scholarly journals Multistability and Modal Interactions in Periodic 2D Coupled Pendulums Array

2016 ◽  
Author(s):  
Diala Bitar ◽  
Najib Kacem ◽  
Noureddine Bouhaddi
Keyword(s):  
Multiple Scales ◽  
Basins Of Attraction ◽  
Nonlinear Phenomenon ◽  
Modal Interactions ◽  
Nonlinear Structure ◽  
Collective Dynamics ◽  
The Stability ◽  
Wave Decomposition ◽  
Multi Mode ◽  
Large Distribution

The collective dynamics of an array of periodic two dimensional (2D) coupled pendulums under harmonic horizontal base excitation is investigated. The coupled differential equations governing the nonlinear vibrations of the considered system have been solved using an analytical-numerical solving procedure, based on the multiple scales method coupled with standing wave decomposition. It allows the identification of complex and wide variety of nonlinear phenomenon exhibited by the periodic nonlinear structure. The frequency responses for several coupled pendulums were calculated in order to analyze the stability, the modal interactions and the bifurcation topologies resulting from the collective dynamics of the coupled pendulums, while highlighting the large number of multimodal solutions for a small number of coupled pendulums. The complexity and the multivaludness of the responses were illustrated by a study of basins of attraction which display the large distribution of the multi-mode branches.

Self Excited Multi-Mode Vibrations of Aircraft Brakes With Nonlinear Negative Damping

1995 ◽  
Author(s):  
Raymond J. Black
Keyword(s):  
Equations Of Motion ◽  
Normal Modes ◽  
Frequency Mode ◽  
Modal Interactions ◽  
Linearized System ◽  
Frictional Interface ◽  
Negative Damping ◽  
The Stability ◽  
Multi Mode ◽  
Lower Frequency Mode

Abstract This paper shows how vibratory modes of a brake/landing gear system can interact strongly when there is nonlinear negative damping being generated at the brake’s friction interface. The approach first considers the normal modes of the linearized system. The nonlinear frictional interface force is then added to the modal equations of motion. The energy added to each of the modes, per cycle of the lowest frequency mode, is then determined. From these functions an amplitude path map and limit cycle amplitudes are determined. Multiple limit cycles are found to exist for certain combinations of damping. Amplitude modulation of the higher-frequency mode at multiples of the lower frequency mode is explained. Time solutions of the motion are obtained and compared to computer simulation results. Results compare closely. The method yields a global view of the stability and modal interactions caused by nonlinear negative damping at the brake’s friction interface.

Nonlinear Response of a Buckled Beam to a Three-to-One Internal Resonance

2003 ◽  
Author(s):  
Samir A. Emam ◽  
Ali H. Nayfeh
Keyword(s):  
Periodic Orbits ◽  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Experimental Results ◽  
Buckled Beam ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Multi Mode

We investigate the nonlinear response of a clamped-clamped buckled beam to a three-to-one internal resonance between the first and third modes when one of them is externally excited. To examine whether the first and third modes are nonlinearly coupled, we use the method of multiple scales to directly attack the partial-differential equation and associated boundary conditions and obtain the equations governing the modulation of their amplitudes and phases. We find that the two modes are nonlinearly coupled. To investigate the large-amplitude dynamics, we use a multi-mode Galerkin discretization to obtain a reduced-order model of the problem. We use a shooting method to compute periodic orbits of the discretized equations and Floquet theory to investigate the stability and bifurcations of these periodic orbits. We note an energy transfer from the first mode, which is externally excited by a primary resonance, to the third mode. We obtain preliminary experimental results of the energy exchange between the first and third modes as a result of a three-to-one internal resonance. More experimental results are being generated.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

scholarly journals A new conceptual model of coral biomineralisation: hypoxia as the physiological driver of skeletal extension

2013 ◽  
Vol 10 (5) ◽  
pp. 2867-2884 ◽  
Author(s):  
S. Wooldridge
Keyword(s):  
Conceptual Model ◽  
Multiple Scales ◽  
High Growth ◽  
High Growth Rate ◽  
Coral Host ◽  
Structural Determinants ◽  
Night Time ◽  
Long Run ◽  
Animal Phyla ◽  
The Stability

Abstract. That corals skeletons are built of aragonite crystals with taxonomy-linked ultrastructure has been well understood since the 19th century. Yet, the way by which corals control this crystallization process remains an unsolved question. Here, I outline a new conceptual model of coral biomineralisation that endeavours to relate known skeletal features with homeostatic functions beyond traditional growth (structural) determinants. In particular, I propose that the dominant physiological driver of skeletal extension is night-time hypoxia, which is exacerbated by the respiratory oxygen demands of the coral's algal symbionts (= zooxanthellae). The model thus provides a new narrative to explain the high growth rate of symbiotic corals, by equating skeletal deposition with the "work-rate" of the coral host needed to maintain a stable and beneficial symbiosis. In this way, coral skeletons are interpreted as a continuous (long-run) recording unit of the stability and functioning of the coral–algae endosymbiosis. After providing supportive evidence for the model across multiple scales of observation, I use coral core data from the Great Barrier Reef (Australia) to highlight the disturbed nature of the symbiosis in recent decades, but suggest that its onset is consistent with a trajectory that has been followed since at least the start of the 1900s. In concluding, I outline how the proposed capacity of cnidarians (which includes modern reef corals) to overcome the metabolic limitation of hypoxia via skeletogenesis also provides a new hypothesis to explain the sudden appearance in the fossil record of calcified skeletons at the Precambrian–Cambrian transition – and the ensuing rapid appearance of most major animal phyla.

