Dynamic Behavior and Stability of a Rotor Under Base Excitation

2006 ◽  
Vol 128 (5) ◽  
pp. 576-585 ◽  
Author(s):  
M. Duchemin ◽  
A. Berlioz ◽  
G. Ferraris
Keyword(s):  
Dynamic Behavior ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Base Excitation ◽  
Flexible Rotor ◽  
Lagrange Equations ◽  
Rotor Systems ◽  
Simple Systems ◽  
Flexible Rotor Systems

The dynamic behavior of flexible rotor systems subjected to base excitation (support movements) is investigated theoretically and experimentally. The study focuses on behavior in bending near the critical speeds of rotation. A mathematical model is developed to calculate the kinetic energy and the strain energy. The equations of motion are derived using Lagrange equations and the Rayleigh-Ritz method is used to study the basic phenomena on simple systems. Also, the method of multiple scales is applied to study stability when the system mounting is subjected to a sinusoidal rotation. An experimental setup is used to validate the presented results.

Theoretical and Experimental Analysis of a Base-Excited Rotor

2006 ◽  
Vol 1 (3) ◽  
pp. 257-263 ◽  
Author(s):  
N. Driot ◽  
C. H. Lamarque ◽  
A. Berlioz
Keyword(s):  
Dynamic Behavior ◽  
Theoretical Study ◽  
Multiple Scales ◽  
Ritz Method ◽  
Flexible Rotor ◽  
Coupled Equations ◽  
Points Of View ◽  
Lateral Displacements ◽  
Support Excitation ◽  
Form Approach

In this study, the dynamic behavior of a flexible rotor system subjected to support excitation (imposed displacements of its base) is analyzed. The effect of an excitation on lateral displacements is investigated from theoretical and experimental points of view. The study focuses on behavior in bending. A mathematical model with two gyroscopic and parametrical coupled equations is derived using the Rayleigh-Ritz method. The theoretical study is based on both the multiple scales method and the normal form approach. An experimental setup is then developed to observe the dynamic behavior permitting the measurement of lateral displacements when the system’s support is subjected to a sinusoidal rotation. The experimental results are favorably compared with the analytical and numerical results.

A Virtual Bearing Method for Optimal Bearing Placement of Flexible Rotor Systems

2017 ◽  
Author(s):  
Shibing Liu ◽  
Bingen Yang
Keyword(s):  
Optimization Problem ◽  
Optimization Method ◽  
Rotor System ◽  
Problem Solution ◽  
Flexible Rotor ◽  
Rotor Systems ◽  
Real Coded Genetic Algorithm ◽  
Minimum Number ◽  
Distributed Transfer Function ◽  
Flexible Rotor Systems

Flexible multistage rotor systems have a variety of engineering applications. Vibration optimization is important to the improvement of performance and reliability for this type of rotor systems. Filling a technical gap in the literature, this paper presents a virtual bearing method for optimal bearing placement that minimizes the vibration amplitude of a flexible rotor system with a minimum number of bearings. In the development, a distributed transfer function formulation is used to define the optimization problem. Solution of the optimization problem by a real-coded genetic algorithm yields the locations and dynamic coefficients of bearings, by which the prescribed operational requirements for the rotor system are satisfied. A numerical example shows that the proposed optimization method is efficient and accurate, and is useful in preliminary design of a new rotor system with the number of bearings unforeknown.

Electrostatically Actuated Coupled MEMS Resonators

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
System Structure ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Beam System ◽  
Mems Resonators ◽  
The Mathematical Model ◽  
Steady State Solutions

This paper deals with the nonlinear response of an electrostatically actuated cantilever beam system composed of two micro beam resonators near natural frequency. The mathematical model of the system is obtained using Lagrange equations. The equations of motion are nondimensionalized and then the method of multiple scales is used to find steady state solutions. Both AC and DC actuation voltages of the first beam are considered, while the influence on the system of DC on the second beam is explored. Graphical representations of the influence of the detuning parameters are provided for a typical micro beam system structure.

scholarly journals Asymmetric Vibrations and Chaos in Spherical Caps under Uniform Time-varying Pressure Fields

2021 ◽  
Author(s):  
Giovanni Iarriccio ◽  
Antonio Zippo ◽  
Francesco Pellicano
Keyword(s):  
Equations Of Motion ◽  
Ritz Method ◽  
Excitation Frequency ◽  
Lagrange Equations ◽  
Reduced Order ◽  
Pressure Fields ◽  
Time Histories ◽  
Implicit Solver ◽  
Spherical Caps ◽  
External Excitation Frequency

Abstract This paper presents a study on nonlinear asymmetric vibrations in shallow spherical caps under pressure loading. The Novozhilov’s nonlinear shell theory is used for modelling the structural strains. A reduced-order model is developed through the Rayleigh-Ritz method and Lagrange equations. The equations of motion are numerically integrated using an implicit solver. The bifurcation scenario is addressed by varying the external excitation frequency. The occurrence of asymmetric vibrations related to quasi-periodic and chaotic motion is shown through the analysis of time histories, spectra, Poincaré maps, and phase planes.

