
Marina Shitikova ◽  
Elena Osipova
Keyword(s):  
Power Series ◽  
Small Parameter ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Governing Equations ◽  
Internal Resonances ◽  
Shear Deformations

In the present paper, the nonlinear free vibrations of fractionally damped plates are studied, equations of motion of which take the rotary inertia and shear deformations into account and involve fivecoupled nonlinear differentialequations in terms of three mutually orthogonal displacements and two angles of rotation. The procedure resulting in decoupling linear parts of equations has been proposed with further utilization of the generalized method of multiple time scales for solving nonlinear governing equations of motion, in so doing the amplitude functions have been expanded into power series in terms of the small parameter and depend on differenttime scales. The occurrence of the internal or combinational resonances in Uflyand-Mindlinplates has been revealed and classified

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