Combination Parametric Resonance of Nonlinear Unbalanced Rotating Shafts

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
M. S. Qaderi ◽  
S. A. A. Hosseini ◽  
M. Zamanian
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Geometrical Nonlinearity ◽  
Primary Resonance ◽  
Rotating Shaft ◽  
Rotating Shafts ◽  
Slow Evolution ◽  
The Stability ◽  
Parametric And External Excitations

In this paper, dynamic response of a rotating shaft with geometrical nonlinearity under parametric and external excitations is investigated. Resonances, bifurcations, and stability of the response are analyzed. External excitation is due to shaft unbalance and parametric excitation is due to periodic axial force. For this purpose, combination resonances of parametric excitation and primary resonance of external force are assumed. Indeed, simultaneous effect of nonlinearity, parametric, and external excitations are investigated using analytical method. By applying the method of multiple scales, four ordinary nonlinear differential equations are obtained, which govern the slow evolution of amplitude and phase of forward and backward modes. Eigenvalues of Jacobian matrix are checked to find the stability of solutions. Both periodic and quasi-periodic motion were observed in the range of study. The influence of various parameters on the response of the system is studied. A main contribution is that the parametric excitation in the presence of nonlinearity can be used to suppress the forward synchronous vibration. Indeed, in the presence of combination parametric excitation, the energy is transferred from forward whirling mode to backward one. This property can be applied in control of rotor unbalance vibrations.

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.

Subcritical Large-Amplitude Vibrations of Imperfect Rotating Shafts

1989 ◽  
Vol 42 (11S) ◽  
pp. S157-S160
Author(s):  
C. E. N. Mazzilli
Keyword(s):  
Dry Friction ◽  
Large Amplitude ◽  
Critical Speed ◽  
Multiple Scales ◽  
Viscous Damping ◽  
Rotating Shaft ◽  
Geometrical Imperfection ◽  
Rotating Shafts ◽  
Large Amplitude Vibrations ◽  
Practical Implications

The effect of a geometrical imperfection, such as the axis flexural deformation, on the large-amplitude vibrations of a horizontal rotating shaft is analyzed with the aid of the Multiple Scales Method. Internal viscous damping and linear elasticity are assumed in the model. It is then seen that no matter how small the imperfection is, a “critical” speed of the order of half the classical critical speed arises, with relevant practical implications. A number of reported large-amplitude cases may eventually be explained this way. It is possible that events such as this will not appear in systems with high dry friction.

Free Vibration and Stability Analysis of Internally Damped Rotating Shafts With General Boundary Conditions

1998 ◽  
Vol 120 (3) ◽  
pp. 776-783 ◽  
Author(s):  
J. Melanson ◽  
J. W. Zu
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Beam Theory ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Rotating Shaft ◽  
Classical Boundary ◽  
Rotating Shafts ◽  
The Stability

Vibration analysis of an internally damped rotating shaft, modeled using Timoshenko beam theory, with general boundary conditions is performed analytically. The equations of motion including the effects of internal viscous and hysteretic damping are derived. Exact solutions for the complex natural frequencies and complex normal modes are provided for each of the six classical boundary conditions. Numerical simulations show the effect of the internal damping on the stability of the rotor system.

Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

Dynamic Stability of the Rotating Shaft Made of Boltzmann Viscoelastic Solid

1986 ◽  
Vol 53 (2) ◽  
pp. 424-429 ◽  
Author(s):  
W. Zhang ◽  
F. H. Ling
Keyword(s):  
Dynamic Stability ◽  
High Speed ◽  
Static Load ◽  
Viscoelastic Solid ◽  
Rotating Shaft ◽  
Special Cases ◽  
Analytical Formulas ◽  
Rotating Shafts ◽  
The Stability ◽  
Inclined Angle

A general theory is developed in this paper for studying the dynamic stability of high-speed nonuniform rotating shafts made of a Boltzmann viscoelastic solid. The equation of motion of the shaft is deduced. The stability criteria are derived by using this equation. The unstable regions for a nonhomogeneous viscoelastic shaft are worked out numerically. Analytical formulas are also given in this paper for determining the planar deflection of the shaft and its inclined angle due to a planar static load. The conclusions for special cases given in the literature known to the authors are all covered by the results in this paper.

