Effects of Bearing and Shaft Asymmetries on the Resonant Oscillations of a Rotor-Dynamic System

1996 ◽  
Vol 118 (1) ◽  
pp. 107-114
Author(s):  
R. Ganesan
Keyword(s):  
Steady State ◽  
Dynamic System ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Mass Unbalance ◽  
Asymmetric Rotor ◽  
Amplitude And Frequency Modulation ◽  
Rotor Dynamic ◽  
Steady State Solutions

Parametric steady-state vibrations of an asymmetric rotor while passing through primary resonance and the associated stability behavior are analyzed. The undamped case is considered and the equations of motion are rewritten in a from suitable for applying the method of multiple scales. Sensitivity to the bearing as well as shaft asymmetries of the oscillations due to unbalance excitation is evaluated. Expressions for amplitude and frequency modulation functions are obtained and are specialized to yield the steady-state solutions near primary resonance. Frequency-amplitude relationships that result from combined parametric and mass unbalance excitations are derived. Stability regions in the parameter space are obtained based on the time evolution of the amplitude and phase of the steady-state motions. The effects of bearing asymmetry on the amplitude and phase of the resonant oscillations are brought out. The sensitivity of vibrational and stability characteristics to various rotor-dynamic system parameters is illustrated through a numerical investigation.

Resonant Oscillations and Stability of Asymmetric Rotors

1993 ◽  
Author(s):  
R. Ganesan ◽  
T. S. Sankar
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Stable Solution ◽  
Primary Resonance ◽  
Mass Unbalance ◽  
The Many ◽  
Uniform Expansions ◽  
Steady State Solutions

Abstract Non-stationary oscillations of an asymmetric rotor while passing through primary resonance and the associated stability behaviour are analyzed. Solutions are developed based on a Jeffcott rotor model and the equations of motion are rewritten in a form suitable for applying the method of multiple scales. The many-variable version using “slow” and “fast” lime scales is applied to obtain the uniform expansions of amplitudes of motion. Similar general expressions for amplitude and frequency modulation functions are explicitly obtained and are specialized to yield steady-state solutions. Frequency-amplitude relationships resulting from combined parametric and mass unbalance excitations, for the nonlinear vibration are derived. Stability regions in the parameter space are obtained for a stable solution in terms of the perturbed steady-state solutions of the governing equations of motion. Also, trivial solutions are examined for stability. The sensitivity of vibration amplitudes to various rotor-dynamic system parameters is illustrated through a numerical study.

Nonlinear Study of Rotor-AMB System Subject to Multi-Parametric Excitations

2013 ◽  
Vol 397-400 ◽  
pp. 359-364
Author(s):  
Lin Li ◽  
Yong Jie Han ◽  
Zheng Yi Ren
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Active Magnetic Bearings ◽  
Jump Phenomenon ◽  
State Response ◽  
Non Linear ◽  
Response Function Method ◽  
Amb System ◽  
Steady State Solutions

The non-linear response of a rotor supported by active magnetic bearings (AMB) under multi-parametric excitations is investigated. The method of multiple scales is applied to analyze the response of two modes of a rotorAMB system near the primary resonance case. The steady-state response of the system is studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening non-linearity. The effects of the different parameters on the steady state solutions are investigated and discussed.

Autoparametric interaction in a double pendulum system

2012 ◽  
Vol 226 (8) ◽  
pp. 1971-1986
Author(s):  
Matthew P Cartmell ◽  
Ivana Kovacic ◽  
Miodrag Zukovic
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Double Pendulum ◽  
Pendulum System ◽  
Right Hand ◽  
The Right ◽  
Steady State Solutions

This article investigates a four-degree-of-freedom mechanical model comprising a horizontal bar onto which two identical pendula are fitted. The bar is suspended from a pair of springs and the left-hand-side pendulum is excited by means of a harmonic torque. The article shows that autoparametric interaction is possible by means of typical external and internal resonance conditions involving the system natural frequencies and excitation frequency, yielding an interesting case when the right-hand-side pendulum does not oscillate, but stays at rest. It is demonstrated that applying the standard method of multiple scales to this system leads to slow-time and subsequently steady-state equations representative of periodic responses; however, in common with previous findings reported in the literature for systems of four or more interacting modes, global solutions are not obtainable. This article then concentrates on discussing a proposed new modification to the method of multiple scales in which the effect of detuning is accentuated within the zeroth-order perturbation equations and it is then demonstrated that the numerical solutions from this approach to multiple scales yield results that are virtually indistinguishable from those obtained from direct numerical integration of the equations of motion. It is also shown that the algebraic structure of the steady-state solutions for the modified multiple scales analysis is identical to that obtained from a harmonic balance analysis for the case when the right-hand-side pendulum is decoupled. This particular decoupling case is prominent from examination of both the original equations of motion and the steady-state solutions irrespective of the analysis undertaken. This article concludes by showing that the translation and rotation of the bar are, in this particular case, mutually coupled and opposite in sign.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

