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.

Localized Modes in Periodic Systems With Nonlinear Disorders

1997 ◽  
Vol 64 (4) ◽  
pp. 940-945 ◽  
Author(s):  
C. W. Cai ◽  
H. C. Chan ◽  
Y. K. Cheung
Keyword(s):  
Degrees Of Freedom ◽  
Maximum Amplitude ◽  
Critical Value ◽  
Periodic Systems ◽  
Algebraic Equations ◽  
Localized Modes ◽  
Transformation Technique ◽  
Nonlinear Algebraic Equations ◽  
Coupling Stiffness ◽  
Localized Mode

The localized modes of periodic systems with infinite degrees-of-freedom and having one or two nonlinear disorders are examined by using the Lindstedt-Poincare (L-P) method. The set of nonlinear algebraic equations with infinite number of variables is derived and solved exactly by the U-transformation technique. It is shown that the localized modes exist for any amount of the ratio between the linear coupling stiffness kc and the coefficient γ of the nonlinear disordered term, and the nonsymmetric localized mode in the periodic system with two nonlinear disorders occurs as the ratio kc/γ, decreasing to a critical value depending on the maximum amplitude.

Nonlinear Mode Localization in Systems Governed by Partial Differential Equations

1996 ◽  
Vol 49 (2) ◽  
pp. 87-99 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Extreme Values ◽  
Nonlinear Partial Differential Equations ◽  
Flexible Structures ◽  
Partial Differential ◽  
Mode Localization ◽  
Infinite Domains ◽  
Nonlinear Mode

The concept of nonlinear normal mode (NNM) is used to study localized oscillations in certain classes of oscillators governed by nonlinear partial differential equations. NNMs are synchronous free oscillations during which all positional coordinates of the system reach their extreme values or pass through the equilibrium position at the same instant of time. Although such motions can be regarded as nonlinear analogs of the linear normal modes of classical vibration theory, not all NNMs are analytic continuations of linear ones. Continuous systems of finite and infinite spatial extent are considered. For periodic assemblies consisting of a finite number of nonlinear structural members, the NNMs are computed asymptotically by solving nonlinear sets of equations possessing regular singular points. Some of the computed NNMs are spatially localized to only a limited number of components of the assembly. The bifurcations giving rise to nonlinear mode localization are examined using the perturbation method of multiple-scales. The implications of nonlinear mode localization on the vibration and shock isolation of periodic flexible structures are discussed. In particular, localized NNMs lead to passive motion confinement of disturbances generated by impulsive loads. Finally, the concept of NNMs is extended to analytically study standing waves with spatially localized envelopes in a class of nonlinear partial differential equations defined over infinite domains. It is shown that NNM-based methodologies can be an effective tool for analyzing such motions.

Performance testing for an active/passive vibration isolation and steering system

1996 ◽  
Author(s):  
Jeanne Sullivan ◽  
James Gooding ◽  
Michelle Idle ◽  
Alok Das ◽  
Terance Hoffman ◽  
...  
Keyword(s):  
Vibration Isolation ◽  
Performance Testing ◽  
Steering System ◽  
Passive Vibration Isolation

scholarly journals Tilt Active Vibration Isolation Using Vertical Pendulum and Piezoelectric Transducer with Parallel Controller

2021 ◽  
Vol 11 (10) ◽  
pp. 4526
Author(s):  
Lihua Wu ◽  
Yu Huang ◽  
Dequan Li
Keyword(s):  
Piezoelectric Transducer ◽  
Vibration Isolation ◽  
Vertical Vibration ◽  
Open Loop ◽  
Negative Effects ◽  
Voltage Controller ◽  
Hysteresis Nonlinearity ◽  
Passive Vibration Isolation ◽  
Active Vibration ◽  
Active Vibration Isolation

Tilt vibrations inevitably have negative effects on some precise engineering even after applying horizontal and vertical vibration isolations. It is difficult to adopt a traditional passive vibration isolation (PVI) scheme to realize tilt vibration isolation. In this paper, we present and develop a tilt active vibration isolation (AVI) device using a vertical pendulum (VP) tiltmeter and a piezoelectric transducer (PZT). The potential resolution of the VP is dependent on the mechanical thermal noise in the frequency bandwidth of about 0.0265 nrad, which need not be considered because it is far below the ground tilt of the laboratory. The tilt sensitivity of the device in an open-loop mode, investigated experimentally using a voltage controller, is found to be (1.63±0.11)×105 V/rad. To compensate for the hysteresis nonlinearity of the PZT, we experimentally established the multi-loop mathematical model of hysteresis, and designed a parallel controller consisting of both a hysteresis inverse model predictor and a digital proportional–integral–differential (PID) adjuster. Finally, the response of the device working in close-loop mode to the tilt vibration was tested experimentally, and the tilt AVI device showed a good vibration isolation performance, which can remarkably reduce the tilt vibration, for example, from 6.0131 μrad to below 0.0103 μrad.