scholarly journals Nonlinear Blade Stability for a Scaled Autogyro Rotor at High Advance Ratios

2020 ◽  
Vol 65 (1) ◽  
pp. 1-19
Author(s):  
Djamel Rezgui ◽  
Mark H. Lowenberg
Keyword(s):  
High Speed ◽  
Flow Speed ◽  
Numerical Continuation ◽  
Control Input ◽  
Nonlinear Phenomenon ◽  
Scaled Model ◽  
Underlying Mechanisms ◽  
Slowed Rotor ◽  
The Stability ◽  
Blade Pitch Angle

Despite current research advances in aircraft dynamics and increased interest in the slowed rotor concept for high-speed compound helicopters, the stability of autogyro rotors remains partially understood, particularly at lightly loaded conditions and high advance ratios. In autorotation, the periodic behavior of a rotor blade is a complex nonlinear phenomenon, further complicated by the fact that the rotor speed is not held constant. The aim of the analysis presented in this article is to investigate the underlying mechanisms that can lead to rotation-flap blade instability at high advance ratios for a teetering autorotating rotor. The stability analysis was conducted via wind tunnel tests of a scaled autogyro model combined with numerical continuation and bifurcation analysis. The investigation assessed the effect of varying the flow speed, blade pitch angle, and rotor shaft tilt relative to the flow on the rotor performance and blade stability. The results revealed that rotor instability in autorotation is associated with the existence of fold bifurcations, which bound the control-input and design parameter space within which the rotor can autorotate. This instability occurs at a lightly loaded condition and at advance ratios close to 1 for the scaled model. Finally, it was also revealed that the rotor inability to autorotate was driven by blade stall.

scholarly journals Analysis of the stability of multi-mode systems with approximating nonlinear control laws

2021 ◽  
Vol 872 (1) ◽  
pp. 012010
Author(s):  
E A Shakhray ◽  
E V Lubentsova ◽  
V F Lubentsov ◽  
M V Meflekh
Keyword(s):  
Nonlinear Control ◽  
Control Laws ◽  
The Stability ◽  
Multi Mode

scholarly journals Stability and Hopf Bifurcation Analysis of an Epidemic Model by Using the Method of Multiple Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Wanyong Wang ◽  
Lijuan Chen
Keyword(s):  
Periodic Solution ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Value Of Time ◽  
Nonlinear Incidence Rate ◽  
The Stability ◽  
Bifurcating Periodic Solution

A delayed epidemic model with nonlinear incidence rate which depends on the ratio of the numbers of susceptible and infectious individuals is considered. By analyzing the corresponding characteristic equations, the effects of time delay on the stability of the equilibria are studied. By choosing time delay as bifurcation parameter, the critical value of time delay at which a Hopf bifurcation occurs is obtained. In order to derive the normal form of the Hopf bifurcation, an extended method of multiple scales is developed and used. Then, the amplitude of bifurcating periodic solution and the conditions which determine the stability of the bifurcating periodic solution are obtained. The validity of analytical results is shown by their consistency with numerical simulations.

Friction Induced Vibrations in Aircraft Disk Brakes

1997 ◽  
Author(s):  
Lisle B. Hagler ◽  
Per G. Reinhall
Keyword(s):  
Friction Coefficient ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Rotational Model ◽  
Stick Slip ◽  
Numeric Simulation ◽  
Constant Friction ◽  
The Stability ◽  
Rotor Stiffness ◽  
Friction Induced Vibration

Abstract This paper presents a detailed analysis of the dynamic behavior of a single rotor/stator brake system. Two separate mathematical models of the brake are considered. First, a non-rotational model is constructed with the purpose of showing that friction induced vibration can occur in the stator without assuming stick-slip behavior and a velocity dependent friction coefficient. Self-induced vibrations are analyzed via the application of the method of multiple scales. The stability boundaries of the primary resonance, as well as the super-harmonics and sub-harmonics are determined. Secondly, rotational effects are investigated by considering a mathematical brake model consisting of a spinning rotor engaging against a flexible stator. Again, a constant friction coefficient is assumed. The stability of steady whirl is determined as a function of the system parameters. We demonstrate that only forward whirl is stable for no-slip motion of the rotor. The interactions between chatter, squeal, and rotor whirl are investigated through numeric simulation. It is shown that rotor whirl can be an important source of the torsional oscillations (squeal) of the stator and that the settling time to no-slip decreases as the ratio of the stator to rotor stiffness is increased.

scholarly journals Multiharmonic Resonance Control Testing of an Internally Resonant Structure

Vibration ◽  
2020 ◽  
Vol 3 (3) ◽  
pp. 217-234
Author(s):  
Alexander D. Shaw ◽  
Thomas L. Hill ◽  
Simon A. Neild ◽  
Michael I. Friswell
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Testing Procedure ◽  
Modal Interactions ◽  
Nonlinear Structure ◽  
Testing Method ◽  
Passive Testing ◽  
Nonlinear Normal ◽  
Resonance Control ◽  
Experimental Characterisation

The experimental characterisation of a nonlinear structure is a challenging process, particularly for multiple degree of freedom and continuous structures. Despite attracting much attention from academia, there is much work needed to create processes that can achieve characterisation in timescales suitable for industry, and a key to this is the design of the testing procedure itself. This work proposes a passive testing method that seeks a desired degree of resonance between forcing and response. In this manner, the process automatically seeks data that reveals greater detail of the underlying nonlinear normal modes than a traditional stepped sine method. Furthermore, the method can target multiple harmonics of the fundamental forcing frequency, and is therefore suitable for structures with complex modal interactions. The method is presented with some experimental examples, using a structure with a 3:1 internal resonance.

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