Imbalance Effects in Rotor Systems Under High Angular Accelerations

1994 ◽  
Author(s):  
R. D. Neilson ◽  
A. D. S. Barr ◽  
N. J. Blandford-Baker
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Angular Acceleration ◽  
Torsional Stiffness ◽  
Flexible Rotor ◽  
Disk System ◽  
Transient Vibration ◽  
Rotor Systems ◽  
Six Degrees Of Freedom ◽  
Transient Problems

To assess correctly the effects of transient vibration in a system with imbalance care is required in modelling the system. This is particularly true in cases of extreme imbalance e.g. a blade-off simulation in turbo-machinery. Generally, however, the imbalance is modelled as a simple mrΩ2 term applied when the blade is released but this does not include all possible terms. This paper presents the detailed equations of motion of a flexible rotor system with distributed imbalance. The equations are presented in a rotating coordinate system. The modelling includes coupling between the torsional, lateral and axial motions. A simpler model of a two disk system is then presented in fixed coordinates. The disks which can move laterally am connected by a massless shaft which has both lateral and torsional stiffness giving the system six degrees of freedom. An analysis is presented showing that the model is the same as the conventional model for steady state circular orbits. Results from a simplified blade-off simulation are then presented and compared to the standard mrΩ2 model. The conclusion drawn from these simulations is that the additional terms should be included for high angular acceleration transient problems.

Robust Optimization of Flexible Rotor Systems With Uncertain Parameters via Interval Analysis Method

2019 ◽  
Author(s):  
Jie Hong ◽  
ZheFu Yang ◽  
YaoYu Ni ◽  
YanHong Ma
Keyword(s):  
Interval Analysis ◽  
High Speed ◽  
Dynamic Performance ◽  
Practical Significance ◽  
Analysis Method ◽  
Flexible Rotor ◽  
Parameter Uncertainties ◽  
Rotor Systems ◽  
Interval Analysis Method ◽  
Flexible Rotor Systems

Abstract Uncertainties in the input parameters are inevitable in any design process. Along with the demands for higher rotational speed and higher efficiency of rotating machinery, parameter uncertainties (e.g. support stiffness, the effective bending stiffness of connecting structures) resulted from the increasing load on rotor systems lead to significant scatter of its dynamic performance. These parameters are “uncertain but bounded” which means the distributions are unknown, but the intervals are always got easier. This paper presents a method to robustly optimize the dynamic performance of flexible rotor systems taking into account parameter uncertainties via interval analysis method. Interval analysis methods for modal properties and dynamic response behavior of rotor systems are developed with the interval variables introduced into the equation of motion. The aim of the robust design method is to optimize the critical speed margins and dynamic load on bearings, in the meanwhile, minimizing the variability of the objective items by the means of reducing their sensitivity to parameter uncertainties. A numerical example is presented, results show that, for the high-speed flexible rotor systems, the optimal choices of design variables could reduce of sensitivity to rotor parameter uncertainties, thus optimizing the variability of dynamic performance, which has important practical significance in engineering.

Reduced Order Model of Two Coupled MEMS Parallel Cantilever Resonators Under DC and AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Frequency Response ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Reduced Order Model ◽  
Lagrange Equations ◽  
Order Model ◽  
Reduced Order ◽  
Dc Voltage ◽  
Ground Plate

This paper deals with a system of two coupled parallel identical MEMS cantilever resonators and a ground plate. Alternating Current (AC) and Direct Current (DC) voltages are applied between the first resonator and ground plate, and a DC voltage applied between the resonators. The AC voltage frequency is near natural frequency of the resonators. The electrostatic forces produced by voltages are nonlinear. System equations of motion are obtained using Lagrange equations, then nondimensionalized. The Method of Multiple Scales (MMS) is used to find the steady state frequency response. The Reduced Order Model (ROM) is used to validate MMS results. Matlab is used to find cantilever frequency response of the resonator tip. The DC voltage between resonators is showed to significantly influence the response of the first resonator.

Elastodynamic Analysis of a Completely Elastic System

1977 ◽  
Vol 99 (3) ◽  
pp. 604-609 ◽  
Author(s):  
D. Kohli ◽  
D. Hunter ◽  
G. N. Sandor
Keyword(s):  
Vibration Analysis ◽  
Elastic System ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Lagrange Equations ◽  
Elastic Supports ◽  
Linear Ordinary Differential Equations ◽  
Elastic Links ◽  
Inertia Forces ◽  
Crank Mechanism

The completely elastic system considered for this vibration analysis consists of an offset slider-crank mechanism having (a) elastic supports and mountings of the mechanism permitting translational vibrations of the shafts and supports, (b) elastic shafts permitting torsional vibrations, (c) elastic links of the mechanism which deform due to external or internal body forces and allow flexural and axial vibrations. Both the effect of the deformations caused by the inertia forces in the mechanism links, shafts, and supports and the effect of change in the inertia forces due to these deformations are taken into account in constructing a general mathematical model for conducting elastodynamic analysis. The rigid displacements (finite and infinitesimal) of the mechanism links due to deformations in the support are evaluated using a truncated Taylor series approximation. Deformation in the links caused by the inertia forces is approximated by a finite number of terms in a Fourier series using the Raleigh-Ritz method. The Lagrange equations of motion are used to obtain coupled time varying linear ordinary differential equations of motion for the vibration analysis of the slider-crank mechanism. The method in general may be applied to any planar or spatial system consisting of elastic links, elastic shafts, and elastic supports. Numerical examples are presented for illustration.

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