Nonlinear Oscillations of Nonlinear Damping Gyros: Resonances, Hysteresis and Multistability

2020 ◽  
Vol 30 (14) ◽  
pp. 2050203
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
L. A. Hinvi ◽  
V. Kamdoum Tamba ◽  
A. A. Koukpémèdji ◽  
...  
Keyword(s):  
Fixed Points ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Approximate Equation ◽  
Nonlinear Damping ◽  
Phase Portraits ◽  
Exact Equation ◽  
Restoring Force ◽  
The Stability

This paper addresses the issues on the dynamics of nonlinear damping gyros subjected to a quintic nonlinear parametric excitation. The fixed points and their stability are analyzed for the autonomous gyros equation. The number of fixed points of the system varies from one to six. The approximate equation of gyros is considered by expanding the nonlinear restoring force and parametric excitation for the study of the dynamics of gyros. Amplitude and frequency of possible resonances are found by using the multiple scales method. Also obtained are the principal parametric resonance and orders 4 and 6 subharmonic resonances. The stability conditions for each of these resonances are also obtained. Chaotic oscillations, multistability, hysteresis, and coexisting attractors are found using the bifurcation diagrams, the Lyapunov exponents, the phase portraits, the Poincaré section and the time histories. The effects of the damping parameter, the angular spin velocity and the parametric nonlinear excitation are analyzed. Results obtained by using the approximate gyros equation are compared to the dynamics obtained with the exact equation of gyros. The analytical investigations are complemented by numerical simulations.

Transverse Vibrations of a Centrally Clamped Rotating Circular Disk

1999 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Adel Jilani ◽  
Piergiuseppe Manzione
Keyword(s):  
Multiple Scales ◽  
Primary Resonance ◽  
Circular Disk ◽  
Qualitative Behavior ◽  
Equilibrium Solutions ◽  
Transverse Loads ◽  
Transverse Vibrations ◽  
Critical Rotational Speed ◽  
The Stability ◽  
Good Agreement

Abstract The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed. The results specialized to the linear case of disks clamped at the center and free at the periphery are in good agreement with those reported in the literature. The natural frequencies of spinning hard and floppy disks are obtained for various nodal diameters and nodal circles. Primary resonance is shown to occur at the critical rotational speed at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, we determine a set of four nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of two interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. They are reduced to an equivalent set of two first-order equations whose equilibrium solutions are determined analytically. The stability characteristics of these solutions is studied; the qualitative behavior of the response is independent of the mode being considered.

On nonlinear perturbation analysis of a structure carrying a circular cylindrical liquid tank under horizontal excitation

2018 ◽  
Vol 25 (5) ◽  
pp. 1058-1079 ◽  
Author(s):  
N. K. A. Attari ◽  
F. R. Rofooei ◽  
Z. Waezi
Keyword(s):  
Differential Equations ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Structural Response ◽  
Single Mode ◽  
Excitation Amplitude ◽  
Combined System ◽  
The Stability

The lateral response of a single degree of freedom structural system containing a rigid circular cylindrical liquid tank under harmonic and earthquake excitations at a 1:2 autoparametric resonance case is considered. The governing nonlinear differential equations of motion for the combined system are solved by means of a multiple scales method considering the first three liquid sloshing modes (1,1), (0,1), and (2,1), under horizontal excitation. The fixed points of the gyroscopic type of governing differential equations are determined and their stability is investigated employing the perturbation method. The obtained results reveal an increase in the stability region for a single-mode response with respect to the excitation amplitude. The saturation phenomenon is observed in the decoupled modes of the system; however, the structural mode and the first anti-symmetric mode of liquid are a combination of the saturated mode and another mode whose scale factor is crucial for the structural response. The results of perturbation analysis are in good agreement with results obtained from numerical methods.

Reduced Order Model of Coaxial Vibrations of Electrostatically Actuated Double Wall Carbon Nanotubes: Frequency Response of Primary Resonance

2018 ◽  
Author(s):  
Ezequiel Juarez ◽  
Dumitru I. Caruntu
Keyword(s):  
Carbon Nanotubes ◽  
Frequency Response ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Approximate Solutions ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
The Stability ◽  
Euler Bernoulli Beam ◽  
Steady State Solutions

In this paper, the Reduced Order Method (ROM) and the Method of Multiple Scales (MMS) are used to investigate the influences of dimensionless damping and voltage parameters on the amplitude-frequency response of an electrostatically actuated double-walled carbon nanotube (DWCNT). The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Soft AC excitation and small viscous damping forces are assumed. Herein, the coaxial case is investigated. In this mode of vibration, the outer and inner carbon nanotubes move synchronously (in-phase) with the same maximum tip deflection. The DWCNT structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the DWCNT is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order MMS solution. The analytical approximate solutions provided by MMS are validated numerically by two-term (2T) Time Reponses and AUTO-07P. The two methods in this paper are found to be in excellent agreement at lower amplitudes. Additionally, the two methods are assessed for their advantages and limitations. The importance of the results in this paper are the effect of damping and voltage on the stability of the DWCNT vibration.

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.

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