scholarly journals A Low-g MEMS Accelerometer with High Sensitivity, Low Nonlinearity and Large Dynamic Range Based on Mode-Localization of 3-DoF Weakly Coupled Resonators

Micromachines ◽  
2021 ◽  
Vol 12 (3) ◽  
pp. 310
Author(s):  
Muhammad Mubasher Saleem ◽  
Shayaan Saghir ◽  
Syed Ali Raza Bukhari ◽  
Amir Hamza ◽  
Rana Iqtidar Shakoor ◽  
...  
Keyword(s):  
Microelectromechanical Systems ◽  
Dynamic Range ◽  
Proof Mass ◽  
Amplitude Ratio ◽  
Silicon On Insulator ◽  
Coupled Resonators ◽  
Mode Localization ◽  
Mems Accelerometer ◽  
Weakly Coupled ◽  
Input Acceleration

This paper presents a new design of microelectromechanical systems (MEMS) based low-g accelerometer utilizing mode-localization effect in the three degree-of-freedom (3-DoF) weakly coupled MEMS resonators. Two sets of the 3-DoF mechanically coupled resonators are used on either side of the single proof mass and difference in the amplitude ratio of two resonator sets is considered as an output metric for the input acceleration measurement. The proof mass is electrostatically coupled to the perturbation resonators and for the sensitivity and input dynamic range tuning of MEMS accelerometer, electrostatic electrodes are used with each resonator in two sets of 3-DoF coupled resonators. The MEMS accelerometer is designed considering the foundry process constraints of silicon-on-insulator multi-user MEMS processes (SOIMUMPs). The performance of the MEMS accelerometer is analyzed through finite-element-method (FEM) based simulations. The sensitivity of the MEMS accelerometer in terms of amplitude ratio difference is obtained as 10.61/g for an input acceleration range of ±2 g with thermomechanical noise based resolution of 0.22 and nonlinearity less than 0.5%.

Localized and Non-Localized Normal Modes in a Strongly Nonlinear Discrete System

1991 ◽  
Author(s):  
Alexander F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Linear Vibration ◽  
Mode Localization ◽  
Linear Mode ◽  
Strongly Nonlinear ◽  
System A ◽  
Coupling Stiffness ◽  
Nonlinear Normal ◽  
Nonlinear Discrete System

Abstract The free oscillations of a strongly nonlinear, discrete oscillator are examined by computing its “nonsimilar nonlinear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing nonlinear perturbation techniques. For an oscillator with weak coupling stiffness and “mistiming,” both localized and nonlocalized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only nonlocalized modes are found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization.

scholarly journals Effects and Prospects of the Vibration Isolation Methods for an Atomic Interference Gravimeter

Sensors ◽  
2022 ◽  
Vol 22 (2) ◽  
pp. 583
Author(s):  
Wenbin Gong ◽  
An Li ◽  
Chunfu Huang ◽  
Hao Che ◽  
Chengxu Feng ◽  
...  
Keyword(s):  
Vibration Isolation ◽  
Prospective Development ◽  
Passive Vibration Isolation ◽  
Isolation Methods ◽  
Challenging Environment ◽  
Research Findings ◽  
Active Vibration ◽  
Vibration Compensation ◽  
Atomic Interference ◽  
Vibration Noise

An atomic interference gravimeter (AIG) is of great value in underwater aided navigation, but one of the constraints on its accuracy is vibration noise. For this reason, technology must be developed for its vibration isolation. Up to now, three methods have mainly been employed to suppress the vibration noise of an AIG, including passive vibration isolation, active vibration isolation and vibration compensation. This paper presents a study on how vibration noise affects the measurement of an AIG, a review of the research findings regarding the reduction of its vibration, and the prospective development of vibration isolation technology for an AIG. Along with the development of small and movable AIGs, vibration isolation technology will be better adapted to the challenging environment and be strongly resistant to disturbance in the future.

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