Nonlinear Forced Vibration Analysis of Smart Curved CNTs Conveying Fluid

2021 ◽  
Author(s):  
Vahid Mohamadhashemi ◽  
Amir Jalali ◽  
Habib Ahmadi
Keyword(s):  
Carbon Nanotube ◽  
Velocity Vector ◽  
Multiple Scales ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Maximum Amplitude ◽  
Beam Theory ◽  
Physical Parameters ◽  
Conveying Fluid

In this study, the nonlinear vibration of a curved carbon nanotube conveying fluid is analyzed. The nanotube is assumed to be covered by a piezoelectric layer and the Euler–Bernoulli beam theory is employed to establish the governing equations of motion. The influence of carbon nanotube curvature on structural modeling and fluid velocity vector is considered and the slip boundary conditions of CNT conveying fluid are included. The mathematical modeling of the structure is developed using Hamilton’s principle and then, the Galerkin procedure is employed to discretize the equation of motion. Furthermore, the frequency response of the system is extracted by applying the multiple scales method of perturbation. Finally, a comprehensive study is carried out on the primary resonance and piezoelectric-based parametric resonance of the system. It is shown that consideration of nanotube curvature may lead to an increase in nonlinearity. Implementing the fluid velocity vector in which nanotube curvature is included highly affects the maximum amplitude of the response and should not be ignored. Furthermore, different system parameters have evident impacts on the behavior of the system and therefore, selecting the reasonable geometrical and physical parameters of the system can be very useful to achieve a favorable response.

Steady-State Characteristics of a Dual-Rotor System With Intershaft Bearing Subjected to Mass Unbalance and Base Motions

2018 ◽  
Author(s):  
Xi Chen ◽  
Mingfu Liao
Keyword(s):  
Steady State ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Axial Rotation ◽  
Rotor System ◽  
Mode Shapes ◽  
Mass Unbalance ◽  
Moving Base ◽  
Lateral Rotation ◽  
Force Vectors

A dual-rotor system with an intershaft bearing subjected to mass unbalance and base motions is established. Using Lagrange’s principle, equations of motion for dual-rotor system relative to moving base are derived. Rotary inertia, gyroscopic inertia, transverse shear deformation, mass unbalance, and six components of deterministic base motions are taken into account. Using state-space vector, steady-state characteristics of dual-rotor system are analyzed through dual-rotor critical speed map, mode shapes, unbalance responses considering base rotations, frequency responses due to base motions, and shaft orbits. The results show that base translations just add external force vectors, while base rotations bring on parametric system matrices and additional force vectors. Base rotations not only change natural frequencies of dual-rotor system, but also break the symmetry of dynamic characteristics in the case of base lateral rotation. Excited by base harmonic translation, resonant frequencies correspond to whirl frequencies. The orbit remains circular under base axial rotation, while it becomes elliptical with a static offset under lateral rotation and then a complicated curve due to harmonic translation. When harmonic frequency of base translation gets close to dual-rotor excitation frequencies, obvious beat vibration appears. Overrall, this flexible approach can ensure calculation accuracy with high efficiency and good expandability.

Nonlinear Vibrations of a Beam-Spring-Large Mass System

2017 ◽  
Author(s):  
Mohammad A. Bukhari ◽  
Oumar R. Barry
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Large Mass ◽  
Mode Shapes ◽  
Resonance Excitation ◽  
Response Curves ◽  
Mass System ◽  
Spring System

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

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.

Multimode Interactions in Suspended Cables

2002 ◽  
Vol 8 (3) ◽  
pp. 337-387 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat ◽  
Char-Ming Chin ◽  
Walter Lacarbonara
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Internal Resonances ◽  
Out Of Plane ◽  
Space Formulation ◽  
Suspended Cables ◽  
One To One ◽  
Plane Mode

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

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